Ergebnisse der Mathematik Volume 34 · Applications to Nonlinear Partial Differential Equations and...

Ergebnisse der Mathematik Volume 34und ihrer Grenzgebiete

3. Folge

A Series of Modern Surveysin Mathematics

Editorial BoardM. Gromov, Bures-sur-Yvette J. Jost, LeipzigJ. Kollár, Princeton G. Laumon, OrsayH. W. Lenstra, Jr., Leiden J. Tits, ParisD. B. Zagier, Bonn G. Ziegler, BerlinManaging Editor R. Remmert, Münster

Michael Struwe

Variational Methods

Applications to NonlinearPartial Differential Equationsand Hamiltonian Systems

Fourth Edition

123

Michael StruweETH ZürichDepartement MathematikRämistr. 1018092 Zürich, Switzerland

ISBN 978-3-540-74012-4 e-ISBN 978-3-540-74013-1

DOI 10.1007/978-3-540-74013-1

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of ModernSurveys in Mathematics ISSN 0071-1136

Library of Congress Control Number: 2008923744

Mathematics Subject Classification (2000): 58E05, 58E10, 58E12, 58E30, 58E35, 34C25, 34C35, 35A15,35K15, 35K20, 35K22, 58F05, 58F22, 58G11

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Preface to the Fourth Edition

Almost twenty years after conception of the first edition, it was a challenge toprepare an updated version of this text on the Calculus of Variations. The fieldhas truely advanced dramatically since that time, to an extent that I find itimpossible to give a comprehensive account of all the many important devel-opments that have occurred since the last edition appeared. Fortunately, anexcellent overview of the most significant results, with a focus on functionalanalytic and Morse theoretical aspects of the Calculus of Variations, can befound in the recent survey paper by Ekeland-Ghoussoub [1]. I therefore haveonly added new material directly related to the themes originally covered.

Even with this restriction, a selection had to be made. In view of the factthat flow methods are emerging as the natural tool for studying variationalproblems in the field of Geometric Analysis, an emphasis was placed on ad-vances in this domain. In particular, the present edition includes the prooffor the convergence of the Yamabe flow on an arbitrary closed manifold ofdimension 3 ! m ! 5 for initial data allowing at most single-point blow-up.Moreover, we give a detailed treatment of the phenomenon of blow-up and dis-cuss the newly discovered results for backward bubbling in the heat flow forharmonic maps of surfaces.

Aside from these more significant additions, a number of smaller changeshave been made throughout the text, thereby taking care not to spoil the fresh-ness of the original presentation. References have been updated, whenever pos-sible, and several mistakes that had survived the past revisions have now beeneliminated. I would like to thank Silvia Cingolani, Irene Fonseca, EmmanuelHebey, and Maximilian Schultz for helpful comments in this regard. Moreover,I am indebted to Gilles Angelsberg, Ruben Jakob, Reto Muller, and MelanieRupflin, for carefully proof-reading the new material.

Zurich, July 2007 Michael Struwe

Preface to the Third Edition

The Calculus of Variations continues to be an area of very rapid growth. Vari-ational methods are indispensable as a tool in mathematical physics and ge-ometry.

Results on Ginzburg-Landau type variational problems inspire research onthe related Seiberg-Witten functional on a Kahler surface and invite specula-tions about possible applications in topology (Ding-Jost-Li-Peng-Wang [1]).

Variational methods are applied in cosmology, as in the recent work ofFortunato-Giannoni-Masiello [1] and Giannoni-Masiello-Piccione [1] on geode-sics in Lorentz manifolds and gravitational lenses.

Applications to Hamiltonian dynamics now include a proof of the Seifertconjecture on brake orbits (Giannoni [1]) and results on homoclinic and hete-roclinic solutions (Coti Zelati-Ekeland-Sere [1], Rabinowitz [1], Sere [1]) withinteresting counterparts in the field of semilinear elliptic equations (Coti-Zelati-Rabinowitz [1], Rabinowitz [13]).

The Calculus of Variations also has advanced on a more technical level.Campa-Degiovanni [1], Corvellec-Degiovanni-Marzocchi [1], Degiovanni-Mar-zocchi [1], Io!e [1], and Io!e-Schwartzman [1] have extended critical pointtheory to functionals on metric spaces, with applications, for instance, to quasi-linear elliptic equations (Arioli [1], Arioli-Gazzola [1], Canino-Degiovanni [1]).

Bolle [1] has proposed a new approach to perturbation theory, as treatedin Section II.7 of this monograph. Numerous applications are studied in Bolle-Ghoussoub-Tehrani [1].

The method of parameter dependence as in Sections I.7 and II.9 has foundfurther striking applications in Chern-Simons theory (Struwe-Tarantello [1])and independently for a related problem in mean field theory (Ding-Jost-Li-Wang [1]). Inspired by these results, Wang-Wei [1] were able to solve a problemin chemotaxis with a similar structure. Jeanjean [1] and Jeanjean-Toland [1]have discovered an abstract setting where parameter dependence may be ex-ploited.

Ambrosetti [1], Ambrosetti-Badiale-Cingolani [1], and Ambrosetti-Badiale[1], [2] have found new applications of variational methods in bifurcation theory,refining the classical results of Bohme [1] and Marino [1]. In Ambrosetti-GarciaAzorero-Peral [1] these ideas are applied to obtain precise existence results forconformal metrics of prescribed scalar curvature close to a constant, which shednew light on the work of Bahri-Coron [1], [2], Chang-Yang [1] quoted in SectionIII.4.11.

The field of critical equations as in Chapter III has been particularly active.Concentration profiles for Palais-Smale sequences as in Theorem III.3.1

have been studied in more detail by Rey [1] and Flucher [1].

viii Preface to the Third Edition

Quite surprisingly, results analogous to Theorem III.3.1 have been dis-covered also for sequences of solutions to critical semilinear wave equations(Bahouri-Gerard [1]).

For the semilinear elliptic equations of critical exponential growth relatedto the Moser-Trudinger inequality on a planar domain the patterns for exis-tence and non-existence results are strikingly analogous to the higher dimen-sional case (Adimurthi [1], Adimurthi-Srikanth-Yadava [1]), and, on a macro-scopic scale, quantization phenomena analogous to Theorem III.3.1 are ob-served for concentrating solutions of semilinear equations with exponentialgrowth (Brezis-Merle [1], Li-Shafrir [1]). However, results of Struwe [17] andOgawa-Suzuki [1] on the one hand and an example by Adimurthi-Prashanth[1] on the other suggest that there may be many qualitatively distinct types ofblow-up behavior for Palais-Smale sequences in this case. Still, Theorem III.3.1remains valid for solutions (Adimurthi-Struwe [1]) and also the analogue of The-orem III.3.4 has been obtained (Struwe [25]). The many similarities and subtledi!erences to the critical semilinear equations in higher dimensions make thisfield particularly attractive for further study.

References have been updated and a small number of mistakes have beenrectified. I am indepted to Gerd Muller, Paul Rabinowitz, and Henry Wentefor their comments.

Zurich, July 1999 Michael Struwe

Preface to the Second Edition

During the short period of five years that have elapsed since the publicationof the first edition a number of interesting mathematical developments havetaken place and important results have been obtained that relate to the themeof this book.

First of all, as predicted in the Preface to the first edition, Morse theory, in-deed, has gone through a dramatic change, influenced by the work by AndreasFloer on Hamiltonian systems and in particular, on the Arnold conjecture.There are now also excellent accounts of these developments and their ramifi-cations; see, in particular, the monograph by Matthias Schwarz [1]. The bookby Hofer-Zehnder [2] on Symplectic Geometry shows that variational methodsand, in particular, Floer theory have applications that range far beyond theclassical area of analysis.

Second, as a consequence of an observation by Stefan Muller [1] whichprompted the seminal work of Coifman-Lions-Meyer-Semmes [1], Hardy spacesand the space BMO are now playing a very important role in weak conver-gence results, in particular, when dealing with problems that exhibit a special(determinant) structure. A brief discussion of these results and some modelapplications can be found in Section I.3.

Moreover, variational problems depending on some real parameter in cer-tain cases have been shown to admit rather surprising a-priori bounds on criticalpoints, with numerous applications. Some examples will be given in SectionsI.7 and II.9.

Other developments include the discovery of Hamiltonian systems withno periodic orbits on some given energy hypersurface, due to Ginzburg andHerman, and the discovery, by Chang-Ding-Ye, of finite time blow-up for theevolution problem for harmonic maps of surfaces, thus completing the resultsin Sections II.8, II.9 and III.6, respectively.

A beautiful recent result of Ye concerns a new proof of the Yamabe theoremin the case of a locally conformally flat manifold. This proof is presented indetail in Section III.4 of this new edition.

In view of their numerous and wide-ranging applications, interest in vari-ational methods is very strong and growing. Out of the large number of recentpublications in the general field of the calculus of variations and its applica-tions some 50 new references have been added that directly relate to one of thethemes in this monograph.

Owing to the very favorable response with which the first edition of thisbook was received by the mathematical community, the publisher has sug-gested that a second edition be published in the Ergebnisse series. It is apleasure to thank all the many mathematicians, colleagues, and friends who

x Preface to the Second Edition

have commented on the first edition. Their enthusiasm has been highly in-spiring. Moreover, I would like to thank, in particular, Matts Essen, MartinFlucher and Helmut Hofer for helpful suggestions in preparing this new edition.

All additions and changes to the first edition were carefully implemented bySuzanne Kronenberg, using the Springer TeX-Macros package, and I gratefullyacknowledge her help.

Zurich, June 1996 Michael Struwe

Preface to the First Edition

It would be hopeless to attempt to give a complete account of the history ofthe calculus of variations. The interest of Greek philosophers in isoperimetricproblems underscores the importance of “optimal form” already in ancientcultures; see Hildebrandt-Tromba [1] for a beautiful treatise of this subject.While variational problems thus are part of our classical cultural heritage, thefirst modern treatment of a variational problem is attributed to Fermat, seeGoldstine [1; p. 1]. Postulating that light follows a path of least possible time,in 1662 Fermat was able to derive the laws of refraction, thereby using methodswhich may already be termed analytic.

With the development of the Calculus by Newton and Leibniz, the basiswas laid for a more systematic development of the calculus of variations. Thebrothers Johann and Jakob Bernoulli and Johann’s student Leonhard Euler, allfrom the city of Basel in Switzerland, were to become the “founding fathers”(Hildebrandt-Tromba [1; p. 21]) of this new discipline. In 1743 Euler [1] sub-mitted “A method for finding curves enjoying certain maximum or minimumproperties”, published in 1744, the first textbook on the calculus of variations.In an appendix to this book Euler [1; Appendix II, p. 298] expresses his beliefthat “every e!ect in nature follows a maximum or minimum rule” (see alsoGoldstine [1; p. 106]), a credo in the universality of the calculus of variations asa tool. The same conviction also shines through Maupertuis’ [1] work on thefamous “least action principle”, also published in 1744. (In retrospect, how-ever, it seems that Euler was the first to observe this important principle. Seefor instance Goldstine [1; p. 67 f. and p. 101 !.] for a more detailed histori-cal account.) Euler’s book was a great source of inspiration for generations ofmathematicians following.

Major contributions were made by Lagrange, Legendre, Jacobi, Clebsch,Mayer, and Hamilton to whom we owe what we now call “Euler-Lagrangeequations”, the “Jacobi di!erential equation” for a family of extremals, or“Hamilton-Jacobi theory”.

The use of variational methods was not at all limited to one-dimensionalproblems in the mechanics of mass-points. In the 19th century variationalmethods also were employed for instance to determine the distribution of anelectrical charge on the surface of a conductor from the requirement that theenergy of the associated electrical field be minimal (“Dirichlet’s principle”; seeDirichlet [1] or Gauss [1]) or were used in the construction of analytic functions(Riemann [1]).

However, none of these applications was carried out with complete rigor.Often the model was confused with the phenomenon that it was supposed todescribe and the fact (?) that for instance in nature there always exists an

xii Preface to the First Edition

equilibrium distribution for an electrical charge on a conducting surface wastaken as su"cient evidence for the corresponding mathematical problem tohave a solution. A typical reasoning reads as follows:

“In any event therefore the integral will be non-negative and hence theremust exist a distribution (of charge) for which this integral assumes its mini-mum value,” (Gauss [1; p. 232], translation by the author).

However, towards the end of the 19th century progress in abstraction and abetter understanding of the foundations of the calculus opened such argumentsto criticism. Soon enough, Weierstrass [1; pp. 52–54] found an example of a vari-ational problem that did not admit a minimum solution. Weierstrass challengedhis colleagues to find a continuously di!erentiable function u: ["1, 1] # IR min-imizing the integral

I(u) =! 1

!1

""""xd

dxu

""""2

dx

subject (for instance) to the boundary conditions u(±1) = ±1. Choosing

u!(x) =arctan(x

! )arctan( 1

! ), ! > 0,

as a family of comparison functions, Weierstrass was able to show that theinfinium of I in the above class was 0; however, the value 0 is not attained.(See also Goldstine [1; p. 371 f.].) Weierstrass’ critique of Dirichlet’s principleprecipitated the calculus of variations into a Grundlagenkrise comparable to thecrisis in set theory and logic after Russel’s discovery of antinomies in Cantor’sset theory or Godel’s incompleteness proof.

However, through the combined e!orts of several mathematicians who didnot want to give up the wonderful tool that Dirichlet’s principle had been –including Weierstrass, Arzela, Frechet, Hilbert, and Lebesgue – the calculus ofvariations was revalidated and emerged from its crisis with new strength andvigor.

Hilbert’s speech at the centennial assembly of the International Congress1900 in Paris, where he proposed his famous 20 problems – two of which weredevoted to questions related to the calculus of variatons – marks this newlyfound confidence.

In fact, following Hilbert’s [1] and Lebesgue’s [1] solution of the Dirichletproblem, a development began which within a few decades brought tremendoussuccess, highlighted by the 1929 theorem of Ljusternik and Schnirelman [1] onthe existence of three distinct prime closed geodesics on any compact surfaceof genus zero, or the 1930/31 solution of Plateau’s problem by Douglas [1], [2]and Rado [1].

The Ljusternik-Schnirelman result (and a previous result by Birkho! [1],proving the existence of one closed geodesic on a surface of genus 0) alsomarks the beginning of global analyis. This goes beyond Dirichlet’s princi-ple as we no longer consider only minimizers (or maximizers) of variational

Preface to the First Edition xiii

integrals, but instead look at all their critical points. The work of Ljusternikand Schnirelman revealed that much of the complexity of a function spaceis invariably reflected in the set of critical points of any variational integraldefined on it, an idea whose importance for the further development of math-ematics can hardly be overestimated, whose implications even today may onlybe conjectured, and whose applications seem to be virtually unlimited. Later,Ljusternik and Schnirelman [2] laid down the foundations of their method in ageneral theory. In honor of their pioneering e!ort any method which seeks todraw information concerning the number of critical points of a functional fromtopological data today often is referred to as Ljusternik-Schnirelman theory.

Around the time of Ljusternik and Schnirelman’s work, another – equallyimportant – approach towards a global theory of critical points was pursuedby Marston Morse [2]. Morse’s work also reveals a deep relation between thetopology of a space and the number and types of critical points of any functiondefined on it. In particular, this led to the discovery of unstable minimalsurfaces through the work of Morse-Tompkins [1], [2] and Shi!man [1], [2].Somewhat reshaped and clarified, in the 50’s Morse theory was highly successfulin topology (see Milnor [1] and Smale [1]). After Palais [1], [2] and Smale [2] inthe 60’s succeeded in generalizing Milnor’s constructions to infinite-dimensionalHilbert manifolds – see also Rothe [1] for some early work in this regard –Morse theory finally was recognized as a useful (and usable) instrument alsofor dealing with partial di!erential equations.

However, applications of Morse theory seemed somewhat limited in view ofprohibitive regularity and non-degeneracy conditions to be met in a variationalproblem, conditions which – by the way – were absent in Morse’s originalwork. Today, inspired by the deep work of Conley [1], Morse theory seems tobe turning back to its origins again. In fact, a Morse-Conley theory is emergingwhich one day may provide a tool as universal as Ljusternik-Schnirelman theoryand still o!er an even better resolution of the relation between the critical setof a functional and topological properties of its domain. However, in spiteof encouraging results, for instance by Benci [4], Conley-Zehnder [1], Jost-Struwe [1], Rybakowski [1], [2], Rybakowski-Zehnder [1], Salamon [1], and – inparticular – Floer [1], a general theory of this kind does not yet exist.

In these notes we want to give an overview of the state of the art in someareas of the calculus of variations. Chapter I deals with the classical directmethods and some of their recent extensions. In Chapters II and III we discussminimax methods, that is, Ljusternik-Schnirelman theory, with an emphasis onsome limiting cases in the last chapter, leaving aside the issue of Morse theorywhose face is currently changing all too rapidly.

Examples and applications are given to semilinear elliptic partial di!er-ential equations and systems, Hamiltonian systems, nonlinear wave equations,and problems related to harmonic maps of Riemannian manifolds or surfacesof prescribed mean curvature. Although our selection is of course biased bythe interests of the author, an e!ort has been made to achieve a good balancebetween di!erent areas of current research. Most of the results are known;

xiv Preface to the First Edition

some of the proofs have been reworked and simplified. Attributions are madeto the best of the author’s knowledge. No attempt has been made to give anexhaustive account of the field or a complete survey of the literature.

General references for related material are Berger-Berger [1], Berger [1],Chow-Hale [1], Eells [1], Nirenberg [1], Rabinowitz [11], Schwartz [2], Zeidler[1]; in particular, we recommend the recent books by Ekeland [2] and Mawhin-Willem [1] on variational methods with a focus on Hamiltonian systems andthe forthcoming works of Chang [7] and Giaquinta-Hildebrandt. Besides, wemention the classical textbooks by Krasnoselskii [1] (see also Krasnoselskii-Zabreiko [1]), Ljusternik-Schnirelman [2], Morse [2], and Vainberg [1]. As forapplications to Hamiltonian systems and nonlinear variational problems, theinterested reader may also find additional references on a special topic in thesefields in the short surveys by Ambrosetti [2], Rabinowitz [9], or Zehnder [1].

The material covered in these notes is designed for advanced graduateor Ph.D. students or anyone who wishes to acquaint himself with variationalmethods and possesses a working knowledge of linear functional analysis andlinear partial di!erential equations. Being familiar with the definitions andbasic properties of Sobolev spaces as provided for instance in the book byGilbarg-Trudinger [1] is recommended. However, some of these prerequisitescan also be found in the appendix.

In preparing this manuscript I have received help and encouragement froma number of friends and colleagues. In particular, I wish to thank Pro!. Her-bert Amann and Hans-Wilhelm Alt for helpful comments concerning the firsttwo sections of Chapter I. Likewise, I am indebted to Prof. Jurgen Moser foruseful suggestions concerning Section I.4 and to Pro!. Helmut Hofer and Ed-uard Zehnder for advice on Sections I.6, II.5, and II.8, concerning Hamiltoniansystems.

Moreover, I am grateful to Gabi Hitz, Peter Bamert, Jochen Denzler, Mar-tin Flucher, Frank Josellis, Thomas Kerler, Malte Schunemann, Miguel Sofer,Jean-Paul Theubet, and Thomas Wurms for going through a set of preliminarynotes for this manuscript with me in a seminar at ETH Zurich during the win-ter term of 1988/89. The present text certainly has profited a great deal fromtheir careful study and criticism.

Special thanks I also owe to Kai Jenni for the wonderful typesetting of thismanuscript with the TEX text processing system.

I dedicate this book to my wife Anne.

Zurich, January 1990 Michael Struwe

Contents

Chapter I. The Direct Methods in the Calculus of Variations . . . . . . . . . . 1

1. Lower Semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Degenerate Elliptic Equations, 4 — Minimal Partitioning Hypersurfaces, 6— Minimal Hypersurfaces in Riemannian Manifolds, 7 — A General LowerSemi-continuity Result, 8

2. Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Semilinear Elliptic Boundary Value Problems, 14 — Perron’s Method in aVariational Guise, 16 — The Classical Plateau Problem, 19

3. Compensated Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Applications in Elasticity, 29 — Convergence Results for Nonlinear EllipticEquations, 32 — Hardy Space Methods, 35

4. The Concentration-Compactness Principle . . . . . . . . . . . . . . . . . . . . . . . 36Existence of Extremal Functions for Sobolev Embeddings, 42

5. Ekeland’s Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Existence of Minimizers for Quasi-convex Functionals, 54

6. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Hamiltonian Systems, 60 — Periodic Solutions of Nonlinear Wave Equations,65

7. Minimization Problems Depending on Parameters . . . . . . . . . . . . . . . 69Harmonic Maps with Singularities, 71

Chapter II. Minimax Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

1. The Finite Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2. The Palais-Smale Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3. A General Deformation Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Pseudo-gradient Flows on Banach Spaces, 81 — Pseudo-gradient Flows onManifolds, 85

4. The Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Closed Geodesics on Spheres, 89

xvi Contents

5. Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Krasnoselskii Genus, 94 — Minimax Principles for Even Functionals, 96 —Applications to Semilinear Elliptic Problems, 98 — General Index Theories,99 — Ljusternik-Schnirelman Category, 100 — A Geometrical S1-Index, 101— Multiple Periodic Orbits of Hamiltonian Systems, 103

6. The Mountain Pass Lemma and its Variants . . . . . . . . . . . . . . . . . . . . . 108Applications to Semilinear Elliptic Boundary Value Problems, 110 — TheSymmetric Mountain Pass Lemma, 112 — Application to Semilinear Equa-tions with Symmetry, 116

7. Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Applications to Semilinear Elliptic Equations, 120

8. Linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Applications to Semilinear Elliptic Equations, 128 — Applications to Hamil-tonian Systems, 130

9. Parameter Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

10. Critical Points of Mountain Pass Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Multiple Solutions of Coercive Elliptic Problems, 147

11. Non-di!erentiable Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

12. Ljusternik-Schnirelman Theory on Convex Sets . . . . . . . . . . . . . . . . . . 162Applications to Semilinear Elliptic Boundary Value Problems, 166

Chapter III. Limit Cases of the Palais-Smale Condition . . . . . . . . . . . . . . . 169

1. Pohozaev’s Non-existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

2. The Brezis-Nirenberg Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Constrained Minimization, 174 — The Unconstrained Case: Local Compact-ness, 175 — Multiple Solutions, 180

3. The E!ect of Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183A Global Compactness Result, 184 — Positive Solutions on Annular-ShapedRegions, 190

4. The Yamabe Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194The Variational Approach, 195 — The Locally Conformally Flat Case, 197— The Yamabe Flow, 198 — The Proof of Theorem 4.9 (following Ye [1]),200 — Convergence of the Yamabe Flow in the General Case, 204 — TheCompact Case u! > 0, 211 — Bubbling: The Case u! ! 0 , 216

Contents xvii

5. The Dirichlet Problem for the Equation of Constant Mean Curvature 220Small Solutions, 221 — The Volume Functional, 223 — Wente’s UniquenessResult, 225 — Local Compactness, 226 — Large Solutions, 229

6. Harmonic Maps of Riemannian Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 231The Euler-Lagrange Equations for Harmonic Maps, 232 — Bochner identity,234 — The Homotopy Problem and its Functional Analytic Setting, 234 —Existence and Non-existence Results, 237 — The Heat Flow for HarmonicMaps, 238 — The Global Existence Result, 239 — The Proof of Theorem 6.6,242 — Finite-Time Blow-Up, 253 — Reverse Bubbling and Nonuniqueness,257

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Sobolev Spaces, 263 — Holder Spaces, 264 — Imbedding Theorems, 264 —Density Theorem, 265 — Trace and Extension Theorems, 265 — PoincareInequality, 266

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268Schauder Estimates, 268 — Lp-Theory, 268 — Weak Solutions, 269 — A Reg-ularity Result, 269 — Maximum Principle, 271 — Weak Maximum Principle,272 — Application, 273

Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274Frechet Di!erentiability, 274 — Natural Growth Conditions, 276

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Glossary of Notations

V, V " generic Banach space with dual V "

$ ·$ norm in V

$ ·$ " induced norm in V ", often also denoted $ ·$%·, ·&: V ' V " # IR dual pairing, occasionally also used to denote scalar

product in IRn

E generic energy functionalDE Frechet derivativeDom(E) domain of E

%v, DE(u)& = DE(u)v = DvE(u) directional derivative of E at u in di-rection v

Lp("; IRn) space of Lebesgue-measurable functions u: " # IRn

with finite Lp-norm

$u$Lp =#!

"|u|p dx

$1/p, 1 ! p < (

L#("; IRn) space of Lebesgue-measurable and essentiallybounded functions u: " # IRn with norm

$u$L! = ess supx$"

|u(x)|

Hm,p("; IRn) Sobolev space of functions u ) Lp("; IRn) with|*ku| ) Lp(") for all k ) INn

0 , |k| ! m, with norm$u$Hm,p =

%0%|k|%m $*ku$Lp

Hm,p0 ("; IRn) completion of C#

0 ("; IRn) in the norm $ · $Hm,p ;if " is bounded an equivalent norm is given by$u$Hm,p

0=

%|k|=m $*ku$Lp

H!m,q("; IRn) dual of Hm,p0 ("; IRn), where 1

p = 1q = 1; q is omit-

ted, if p = q = 2Dm,p("; IRn) completion of C#

0 ("; IRn) in the norm $u$Dm,p =%|k|=m $*ku$Lp

xx Glossary of Notations

Cm,#("; IRn) space of m times continuously di!erentiable func-tions u: " # IRn whose mth order derivatives areHolder continuous with exponent 0 ! # ! 1

C#0 ("; IRn) space of smooth functions u: " # IRn with compact

support in "

supp(u) = {x ) " ; u(x) += 0} support of a function u: " # IRn

"& ,, " the closure of "& is compact and contained in "

restriction of a measureLn Lebesgue measure on IRn

B$(u; V ) = {v ) V ; $u " v$ < $} open ball of radius $ around u )V ; in particular, if V = IRn, then B$(x0) =B$(x0; IRn), B$ = B$(0)

Re real partIm imaginary partc, C generic constantsCross-references (N.x.y) refers to formula (x, y) in Chapter N

(x.y) within Chapter N refers to formula (N.x.y)

Chapter I

The Direct Methodsin the Calculus of Variations

Many problems in analysis can be cast into the form of functional equationsF (u) = 0, the solution u being sought among a class of admissible functionsbelonging to some Banach space V .

Typically, these equations are nonlinear; for instance, if the class of ad-missible functions is restricted by some (nonlinear) constraint.

A particular class of functional equations is the class of Euler-Lagrangeequations

DE(u) = 0

for a functional E on V , which is Frechet-di!erentiable with derivative DE.We say such equations are of variational form.

For equations of variational form an extensive theory has been developed,and variational principles play an important role in mathematical physics anddi!erential geometry, optimal control and numerical anlysis.

We briefly recall the basic definitions that will be needed in this and the follow-ing chapters, see Appendix C for details: Suppose E is a Frechet-di!erentiablefunctional on a Banach space V with normed dual V ! and duality pairing!·, ·" : V # V ! $ IR, and let DE : V $ V ! denote the Frechet-derivative ofE. Then the directional (Gateaux-) derivative of E at u in the direction of vis given by

d

d!E(u + !v)

! = 0= !v, DE(u)" = DE(u) v.

For such E, we call a point u % V critical if DE(u) = 0; otherwise, u is calledregular. A number " % IR is a critical value of E if there exists a critical point uof E with E(u) = ". Otherwise, " is called regular. Of particular interest (alsoin the non-di!erentiable case) will be relative minima of E, possibly subjectto constraints. Recall that for a set M & V a point u % M is an absoluteminimizer for E on M if for all v % M there holds E(v) ' E(u). A pointu % M is a relative minimizer for E on M if for some neighborhood U of u inV it is absolutely E-minimizing in M (U . Moreover, in the di!erentiable case,we shall also be interested in the existence of saddle points, that is, criticalpoints u of E such that any neighborhood U of u in V contains points v, wsuch that E(v) < E(u) < E(w). In physical systems, saddle points appear asunstable equilibria or transient excited states.

2 Chapter I. The Direct Methods in the Calculus of Variations

In this chapter we review some basic methods for proving the existenceof relative minimizers. Somewhat imprecisely we summarily refer to thesemethods as the direct methods in the calculus of variations. However, besidesthe classical lower semi-continuity and compactness method we also include thecompensated compactness method of Murat and Tartar, and the concentration-compactness principle of P.L. Lions. Moreover, we recall Ekeland’s variationalprinciple and the duality method of Clarke and Ekeland.

Applications will be given to problems concerning minimal hypersurfaces,semilinear and quasi-linear elliptic boundary value problems, finite elasticity,Hamiltonian systems, and semilinear wave equations.

From the beginning it will be apparent that in order to achieve a satisfac-tory existence theory the notion of solution will have to be suitably relaxed.Hence, in general, the above methods will at first only yield generalized or“weak” solutions of our problems. A second step often will be necessary toshow that these solutions are regular enough to be admitted as classical solu-tions. The regularity theory in many cases is very subtle and involves a delicatemachinery. It would go beyond the scope of this book to cover this topic com-pletely. However, for the problems that we will mostly be interested in, theregularity question can be dealt with rather easily. The reader will find thismaterial in Appendix B. References to more advanced texts on the regularityissue will be given where appropriate.

1. Lower Semi-continuity

In this section we give su"cient conditions for a functional to be bounded frombelow and to attain its infimum.

The discussion can be made largely independent of any di!erentiability as-sumptions on E or structure assumptions on the underlying space of admissiblefunctions M . In fact, we have the following classical result.

1.1 Theorem. Let M be a topological Hausdor! space, and suppose E : M $IR ) +* satisfies the condition of bounded compactness:

For any # % IR the setK! = {u % M ; E(u) + #}(1.1)

is compact (Heine-Borel property).

Then E is uniformly bounded from below on M and attains its infimum. Theconclusion remains valid if instead of (1.1) we suppose that any sub-level setK! is sequentially compact.

1. Lower Semi-continuity 3

Remark. Necessity of condition (1.1) is illustrated by simple examples: Thefunction E: [,1, 1] $ IR given by E(x) = x2 if x -= 0, E(x) = 1 if x = 0, orthe exponential function E(x) = exp(x) on IR are bounded from below but donot admit a minimizer. Note that the space M in the first example is compactwhile in the second example the function E is smooth – even analytic.

Proof of Theorem 1.1. Suppose (1.1) holds. We may assume E -. +*. Let

#0 = infM

E ' ,*,

and let (#m) be the strictly decreasing sequence

#m / #0 (m $ *) .

Let Km = K!m . By assumption, each Km is compact and non-empty. More-over, Km 0 Km+1 for all m. By compactness of Km there exists a pointu %

!m"IN Km, satisfying

E(u) + #m, for all m.

Passing to the limit m $ * we obtain that

E(u) + #0 = infM

E,

and the claim follows.If instead of (1.1) each K! is sequentially compact, we choose a minimizing

sequence (um) in M such that E(um) $ #0. Then for any # > #0 the sequence(um) will eventually lie entirely within K!. By sequential compactness of K!therefore (um) will accumulate at a point u %

!!>!0

K! which is the desiredminimizer.

Note that if E : M $ IR satisfies (1.1), then for any # % IR the set

{u % M ; E(u) > #} = M \ K!

is open, that is, E is lower semi-continuous. (Respectively, if each K! is sequen-tially compact, then E will be sequentially lower semi-continuous.) Conversely,if E is (sequentially) lower semi-continuous and for some # % IR the set K! is(sequentially) compact, then K! will be (sequentially) compact for all # + #and again the conclusion of Theorem 1.1 will be valid.

Note that the lower semi-continuity condition can be more easily fulfilledthe finer the topology on M . In contrast, the condition of compactness of thesub-level sets K! , # % IR, calls for a coarse topology and both conditions arecompeting. In practice, there is often a natural weak Sobolev space topologywhere both conditions can be simultaneously satisfied. However, there aremany interesting cases where condition (1.1) cannot hold in any reasonabletopology (even though relative minimizers may exist). Later in this chapter we


shall see some examples and some more delicate ways of handling the possibleloss of compactness. See Section 4; see also Chapter III.

In applications, the conditions of the following special case of Theorem 1.1can often be checked more easily.

1.2 Theorem. Suppose V is a reflexive Banach space with norm 1 ·1 , and letM & V be a weakly closed subset of V . Suppose E : M $ IR ) +* is coerciveand (sequentially) weakly lower semi-continuous on M with respect to V , thatis, suppose the following conditions are fullfilled:(1#) E(u) $ * as 1u1 $ *, u % M .(2#) For any u % M , any sequence (um) in M such that um $ u weakly in Vthere holds:

E(u) + lim infm$%

E(um) .

Then E is bounded from below on M and attains its infimum in M .

The concept of minimizing sequences o!ers a direct and (apparently) construc-tive proof.

Proof. Let #0 = infM E and let (um) be a minimizing sequence in M , that is,satisfying E(um) $ #0. By coerciveness, (um) is bounded in V . Since V isreflexive, by the Eberlein-Smulian theorem (see Dunford-Schwartz [1; p. 430])we may assume that um $ u weakly for some u % V . But M is weakly closed,therefore u % M , and by weak lower semi-continuity

E(u) + lim infm$%

E(um) = #0 .

Examples. An important example of a sequentially weakly lower semi-continous functional is the norm in a Banach space V . Closed and convexsubsets of Banach spaces are important examples of weakly closed sets. If V isthe dual of a separable normed vector space, Theorem 1.2 and its proof remainvalid if we replace weak by weak!-convergence.

We present some simple applications.

Degenerate Elliptic Equations

1.3 Theorem. Let % be a bounded domain in IRn, p % [2,*[ with conjugateexponent q satisfying 1

p + 1q = 1, and let f % H&1,q(%), the dual of H1,p

0 (%),be given. Then there exists a weak solution u % H1,p

0 (%) of the boundary valueproblem

,2 · (|2u|p&22u) = f in %(1.2)u = 0 on &%(1.3)

in the sense that u satisfies the equation


(1.4)"

"(2u|2u|p&22', f' )dx = 0 , 3' % C%

0 (%) .

Proof. Note that the left part of (1.4) is the directional derivative of the C1-functional

E(u) =1p

"

"|2u|p dx ,

"

"fu dx

on the Banach space V = H1,p0 (%) in direction '; that is, problem (1.2), (1.3)

is of variational form.Note that H1,p

0 (%) is reflexive. Moreover, E is coercive. In fact, we have

E(u) ' 1p1u1p

H1,p0

, 1f1H!1,q 1u1H1,p0

' 1p

#1u1p

H1,p0

, c1u1H1,p0

$

' c&11u1p

H1,p0

, C.

Finally, E is (sequentially) weakly lower semi-continuous: It su"ces to showthat for um $ u weakly in H1,p

0 (%) we have

"

"f um dx $

"

"f u dx .

Since f % H&1,q(%) , however, this follows from the very definition of weakconvergence. Hence Theorem 1.2 is applicable and there exists a minimizeru % H1,p

0 (%) of E, solving (1.4).

Note that for p ' 2 the p-Laplacian is strongly monotone in the sense that

"

"

%|2u|p&22u , |2v|p&22v

&· (2u ,2v) dx ' c1u , v1p

H1,p0

.

In particular, the solution u to (1.4) is unique.If f is more regular, say f % Cm,!(%), we would expect the solution u of

(1.4) to be more regular as well. This is true if p = 2, see Appendix B, butin the degenerate case p > 2, where the uniform ellipticity of the p-Laplaceoperator is lost at zeros of |2u|, the best that one can hope for is u % C1,!(%);see Uhlenbeck [1], Tolksdorf [2; p. 128], Di Benedetto [1].

In Theorem 1.3 we have applied Theorem 1.2 to a functional on a reflexivespace. An example in a non-reflexive setting is given next.


Minimal Partitioning Hypersurfaces

For a domain % & IRn let BV (%) be the space of functions u % L1(%) suchthat

"

"|Du| = sup

'"

"

n(

i=1

uDigi dx ;

g = (g1, . . . , gn) % C10 (%; IRn), |g| + 1

)< * ,

endowed with the norm

1u1BV = 1u1L1 +"

"|Du| .

BV (%) is a Banach space, embedded in L1(%), and – provided % is boundedand su"ciently smooth – by Rellich’s theorem the injection BV (%) ($ L1(%)is compact; see for instance Giusti [1; Theorem 1.19, p. 17]. Moreover, thefunction u 4$

*" |Du| is lower semi-continuous with respect to L1-convergence.

Let )G be the characteristic function of a set G & IRn; that is, )G(x) = 1if x % G, )G(x) = 0 else. Also let Ln denote the n-dimensional Lebesguemeasure.

1.4 Theorem. Let % be a smooth, bounded domain in IRn. Then there existsa subset G & % such that

(1#) Ln(G) = Ln(% \ G) =12Ln(%)

and such that its perimeter with respect to %,

(2#) P (G,%) ="

"|D)G| ,

is minimal among all sets satisfying (1#).

Proof. Let M = {)G ; G & % is measurable and satisfies (1#)}, endowed withthe L1-topology, and let E : M $ IR ) +* be given by

E(u) ="

"|Du| .

Since 1)G1L1 + Ln(%), the functional E is coercive on M with respect to thenorm in BV (%). Since bounded sets in BV (%) are relatively compact in L1(%)and since M is closed in L1(%), by weak lower semi-continuity of E in L1(%)the sub-level sets of E are compact. The conclusion now follows from Theorem1.1.

The support of the distribution D)G, where G has minimal perimeter (2#)with respect to %, can be interpreted as a minimal bisecting hypersurface,


dividing % into two regions of equal volume. The regularity of the dividinghypersurface is intimately connected with the existence of minimal cones inIRn. See Giusti [1] for further material on functions of bounded variation, setsof bounded perimeter, the area integrand, and applications.

A related setting for the study of minimal hypersurfaces and related objectsis o!ered by geometric measure theory. Also in this field variational principlesplay an important role; see for instance Almgren [1], Morgan [1], or Simon [1]for introductory material and further references.

Our next example is concerned with a parametric approach.

Minimal Hypersurfaces in Riemannian Manifolds

Let % be a bounded domain in IRn, and let S be a compact subset in IRN .Also let u0 % H1,2(%; IRN ) with u0(%) & S be given. Define

H1,2(%; S) =+u % H1,2(%; IRN ) ; u(%) & S almost everywhere

,

and letM =

+u % H1,2(%; S) ; u , u0 % H1,2

0 (%; IRN ),

.

Then, by Rellich’s theorem, M is closed in the weak topology of V =H1,2(%; IRN ). For u = (u1, . . . , uN ) % H1,2(%; S) let

E(u) ="

"gij(u)2ui2uj dx ,

where g = (gij)1'i,j'N is a given positively definite symmetric matrix withcoe"cients gij(u) depending continuously on u % S, and where, by convention,we tacitly sum over repeated indices 1 + i, j + N . Note that since S iscompact g is uniformly positive definite on S, and there exists * > 0 such thatE(u) ' * 12u12

L2 for u % H1,2(%; S). In addition, since S and % are bounded,we have that 1u1L2 + c uniformly, for u % H1,2(%; S). Hence E is coercive onH1,2(%; S) with respect to the norm in H1,2(%; IRN ).

Finally, E is lower semi-continuous in H1,2(%; S) with respect to weakconvergence in H1,2(%; IRN ). Indeed, if um $ u weakly in H1,2(%; IRN ), byRellich’s theorem um $ u strongly in L2 and hence a subsequence (um) con-verges almost everywhere. By Egorov’s theorem, given + > 0 there is an excep-tional set %# of measure Ln(%#) < + such that um $ u uniformly on % \ %#.We may assume that %# & %#" for + + +(. By weak lower semi-continuity ofthe semi-norm on H1,2(%; IRN ), defined by

|v|2 ="

"\"!

gij(u)2vi2vj dx,

then


"

"\"!

gij(u)2ui2uj dx

+ lim infm$%

"

"\"!

gij(u)2uim2uj

m dx

= lim infm$%

"

"\"!

gij(um)2uim2uj

m dx

+ lim infm$%

E(um) .

Passing to the limit + $ 0, from Beppo Levi’s theorem we obtain

E(u) = lim#$0

"

"\"!

gij(u)2ui2uj dx

+ lim infm$%

E(um) .

Applying Theorem 1.2 to E on M we obtain

1.5 Theorem. For any boundary data u0 % H1,2(%; S) there exists an E-minimal extension u % M .

In di!erential geometry Theorem 1.5 arises in the study of harmonic mapsu : % $ S from a domain % into an N -dimensional manifold S with metricg for prescribed boundary data u = u0 on &%. Like in the previous example,the regularity question is related to the existence of special harmonic maps; inthis case, singularities of harmonic maps from % into S are related to harmonicmappings of spheres into S. For further references see Eells-Lemaire [1], [2],Hildebrandt [3], Jost [2]. For questions concerning regularity see Giaquinta-Giusti [1], Schoen-Uhlenbeck [1], [2].

A General Lower Semi-continuity Result

We now conclude this short list of introductory examples and return to thedevelopment of the variational theory. Note that the property of E being lowersemi-continuous with respect to some weak kind of convergence is at the coreof the above existence results. In Theorem 1.6 below we establish a lower semi-continuity result for a very broad class of variational integrals, including andgoing beyond those encountered in Theorem 1.5, as Theorem 1.6 would alsoapply in the case of unbounded targets S and possibly degenerate or singularmetrics g.

We consider variational integrals

(1.5) E(u) ="

"F (x, u,2u) dx

involving (vector-valued) functions u : % & IRn $ IRN .


1.6 Theorem. Let % be a domain in IRn, and assume that F : % # IRN #IRnN $ IR is a Caratheodory function satisfying the conditions(1#) F (x, u, p) ' ,(x) for almost every x, u, p, where , % L1(%).(2#) F (x, u, ·) is convex in p for almost every x, u.Then, if um, u % H1,1

loc (%) and um $ u in L1(%(), 2um $ 2u weakly inL1(%() for all bounded %( && %, it follows that

E(u) + lim infm$%

E(um) ,

where E is given by (1.5).

Notes. In the scalar case N = 1, weak lower semi-continuity results like Theo-rem 1.6 were first stated by Tonelli [1] and Morrey [1]; these results were thenextended and simplified by Serrin [1], [2] who showed that for non-negative,smooth functions F (x, u, p):% # IR # IRn $ IR, which are convex in p , thefunctional E given by (1.5) is lower semi-continuous with respect to conver-gence in L1

loc(%). A corresponding result in the vector-valued case N > 1subsequently was derived by Morrey [4; Theorem 4.1.1]; however, Eisen [1] notonly pointed out a gap in Morrey’s proof but also gave an example showingthat for N > 1 in general, Theorem 1.6 ceases to be true without the assump-tion that the L1-norms of 2um are uniformly locally bounded. Theorem 1.6 isdue to Berkowitz [1] and Eisen [2]. Related results can be found for instancein Morrey [4; Theorem 1.8.2], or Giaquinta [1]. Our proof is modeled on Eisen[2].

Proof. We may assume that%E(um)

&is finite and convergent. Moreover,

replacing F by F , , we may assume that F ' 0. Let %( && % be given. Byweak local L1-convergence 2um $ 2u, for any m0 % IN there exists a sequence(P l)l)m0 of convex linear combinations

P l =l(

m=m0

#lm2um , 0 + #l

m + 1 ,l(

m=m0

#lm = 1 , l ' m0

such that P l $ 2u strongly in L1(%() and pointwise almost everywhere asl $ *; see for instance Rudin [1; Theorem 3.13]. By convexity, for any m0,any l ' m0, and almost every x % %( :

F%x, u(x), P l(x)

&= F

-x, u(x),

l(

m=m0

#lm2um(x)

.

+l(

m=m0

#lmF (x, u(x),2um(x)) .

Integrating over %( and passing to the limit l $ *, from Fatou’s lemma weobtain:


"

""F (x, u(x),2u(x)) dx + lim inf

l$%

"

""F%x, u(x), P l(x)

&dx

+ supm)m0

"

""F%x, u(x),2um(x)

&dx .

Since m0 was arbitrary, this implies that"

""F%x, u(x),2u(x)

&dx + lim sup

m$%

"

""F (x, u(x),2um(x)) dx ,

for any bounded %( && %.

Now we need the following result (Eisen [2; p. 75]).

1.7 Lemma. Under the hypotheses of Theorem 1.6 on F, um, and u there existsa subsequence (um) such that:

F (x, um(x),2um(x)) , F (x, u(x),2um(x)) $ 0

in measure, locally in %.

Proof of Theorem 1.6 (completed). By Lemma 1.7 for any %( && %, any ! > 0,and any m0 % IN there exists m ' m0 and a set %(

$,m & %( with Ln%%($,m

&< !

such that

(1.6) |F%x, um(x),2um(x)

&, F

%x, u(x),2um(x)

&| < !

for all x % %( \%($,m. Replacing ! by !m = 2&m and passing to a subsequence, if

necessary, we may assume that for each m there is a set %($m,m & %( of measure

< !m such that (1.6) is satisfied (with !m) for all x % %( \ %($m,m. Hence, for

any given ! > 0, if we choose m0 = m0(!) > | log2 !|, %($ =

/m)m0

%($m,m,

this set has measure Ln(%($) < ! and inequality (1.6) holds uniformly for all

x % %( \%($, and all m ' m0(!). Moreover, for ! <+ by construction %(

$ & %(#.

Cover % by disjoint bounded sets %(k) && %, k % IN. Let ! > 0 be givenand choose a sequence !(k) > 0, such that

0k"IN Ln

%%(k)

&!(k) + !. Passing

to a subsequence, if necessary, for each %(k) and !(k) we may choose m(k)0 and

%(k)$ & %(k) such that Ln

#%(k)$

$< !(k) and

|F (x, um(x),2um(x)) , F (x, u(x),2um(x)) | < !(k)

uniformly for x % %(k)\%(k)$ , m ' m(k)

0 . Moreover, we may assume that %(k)$ &

%(k)# , if ! < +, for all k. Then for any K % IN, letting %K = )K

k=1%(k), %K

$ =)K

k=1%(k)$ , we have


"

"K\"K"

F (x, u,2u) dx

+ lim supm$%

"

"K\"K"

F (x, u,2um) dx

+ lim supm$%

"

"K\"K"

F (x, um,2um) dx + !

+ lim supm$%

E(um) + ! = lim infm$%

E(um) + ! .

Letting ! $ 0 and then K $ *, the claim follows from Beppo Levi’s theorem,since F ' 0 and since %K \%K

$ increases when ! 5 0, followed by K 6 *.

Proof of Lemma 1.7. We basically follow Eisen [2]. Suppose by contradictionthat there exist %( && % and ! > 0 such that, letting

%m = {x % %( ; |F (x, um,2um) , F (x, u,2um) | ' !} ,

there holdslim infm$%

Ln(%m) ' 2! .

The sequence (2um), being weakly convergent, is uniformly bounded in L1(%().In particular,

Ln{x % %( ; |2um(x)| ' l} + l&1

"

"(|2um| dx + C

l+ ! ,

if l ' l0(!) is large enough. Setting %m := {x % %m ; |2um(x)| + l0(!)}therefore there holds

lim infm$%

Ln#%m

$' !.

Hence also for %M =1

m)M

%m we have

Ln(%M ) ' ! ,

uniformly in M % IN. Moreover, %( 0 %M 0 %M+1 for all M and therefore%% :=

2

M"IN

%M & %( has Ln(%%) ' !. Finally, neglecting a set of measure

zero and passing to a subsequence, if necessary, we may assume that F (x, z, p)is continuous in (z, p), that um(x), u(x), 2um(x) are unambiguously definedand finite while um(x) $ u(x) as m $ * at every point x % %%.

Note that every point x % %% by construction belongs to infinitely many ofthe sets %m. Choose such a point x. Relabeling, we may assume x %

!m"IN %m.


By uniform boundedness |2um(x)| + C there exists a subsequence m $ * anda vector p % IRnN such that 2um(x) $ p (m $ *). But then by continuity

F%x, um(x),2um(x)

&$ F

%x, u(x), p

&

while alsoF%x, u(x),2um(x)

&$ F

%x, u(x), p

&

which contradicts the characterization of %m given above.

1.8 Remarks. The following observations may be useful in applications.(1#) Theorem 1.6 also applies to functionals involving higher (mth-) orderderivatives of a function u by letting U = (u,2u, . . . ,2m&1u) denote the(m , 1)-jet of u. Note that convexity is only required in the highest-orderderivatives P = 2mu.(2#) If (um) is bounded in H1,1(%() for any %( && %, by Rellich’s theoremand repeated selection of subsequences there exists a subsequence (um) whichconverges strongly in L1(%() for any %( && %.

Local boundedness in H1,1 of a minimizing sequence (um) for E can beinferred from a coerciveness condition like

(1.7) F (x, z, p) ' |p|µ , ,(x), µ ' 1, , % L1 .

The delicate part in the hypotheses concerning (um) is the assumption that(2um) converges weakly in L1

loc. In case µ > 1 in (1.7) this is clear, but in caseµ = 1 the local L1-limit of a minimizing sequence may lie in BVloc instead ofH1,1

loc . See Theorem 1.4, for example; see also Section 3.(3#) By convexity in p, continuity of F in (u, p) for almost every x is equivalentto the following condition, which is easier to check in applications:

F (x, ·, ·) is continuous, separately in u % IRN and p % IRnN , for almostevery x % %.

Indeed, for any fixed x, u, p and all e % IRnN , |e| = 1, # % [0, 1], lettingq = p+#e, p+ = p+e, p& = p,e and writing F (x, u, p) = F (u, p) for brevity,by convexity we have

F (u, q) = F (u,#p+ + (1 , #)p) + #F (u, p+) + (1 , #)F (u, p) ,

F (u, p) = F (u,1

1 + #q +

#

1 + #p&) + 1

1 + #F (u, q) +

#

1 + #F (u, p&) .

Hence

# (F (u, p) , F (u, p+)) + F (u, p) , F (u, q) + # (F (u, p&) , F (u, p))

and it follows that

sup|q&p|'1

|F (u, q) , F (u, p)||q , p| + sup

|q&p|=1|F (u, q), F (u, p)| .

2. Constraints 13

Since the sphere of radius 1 around p lies in the convex hull of finitely manyvectors q0, q1, . . . , qnN , by continuity of F in u and convexity in p the right-hand side of this inequality remains uniformly bounded in a neighborhood of(u, p). Hence F (·, ·) is locally Lipschitz continous in p, locally uniformly in(u, p) % IRN # IRnN . Therefore, if um $ u , pm $ p we have

|F (um, pm) , F (u, p) | + |F (um, pm) , F (um, p) | + |F (um, p) , F (u, p) |+ c|pm , p| + o(1) $ 0 as m $ *,

where o(1) $ 0 as m $ *, as desired.(4#) In the scalar case (N = 1), if F is C2 for example, the existence of aminimizer u for E implies that the Legendre condition

n(

!,%=1

Fp#p$ (x, u, p) -!-% ' 0, for all - % IRn

holds at all points (x, u = u(x), p = 2u(x)), see for instance Giaquinta [1; p. 11f.]. This condition in turn implies the convexity of F in p.

The situation is quite di!erent in the vector-valued case N > 1. In thiscase, in general only the Legendre-Hadamard condition

N(

i,j=1

(

!,%=1

Fpi#pj

$(x, u, p)-!-%.i.j ' 0 , for all - % IRn, . % IRN

will hold at a minimizer, which is much weaker then convexity (Giaquinta [1;p. 12]).

In fact, in Section 3 below we shall see how, under certain additionalstructure conditions on F , the convexity assumption in Theorem 1.6 can beweakened in the vector-valued case.

2. Constraints

Applying the direct methods often involves a delicate interplay between thefunctional E, the space of admissible functions M , and the topology on M . Inthis section we will see how, by means of imposing constraints on admissiblefunctions and/or by a suitable modification of the variational problem, thedirect methods can be successfully employed also in situations where their useseems highly unlikely at first.

Note that we will not consider constraints that are dictated by the prob-lems themselves, such as physical restrictions on the response of a mechanicalsystem. Constraints of this type in general lead to variational inequalities, andwe refer to Kinderlehrer-Stampacchia [1] for a comprehensive introduction tothis field. Instead, we will show how certain variational problems can be solved


by adding virtual – that is, purely technical – constraints to the conditionsdefining the admissible set, thus singling out distinguished solutions.

Semilinear Elliptic Boundary Value Problems

We start by deriving the existence of positive solutions to non-coercive, semi-linear elliptic boundary value problems by a constrained minimization method.Such problems are motivated by studies of flame propagation (see for exampleGel’fand [1; (15.5), p. 357]) or arise in the context of the Yamabe problem (seeSection III.4).

Let % be a smooth, bounded domain in IRn, and let p > 2. If n ' 3 we alsoassume that p satisfies the condition p < 2! = 2n

n&2 . For * % IR consider theproblem

,/u + *u = u|u|p&2 in % ,(2.1)u > 0 in % ,(2.2)u = 0 on &% .(2.3)

Also let 0 < *1 < *2 + *3 + . . . denote the eigenvalues of the operator ,/ onH1,2

0 (%). Then we have the following result:

2.1 Theorem. For any * > ,*1 there exists a positive solution u % C2(%) (C0(%) to problem (2.1)–(2.3).

Proof. Observe that Equation (2.1) is the Euler-Lagrange equation of thefunctional

E(u) =12

"

"

%|2u|2 + *|u|2

&dx , 1

p

"

"|u|p dx

on H1,20 (%), which is neither bounded from above nor from below on this

space. However, using the homogeneity of (2.1) a solution of problem (2.1)–(2.3) can also be obtained by solving a constrained minimization problem forthe functional

E(u) =12

"

"

%|2u|2 + *|u|2

&dx

on the Hilbert space H1,20 (%) , restricted to the set

M = {u % H1,20 (%) ;

"

"|u|p dx = 1} .

We verify that E : M $ IR satisfies the hypotheses of Theorem 1.2. Bythe Rellich-Kondrakov theorem the injection H1,2

0 (%) ($ Lp(%) is completelycontinuous for p < 2!, if n ' 3, respectively for any p < *, if n = 1, 2; seeTheorem A.5 of Appendix A. Hence M is weakly closed in H1,2

0 (%).

2. Constraints 15

Recall the Rayleigh-Ritz characterization

(2.4) *1 = infu#H

1,20 (%)

u$=0

*" |2u|2 dx*" |u|2 dx

of the smallest Dirichlet eigenvalue. This gives the estimate

(2.5) E(u) ' 12min+1,%1 +

*

*1

&,1u12

H1,20

.

From this, coerciveness of E for * > ,*1 is immediate.Weak lower semi-continuity of E follows from weak lower semi-continuity

of the norm in H1,20 (%) and the Rellich-Kondrakov theorem. By Theorem 1.2

therefore E attains its infimum at a point u in M. Note that since E(u) = E(|u|)we may assume that u ' 0.

To derive the variational equation for E first note that E is continuouslyFrechet-di!erentiable in H1,2

0 (%) with

!v, DE(u)" ="

"

%2u2v + *uv

&dx .

Moreover, letting

G(u) ="

"|u|p dx , 1 ,

for p + 2! also G : H1,20 (%) $ IR is continuously Frechet-di!erentiable with

!v, DG(u)" = p

"

"u|u|p&2v dx .

In particular, at any point u % M

!u, DG(u)" = p

"

"|u|p dx = p -= 0 ,

and by the implicit function theorem the set M = G&1(0) is a C1-submanifoldof H1,2

0 (%).Now, by the Lagrange multiplier rule, there exists a parameter µ % IR such

that

!v, (DE(u) , µDG(u))" ="

"

%2u2v + *uv , µu|u|p&2v

&dx

= 0, for all v % H1,20 (%) .

Inserting v = u into this equation yields that

2E(u) ="

"

%|2u|2 + *|u|2

&dx = µ

"

"|u|p dx = µ .


Since u % M cannot vanish identically, from (2.5) we infer that µ > 0. Scalingwith a suitable power of µ, we obtain a weak solution u = µ

1p!2 · u % H1,2

0 (%)of (2.1), (2.3) in the sense that

(2.6)"

"

%2u2v + *uv , u|u|p&2v

&dx = 0 , for all v % H1,2

0 (%) .

Moreover, (2.2) holds in the weak sense u ' 0, u -= 0. To finish the proof we usethe regularity result Lemma B.3 of Appendix B and the observations followingit to obtain that u % C2(%). Finally, by the strong maximum principle u > 0in % ; see Theorem B.4.

Observe that, at least for the kind of nonlinear problems considered here, byLemma B.3 of Appendix B the regularity theory is taken care of and in the fol-lowing we may concentrate on proving existence of (weak) solutions. However,additional structure conditions may imply further useful properties of suitablesolutions. An example is symmetry.

2.2 Symmetry. By a result of Gidas-Ni-Nirenberg [1; Theorem 2.1, p. 216,and Theorem 1, p. 209], if % is convex and symmetric with respect to a hy-perplane, say x1 = 0, any positive solution u of (2.1), (2.3) is even in x1, thatis, u(x1, x() = u(,x1, x() for all x = (x1, x() % %, and &u

&x1< 0 at any point

x = (x1, x() % % with x1 > 0. In particular, if % is a ball, any positive so-lution u is radially symmetric. The proof of this result uses a variant of theAlexandrov-Hopf reflection principle and the maximum principle. This methodlends itself to numerous applications in many di!erent contexts; in Section III.4we shall see that it is even possible to derive a-priori bounds from this methodin the setting of a parabolic equation on the sphere.

Perron’s Method in a Variational Guise

In the previous example the constraint built into the definition of M had thee!ect of making the restricted functional E = E|M coercive. Moreover, thisconstraint only changed the Euler-Lagrange equations by a factor which couldbe scaled away using the homogeneity of the right-hand side of (2.1).

In the next application we will see that sometimes also inequality con-straints can be imposed without changing the Euler-Lagrange equations at aminimizer.

2.3 Weak sub- and super-solutions. Suppose % is a smooth, bounded domainin IRn, and let g : % # IR $ IR be a Caratheodory function with the propertythat |g(x, u)| + C(R) for any R > 0 and all u such that |u(x)| + R almosteverywhere. Given u0 % H1,2

0 (%), we then consider the equation

,/u = g(·, u) in % ,(2.7)u = u0 on &% .(2.8)

2. Constraints 17

By definition u % H1,2(%) is a (weak) sub-solution to (2.7–2.8) if u + u0 on&% and

"

"2u2' dx ,

"

"g( · , u)' dx + 0 for all ' % C%

0 (%) , ' ' 0 .

Similarly u % H1,2(%) is a (weak) super-solution to (2.7–2.8) if in the abovethe reverse inequalities hold.

2.4 Theorem. Suppose u % H1,2(%) is a sub-solution while u % H1,2(%) is asuper-solution to problem (2.7–2.8) and assume that with constants c, c % IRthere holds ,* < c + u + u + c < *, almost everywhere in %. Thenthere exists a weak solution u % H1,2(%) of (2.7–2.8), satisfying the conditionu + u + u almost everywhere in %.

Proof. With no loss of generality we may assume u0 = 0. Let G(x, u) =* u0 g(x, v) dv denote a primitive of g. Note that (2.7–2.8) formally are the

Euler-Lagrange equations of the functional

E(u) =12

"

"|2u|2 dx ,

"

"G(x, u) dx .

However, our assumptions do not allow the conclusion that E is finite or evendi!erentiable on V := H1,2

0 (%) – the smallest space where we have any chanceof verifying coerciveness. Instead we restrict E to

M =+u % H1,2

0 (%) ; u + u + u almost everywhere,

.

Since u, u % L% by assumption, also M & L% and G%x, u(x)

&+ c for all

u % M and almost every x % %.Now we can verify the hypotheses of Theorem 1.2: Clearly, V = H1,2

0 (%)is reflexive. Moreover, M is closed and convex, hence weakly closed. Since Mis essentially bounded, our functional E(u) ' 1

21u12H1,2

0 ("), c is coercive on

M . Finally, to see that E is weakly lower semi-continuous on M , it su"ces toshow that "

"G(x, um) dx $

"

"G(x, u) dx

if um $ u weakly in H1,20 (%), where um, u % M . But – passing to a sub-

sequence, if necessary – we may assume that um $ u pointwise almost ev-erywhere; moreover, |G

%x, um(x)

&| + c uniformly. Hence we may appeal to

Lebesgue’s theorem on dominated convergence.From Theorem 1.2 we infer the existence of a relative minimizer u % M .

To see that u weakly solves (2.7), for ' % C%0 (%) and ! > 0 let v$ =

min+u, max{u, u + !'}

,= u + !', '$ + '$ % M with

'$ = max+0, u + !', u

,' 0 ,

'$ = max+0, u , (u + !')

,' 0 .


Note that '$,'$ % H1,20 ( L%(%).

E is di!erentiable in direction v$, u . Since u minimizes E in M we have

0 + !(v$ , u), DE(u)" = !!', DE(u)" , !'$, DE(u)"+ !'$, DE(u)" ,

so that!', DE(u)" ' 1

!

3!'$, DE(u)" , !'$, DE(u)"

4.

Now, since u is a supersolution to (2.7), we have

!'$, DE(u)" = !'$, DE(u)" + !'$, DE(u) , DE(u)"' !'$, DE(u), DE(u)"

="

""

+2(u , u)2(u + !', u)

,%g(x, u), g(x, u)

&(u + !', u)

,dx

' !

"

""

2(u , u)2' dx , !

"

""

55g(x, u), g(x, u)55 |'| dx ,

where %$ =+x % % ; u(x) + !'(x) ' u(x) > u(x)

,. Note that Ln(%$) $ 0 as

! $ 0. Hence by absolute continuity of the Lebesgue integral we obtain that

!'$, DE(u)" ' o(!) ,

where o(!)/! $ 0 as ! $ 0. Similarly, we conclude that

!'$, DE(u)" + o(!) ,

whence!', DE(u)" ' 0

for all ' % C%0 (%). Reversing the sign of ' and since C%

0 (%) is dense inH1,2

0 (%) we finally see that DE(u) = 0, as claimed.

2.5 A special case. Let % be a smooth bounded domain in IRn, n ' 3, and let

(2.9) g(x, u) = k(x)u , u|u|p&2 ,

where p = 2nn&2 , and where k is a continuous function such that

1 + k(x) + K < *

uniformly in %. Suppose u0 % C1(%) satisfies u0 ' 1 on &%.Then u . 1 is a sub-solution while u . c for large c > 1 is a super-solution toEquations (2.7)–(2.8). Consequently, (2.7)–(2.8) admits a solution u ' 1.

2.6 Remark. The sub-super-solution method can also be applied to equationson manifolds. In the context of the Yamabe problem it has been used byLoewner-Nirenberg [1] and Kazdan-Warner [1]; see Section III.4. The non-linear term in this case is precisely (2.9).

2. Constraints 19

The Classical Plateau Problem

One of the great successes of the direct methods in the calculus of variationswas the solution of Plateau’s problem for minimal surfaces.

Let 0 be a smooth Jordan curve in IR3. From his famous experimentswith soap films Plateau became convinced that any such curve is spanned bya (not necessarily unique) surface of least area.

Fig. 2.1. Minimal surfaces of various topological types (disk, Mobius band, annulus, torus)

In the classical mathematical model the topological type of the surface isspecified to be that of the disk

% = {z = (x, y) ; x2 + y2 < 1} .

A naive approach to Plateau’s conjecture would be to attempt to minimize thearea

A(u) ="

"

6det(2ut2u) dz =

"

"

7|ux|2|uy|2 , (ux · uy)2 dz

among “surfaces” u % H1,2 ( C0(%, IR3) satisfying the Plateau boundary con-dition

(2.10) u|&%

: &% $ 0 is a (weakly) monotone parametrizationpreserving the given orientation of 0 .

However, A is invariant under arbitrary changes of parameter. Hence there isno chance of achieving bounded compactness in the original variational problemand some work was necessary in order to recast this problem in a way which isaccessible by direct methods. Without entering into details let us briefly reportthe main ideas.

It had already beeen observed by Lagrange that if a (smooth) surface S is(locally) area-minimizing for fixed boundary 0 , necessarily the mean curvatureof S vanishes. In isothermal coordinates u(x, y) on S this amounts to theequation

(2.11) /u = 0 .


(See Nitsche [2] or Osserman [1].) Moreover, our choice of parameter impliesthe conformality relations

(2.12) |ux|2 , |uy|2 = 0 = ux · uy in % ,

in addition to the Plateau boundary condition (2.10).We now take equations (2.10)–(2.12) as a definition for a minimal surface

spanning 0 .

2.7 The variational problem. In their 1930 break-through papers, Douglas[1] and Rado [1] ingeniously proposed to solve (2.10)–(2.12) by minimizingDirichlet’s integral

E(u) =12

"

"|2u|2 dz =

12

"

"(|ux|2 + |uy|2) dz

over the class

C(0 ) = {u % H1,2(%, IR3) ; u&%

% C0(&%, IR3) satisfies (2.10)} .

It is easy to see thatE(u) ' A(u) ,

and equality holds if and only if u is conformal. Actually, we have

infC(' )

A(u) = infC(' )

E(u) .

This can be derived for instance from Morrey’s “!-conformality lemma” (Mor-rey [2; Theorem 1.2]). In Struwe [18; Appendix A] also a direct (variational)proof is given. Thus, a minimizer of E also will minimize A – hence it willsatisfy (2.12) and solve the original minimization problem.

The solution of Plateau’s problem is therefore reduced to the followingtheorem.

2.8 Theorem. For any C1-embedded curve 0 there exists a minimizer u ofDirichlet’s integral E in C(0 ).

Note that C(0 ) -= 7 if 0 % C1. (Actually it su"ces to assume that 0 is arectifiable Jordan curve; see Douglas [1], Rado [1].)

To show Theorem 2.8, observe that in replacing A by E we have succeededin reducing the symmetries of the problem drastically. However, E is stillconformally invariant, that is

E(u) = E(u 8 g)

for all g % G, where

G ='

g : z 4$ g(z) = ei( a + z

1 , az; a % C, |a| < 1, 0 + , < 21

)

2. Constraints 21

denotes the conformal group of Mobius transformations of the disc, viewed as asubset of C. The action of G is non-compact in the sense that for any u % C(0 )the orbit {u 8 g ; g % G} weakly accumulates also at constant functions; see forinstance Struwe [17; Lemma I.4.1] for a detailed proof. Hence C(0 ) cannot beweakly closed in H1,2(%, IR3) and Theorem 1.2 cannot yet be applied.

Fortunately, we can also get rid of conformal invariance of E. Note thatfor any oriented triple ei(1 , ei(2 , ei(3 % &% , 0 + ,1 < ,2 < ,3 < 21 thereexists a unique g % G such that g

#e

2&ik3

$= ei(k , k = 1, 2, 3.

Fix a parametrization 2 of 0 (we may assume that 2 is a C1-di!eomor-phism 2 : &% $ 0 ) and let

C!(0 ) =8u % C(0 ) ; u

#e

2&ik3

$= 2

#e

2&ik3

$, k = 1, 2, 3

9,

endowed with the H1,2-topology. This is our space of admissible functions,normalized with respect to G. Note that for any u % C(0 ) there is g % G suchthat u 8 g % C!(0 ).

The following result now is a consequence of the classical “Courant-Lebesgue lemma”.

2.9 Lemma. The set C!(0 ) is weakly closed in H1,2.

Fig. 2.2.

Proof. The proof in a subtle way uses a convexity argument as in the precedingexample. To present this argument explicitly we use the fixed parametrization2 to associate with any u % C!(0 ) a continuous map - : IR $ IR, such that

2#ei)(()

$= u

%ei(&

, -(0) = 0 .

By (2.10) the functions - obtained in this manner are continuous, monotoneand - , id is 21-periodic; moreover, -

%2*k3

&= 2*k

3 , for all k % ZZ by ourthree-point normalization.


Now let

M =+- :IR $ IR ; - is continuous and monotone,

-(,+ 21) = -(,) + 21, -

:21k

3

;=

21k

3, for all , % IR, k % ZZ

,.

Note that M is convex. Let (um) be a sequence in C!(0 ) with associatedfunctions -m % M, and suppose um $ u weakly in H1,2(%). Since each -m ismonotone and satisfies the estimate 0 + -m(,) + 21, for all , % [0, 21], thefamily (-m) is bounded in BV

%[0, 21]

&. Hence (a subsequence) -m $ - almost

everywhere on [0, 21] and therefore – by periodicity – almost everywhere on IR,where - is monotone, - , id is 21-periodic, and - satisfies -

%2*k3

&= 2*k

3 , forall k % ZZ.

Now if - is continuous, it follows from monotonicity that -m $ - uniformly.Thus, by continuity of 2, also um converges uniformly to u on &%, and it followsthat u|&" is continuous and satisfies (2.10). That is, u % C!(0 ), and the proofis complete in this case.

In order to exclude the remaining case, assume by contradiction that - isdiscontinuous at some point ,0. We choose k % ZZ such that |,0 , 2*k

3 | + *3

and let42*(k&1)

3 , 2*(k+1)3

3=: I0. By monotonicity, for almost every ,1,,2 % I0

such that ,1 < ,0 < ,2 we have

21(k , 1)3

+ limm$%

-m(,1) = -(,1) + lim($(!

0

-(,)

< lim($(+

0

-(,) + -(,2) = limm$%

-m(,2) +21(k + 1)

3.

For such ,1, ,2 % I0 denote I1 = {, % I0 ; , + ,1}, I2 = {, % I0 ; , ' ,2}.Then by monotonicity of -m and using the fact that 2 is a di!eomorphism weobtain

lim supm$%

:inf

("I1, +"I2|2#ei)m(()

$, 2#ei)m(+)

$|;

' inf(,+"I0,(<(0<+

|2#ei)(()

$, 2#ei)(+)

$| > 0 .

In particular, there exists ! > 0 independent of ,1,,2 such that

(2.13) |um(ei() , um(ei+)| ' ! > 0

for all , % I1, 3 % I2 if m ' m0(,1,,2) is su"ciently large.Now let z0 = ei(0 and for 4 > 0 denote

U, = {z % % ; |z , z0| < 4}, C, = {z % % ; |z , z0| = 4} .

Note that for all 4 < 1 any point z = ei( % C, ( &% satisfies , % I0.Following Courant [1; p. 103], we will use uniform boundedness of (um) in

H1,2 to show that for suitable numbers 40 %]0, 1[, 4m % [420, 40] the oscillation

of um on C,m can be made arbitrarily small, uniformly in m % IN.

2. Constraints 23

First note that by Fubini’s theorem, if we denote arc length on C, by s, fromthe estimate

* > c '"

"|2um|2 dz '

" 1

0

"

C'

| &&s

um|2 ds d4

we obtain that"

C'

| &&s

um|2 ds < *

for almost every 4 < 1 and all m % IN.Choosing 40 < 1 we may refine this estimate as follows:

"

"|2um|2 dz '

" ,0

,20

-4

"

C'

| &&s

um|2 ds

.d4

4

' | log 40| ess inf,20',',0

-4

"

C'

| &&s

um|2 ds

..

Suppose 4m % [420, 40] is such that

4m

"

C'm

| &&s

um|2 ds + 2 ess inf,20',',0

-4

"

C'

| &&s

um|2 ds

.

and denote

C = supm"IN

"

"|2um|2 dz < * .

Fix zj = ei(j , j = 1, 2, the points of intersection of C,20 with &%, ,1 < ,0 < ,2.Also denote zm

j = ei(mj , j = 1, 2, with ,m

1 < ,0 < ,m2 the points of intersection

of C,m with &%.Then ,m

1 % I1, ,m2 % I2 while by Holder’s inequality

(2.14)|um(zm

1 ) , um(zm2 )|2 +

-"

C'm

| &&s

um| ds

.2

+ 14m

"

C'm

| &&s

um|2 ds + 21C

| log 40|< !

if 40 > 0 is su"ciently small. This estimate being uniform in m, for largem ' m0(,1,,2) we obtain a contradiction to (2.13) and the proof is complete.


Remark. From (2.14), monotonicity (2.10), the assumption that 0 is a Jordancurve – that is, a homeomorphic image of the circle S1 – and the three-pointcondition, also a direct proof of equi-continuity of the sub-level sets of E inC!(0 ) can be given; see Courant [1; Lemma 3.2, p. 103] or Struwe [17; LemmaI.4.3]. Moreover, note that (2.14) implies a uniform estimate for the modulusof continuity of a function u % H1,2

0 (%) on a sequence of concentric circulararcs around any fixed center z0 % % in terms of its Dirichlet integral. Thisobservation was used by Lebesgue [1] to obtain an equi-continuous minimizingsequence for Dirichlet’s integral in his solution of the classical Dirichlet problem;see also Section 5.7.

Proof of Theorem 2.8. By (2.10) and the generalized Poincare inequality The-orem A.9 of Appendix A, for u % C!(0 ) there holds

"

"|u|2 dz + c

:"

"|2u|2 dz +

"

&"|u|2 do

;+ cE(u) + c(0 ).

Thus E is coercive on M = C!(0 ) in H1,2(%; IR3). Moreover E is weakly lowersemi-continuous on H1,2(%; IR3). By Theorem 1.2 and Lemma 2.9 thereforethe functional E achieves its infimum in C!(0 ), which by conformal invarianceequals that in C(0 ).

2.10 Regularity. As in the preceding examples we may ask what regularityproperties the minimizer u and the parametrized surface possess. We note afew results:(1#) u &" is strictly monotone (Douglas [1], Rado [1]).(2#) If 0 % Cm,!, m ' 1, 0 < # < 1, then also u % Cm,!(%, IR3); this result isdue to Hildebrandt [1] (for m ' 4) with later improvements by Nitsche [1] .While remarks (1#), (2#) apply to arbitrary solutions of the Plateau problem(2.10)–(2.12), minimality is crucial for the next observations concerning thegeometric regularity of the parametrized solution surface.(3#) A minimizer u % C!(0 ) parametrizes an immersed minimal surface S =u(%) & IR3; see Osserman [2], Gulliver [1], Alt [1], Gulliver-Osserman-Royden[1]; if 0 is anlytic, S is immersed up to the boundary; see Gulliver-Lesley [1].If 0 is extreme, that is 0 & &K where K & IR3 is convex, S is embedded; seeMeeks-Yau [1]. Existence of embedded minimal surfaces bounded by extremecurves independently was obtained by Tomi-Tromba [1] and Almgren-Simon [1].

2.11 Note. In the history of the calculus of variations it seems that Plateau’sproblem has played a very prominent role. Important developments in thegeneral stream of ideas often were prompted by insights gained from the studyof minimal surfaces. As an example, consider the classical mountain pass lemma(see also Chapter II.1) which was used by Courant [1; Chapter VI.6–7] toestablish the existence of unstable minimal surfaces, previously obtained by

3. Compensated Compactness 25

Morse-Tompkins [1] and Shi!man [1] by a less direct, topological reasoning.However, since this material has been covered very extensively elsewhere (seeStruwe [18]), here we will confine ourselves to the above remarks.

For an introduction to Plateau’s problem and minimal surfaces, see forinstance, Osserman [1], or consult the encyclopedic book by Nitsche [2].

A truely remarkable – popular and profound – book on the subject isavailable by Hildebrandt-Tromba [1].

3. Compensated Compactness

As noted in Remark 1.8, it is conceivable that in the vector-valued case lowersemi-continuity results may hold true under a weaker convexity assumptionthan in Theorem 1.6, provided suitable structure conditions are satisfied bythe functional in variation. Weakening the convexity hypothesis is necessary,for instance, in dealing with problems arising in 3-dimensional elasticity, wherewe encounter energy functionals

*" W (2u) dx with a stored energy function

W depending on the determinant, the minors and the eigenvalues of the de-formation gradient 2u. Since infinite volume distortion for elastic materialswill a!ord an infinite amount of energy, it is natural to suppose that W $ *if either det(2u) $ 0 or det(2u) $ *; hence W cannot be convex in 2u.However, there is a large class of materials that can be described by polyconvexstored energy functions, which are of the form

W (2u) = f(subdeterminants of 2u),

where f is convex in each of its variables. John Ball [1] was the first to seethat lower semi-continuity results will hold for such functionals. The di"culty,of course, lies in proving, for instance, weak convergence det(2um) $ det(2u)for a sequence um $ u weakly in H1,3(%, IR3). Questions of this type had beeninvestigated by Reshetnyak [1], [2]. A general frame for studying such problemsis provided by the compensated compactness scheme of Murat and Tartar.

The basic principle of the compensated compactness method is given inthe following lemma; see Tartar [2; p. 270 f.].

3.1 The compensated compactness lemma. Let % be a domain in IRn andsuppose that(1#) um =

%u1

m, . . . , uNm

&$ u weakly in L2(%; IRN ).

(2#) The set80

j,k ajk&uj

m&xk

; m % IN9

is relatively compact in H&1loc (%; IRL) for

a set of vectors ajk % IRL; 1 + j + N, 1 + k + n. Let

5 =+* % IRN ;

(

j,k

ajk*j-k = 0 for some - % IRn \ {0},


and let Q be a (real) quadratic form such that Q(*) ' 0 for all * % 5. RegardingQ(um) % L1(%) as Radon measures Q(um)dx %

%C0(%()

&!, we may assumethat

%Q(um)

&converges weak!, locally.

Then on any %( && % we have

weak! , limm$%

Q%um

&' Q(u)

in the sense of measures. In particular, if Q(*) = 0 for all * % 5, then

weak! , limm$%

Q%um

&= Q(u)

locally, in the sense of measures.

Proof. Choose ' % C0(%) with compact support and such that 0 + ' + 1. Wemust show that

lim infm$%

"

"Q%um

&'2 dx '

"

"Q(u)'2 dx .

By translation we may assume that u = 0. Moreover, upon replacing um byum' we may assume that the supports of um lie in a fixed cube K && %and that ' . 1 on K. By translation and scaling, moreover, we can obtainK = [0, 21]n. Let

um(x) =(

!"ZZn

µm,!ei!·x , µm,! =

%µ1

m,!, . . . , µNm,!

&% CN ,

be the Fourier expansion for um. Since um is real, we have µm,! = µm,&!. Theassertion then is equivalent to showing that

lim infm$%

1(21)n

"

KQ%um

&dx = lim inf

m$%

(

!"ZZn

:Q%Re µm,!

&+ Q

%Im µm,!

&;' 0 .

By weak convergence um $ 0 in L2 we infer that0!"ZZn |µm,!|2 + c < * and

µm,! $ 0 as m $ *, uniformly on bounded sets of indices #.Moreover, by (2#) the set

<=

>(

!"ZZn

(

j,k

ajkµjm,! #k ei!·x ; m % IN

?@

A

is relatively compact in H&1, which implies that

(

##ZZn

|#|%#0

5550

j,k ajkµjm,! #k

5552

1 + |#|2$ 0,


as #0 $ *, uniformly in m % IN. But this means that µm,! can be decomposedµm,! = *m,! + 6m,! with Re *m,!, Im *m,! % 5 and

0|!|)!0

556m,!

552 $ 0 as#o $ *, uniformly in m % IN.

Indeed, for any # % ZZn, m % IN decompose µm,! = *m,! + 6m,!, whereRe *m,!, Im *m,! % 5 and |6m,!|2 is minimal among all decompositions of thiskind. We claim that for any ! > 0 there exist constants C = C(!), #0 = #0(!)such that for |#| ' #0 we can bound

(3.1) |6m,!|2 + C

5555(

j,k

ajkµjm,!

#k61 + |#|2

55552

+ !55µm,!

552

uniformly in m % IN.Otherwise there exists ! > 0 and a sequence # = #(l), l % IN, with

|#(l)| ' l, and m = m(l) such that for all l there holds

(3.2) |µm,!|2 ' |6m,!|2 ' l

5555(

j,k

ajkµjm,!

#k61 + |#|2

55552

+ ! |µm,!|2 .

(The first inequality follows from the choice of 6m,! above.) Let -(l), .(l) bethe unit vectors

-(l) =#(l)6

1 + |#(l)|2% Sn&1, .(l) =

µm(l),!(l)55µm(l),!(l)

55 % SN&1 ,

and denote by A(l): IRN $ IRL the linear map

. 4$(

j,k

ajk.j-k(l) .

We may assume that -(l) $ - and A(l) $ A as l $ *. Likewise, we maysuppose that .(l) $ .. Passing to the limit in (3.2) it follows that . % ker A;that is, . % 5. Projecting µm,! onto ker A for all # = #(l), m = m(l) wehence obtain a decomposition µm,! = *m,! + 6m,! with Re *m,!, Im *m,! %ker A & 5 and

556m,!

552 + C55Aµm,!

552 + C55A(l)µm,!

552 + o(1)55µm,!

552

= C

555555

(

j,k

ajkµjm,!

#k61 + |#|2

555555

2

+ o(1)55µm,!

552 ,

where o(1) $ 0 (l $ *). But by defintion556m,!

552 +556m,!

552, and we obtain thedesired contradiction to assumption (3.2). Hence for any ! > 0, any #0 ' #0(!):


lim infm$%

(

!"ZZn

:Q%Re µm,!

&+ Q

%Im µm,!

&;

= lim infm$%

(

|!|)!0

:Q%Re µm,!

&+ Q

%Im µm,!

&;

' lim infm$%

(

|!|)!0

:Q%Re *m,!

&+ Q

%Im *m,!

&, c!

55µm,!

552;

+ o(1)

' o(1) , c! ,

where o(1) $ 0 as #0 $ *. Thus, the assertion of the lemma follows as wefirst let #0 $ * and then ! $ 0.

As an application we mention the following well-known result.

3.2 The Div-Curl Lemma. Suppose um $ u, vm $ v weakly in L2(%; IR3) ona domain % & IR3 while the sequences (div um) and (curl vm) are relativelycompact in H&1(%). Then for any ' % C%

0 (%) we have"

"um · vm' dx $

"

"u · v' dx

as m $ *.

Proof. Let wm = (um, vm) % L2(%; IR6), and determine coe"cients ajk % IR4

such that0

jk ajk&wj

m&xk

= (div um, curl vm). Let Q be the quadratic formQ(u, v) = u·v, acting on vectors w = (u, v) % IR6. Note ajk =

%+jk, (!ijk)1'i'3

&

where +jk = 1 if j = k, and +jk = 0 else, !123 = 1 and !ijk = ,!jik = !jki.Hence

5 =+* = (µ, 6) % IR6 ; 9- % IR3 \ {0} : (- · µ , - : 6) = 0

,

=+* = (µ, 6) % IR6 ; µ · 6 = 0

,,

and Q . 0 on 5. Thus the assertion follows from Lemma 3.1.

The div-curl Lemma 3.2 shows how additional bounds on some derivativesallow one to prove continuity of nonlinear expressions (bi-linear in the aboveexample) under weak convergence.

Phrased somewhat di!erently, the reason for the convergence result statedin the div-curl lemma to hold is an implicit divergence structure. This stucturecan be brought out more clearly in the language of di!erential forms. Forsimplicity, we assume that all functions involved are periodic of period 1 in eachvariable. In this case, we may regard % = [0, 1]n as a fundamental domain forthe flat n-dimensional torus Tn = IRn/ZZn. Let d, d! be the exterior di!erentialand co-di!erential, respectively. We consider 1-forms um $ u in L2, vm $ v


in L2 weakly as m $ * with (d!um), (dvm) relatively compact in H&1. Wemay assume u = 0, v = 0. By Hodge decomposition we have

um = dam + d!bm + cm,

vm = dfm + d!gm + hm,

where cm, hm are harmonic 1-forms, dbm = dgm = 0, and with am $ 0, bm $0, fm $ 0, gm $ 0 weakly in H1,2(Tn), cm $ 0, hm $ 0 weakly in L2(Tn).Since the space of harmonic 1-forms on Tn is compact, in fact, cm $ 0 andhm $ 0 smoothly as m $ *. Moreover, since the sequences

/am = d!um,/gm = dvm

are relatively compact in H&1, it follows that (dam), (d!gm) are precompact inL2, and we conclude that

um = d!bm + o(1), vm = dfm + o(1),

where o(1) $ 0 in L2 as m $ *. Moreover, using the Hodge ;-operator anddenoting by “.” contraction of forms, we have

um · vm dx + o(1) = ;(d!bm · dfm) = (d ; bm) : dfm = d%(;bm) : dfm

&,

thus exhibiting the asserted divergence structure. Since by Rellich’s compact-ness result bm $ 0 strongly in L2 as m $ *, it is now trivial to pass to thelimit in the expression

"

T n

um · vm' dx ="

T n

d%(;bm : dfm

&' + o(1)

= (,1)n

"

T n

(;bm) : dfm : d' + o(1) = o(1),

where o(1) $ 0 as m $ *.A divergence structure is also the crucial ingredient in the applications

that follow.

Applications in Elasticity

The most important applications of the compensated compactness methodso far are in elasticity and hyperbolic systems, see Ball [1], [2], DiPerna [1].DiPerna-Majda have applied compensated compactness methods to obtain theexistence of weak solutions to the Euler equations for incompressible fluids, seefor instance DiPerna-Majda [1]. Our interest lies with the extensions of the di-rect methods that compensated compactness implies. Thus we will concentrateon Ball’s lower semi-continuity results for polyconvex materials in elasticity.


3.3 Theorem. Suppose W is a function on (3 # 3)-matrices 7, given by

W (7) = g(7, adj 7, det 7)

where g is a convex non-negative function in the sub-determinants of 7. Let %be a domain in IR3 and let um, u % H1,3

loc (%; IR3). Suppose that um $ u weaklyin H1,3(%(; IR3) while det(2um) $ h weakly in L1(%() for all %( && %, whereh % L1

loc(%). Then"

"W (2u) dx + lim inf

m$%

"

"W (2um) dx .

Proof. The proof of Theorem 1.6 can be carried over once we show that underthe hypotheses made

adj(2um) $ adj(2u)det(2um) $ det(2u)

weakly in L1(%() for all %( && %. The first assertion is a consequence of thedivergence structure of the adjoint matrix Am = adj (2um). Indeed, if indicesi, j are counted modulo 3 we have

Aijm =

&ui+1m

&xj+1

&ui+2m

&xj+2, &ui+2

m

&xj+1

&ui+1m

&xj+2

=&

&xj+1

:ui+1

m&ui+2

m

&xj+2

;, &

&xj+2

:ui+1

m&ui+2

m

&xj+1

;.

Fix %( && %. Note that%Aij

m

&is bounded in L3/2(%(). Hence we may as-

sume that Aijm $ Aij weakly in L3/2(%(). Moreover, by Rellich’s theorem

um $ u in L3(%(), whence um2um $ u2u weakly in L3/2(%(). By continu-ity of the distributional derivative with respect to weak convergence thereforeAij

m $%adj(2u)

&ij in the sense of distributions. Finally, by uniqueness of thedistributional limit, Aij =

%adj(2u)

&ij , and adj(2um) $ adj(2u) weakly inL3/2(%(), in particular weakly in L1(%(), as claimed.

Similarly, expanding the determinant along the first row, we have

det(2um) =&u1

m

&x1

B&u2

m

&x2

&u3m

&x3, &u2

m

&x3

&u3m

&x2

C

, &u1m

&x2

B&u2

m

&x1

&u3m

&x3, &u2

m

&x3

&u3m

&x1

C

+&u1

m

&x3

B&u2

m

&x1

&u3m

&x2, &u2

m

&x2

&u3m

&x1

C.


Again, a divergence structure emerges, if we rewrite this as

det(2um) =&

&x1

:u1

m

B&u2

m

&x2

&u3m

&x3, &u2

m

&x3

&u3m

&x2

C;

, &

&x2

:u1

m

B&u2

m

&x1

&u3m

&x3, &u2

m

&x3

&u3m

&x1

C;

+&

&x3

:u1

m

B&u2

m

&x1

&u3m

&x2, &u2

m

&x2

&u3m

&x1

C;.

Thus convergence det(2um) $ det(2u) in the sense of distributions followsby weak convergence in L3/2(%() of the terms in brackets [- -], proved above,strong convergence um $ u in L3(%(), and weak continuity of the distributionalderivative. Finally, by uniqueness of the weak limit in the distribution sense, itfollows that det(2u) = h and det(2um) $ det(2u) weakly in L1

loc, as claimed.

Observe that in the language of di!erential forms the divergence structure of aJacobian or its minors is even more apparent. In fact, for any smooth functionu = (u1, u2, u3):% & IR3 $ IR3 we have

det(2u) dx = ; det(2u) = du1 : du2 : du3,

where d denotes exterior derivative and where ; denotes the Hodge star operator(which in this case converts a function on % into a 3-form). Now dd = 0, andtherefore

du1 : du2 : du3 = d(u1du2 : du3),

which immediately implies the asserted divergence structure. Moreover, thisresult (and therefore Theorem 3.3) generalizes to any dimension n, for um $ uweakly in H1,n

loc (%; IRn) with det(2um) $ h weakly in L1loc(%) as m $ *.

The assumption det(2um) $ h % L1loc(%) at first sight may appear

rather awkward. However, examples by Ball-Murat [1] show that weak H1,3-convergence in general does not imply weak L1-convergence of the Jacobian.This di"culty does not arise if we assume weak convergence in H1,3+$ for some! > 0.

Hence, adding appropriate growth conditions on W to ensure coercivenessof the functional

*" W (2u) dx on the space H1,3+$(%; IR3) for some ! > 0,

from Theorem 3.3 the reader can derive existence theorems for deformationsof elastic materials involving polyconvex stored energy functions. As a fur-ther reference for such results, see Ciarlet [1] or Dacorogna [1], [2]. Recently,more general results on weak continuity of determinants and correspondingexistence theorems in nonlinear elasticity have been obtained by Giaquinta-Modica-Soucek [1] and Muller [1], [2], [3].

The regularity theory for problems in nonlinear elasticity is still evolving.Some material can be found in the references cited above. In particular, the


question of cavitation of elastic materials has been studied. See for instanceGiaquinta-Modica-Soucek [1].

Convergence Results for Nonlinear Elliptic Equations

We close this section with another simple and useful example of how com-pensated compactness methods may be applied in a nonlinear situation. Thefollowing result is essentially “Murat’s lemma” from Tartar [2, p. 278]:

3.4 Theorem. Suppose um % H1,20 (%) is a sequence of solutions to an elliptic

equation,/um = fm

um = 0in %

on &%

in a smooth and bounded domain % in IRn. Suppose um $ u weakly in H1,20 (%)

while (fm) is bounded in L1(%). Then for a subsequence m $ * we have2um $ 2u in Lq(%) for any q < 2, and 2um $ 2u pointwise almost every-where.

Proof. Choose p > n and let 'm % H1,p0 (%) satisfy

1'm1H1,p0

+ 1"

"(2um ,2u)2'm dx = sup

-"H1,p0 ("), *-*

H1,p0

'1

"

"(2um ,2u)2' dx .

By the Calderon-Zygmund inequality in Lp, see Simader[1], the latter

sup-"H1,p

0 ("), *-*H

1,p0

'1

"

"(2um ,2u)2' dx ' c&11um , u1H1,q

0

where 1p + 1

q = 1.On the other hand, by Sobolev’s embedding H1,p

0 (%) ($ C1&np (%). Hence

by the Arzela-Ascoli theorem we may assume that 'm $ ' weakly in H1,p0 (%)

and uniformly in %. (See Theorem A.5.) Thus"

"(2um ,2u)2'm dx =

"

"(2um ,2u)(2'm ,2') dx + o(1)

= liml$%

"

"(2um ,2ul)(2'm ,2') dx + o(1)

= liml$%

"

"(fm , fl)('m , ') dx + o(1)

+ 2 supl"IN

1fl1L11'm , '1L& + o(1) = o(1),

where o(1) $ 0 as m $ *.


It follows that 2um $ 2u in Lq0(%) for some q0 ' 1. But then, byHolder’s inequality, for any q < 2 we have

12um ,2u1Lq + 12um ,2u1.L112um ,2u11&.L2 $ 0 ,

where 1q = 2 + 1&.

2 .

Results like Theorem 3.4 are needed if one wants to solve nonlinear partialdi!erential equations

(3.3) ,/u = f(x, u,2u)

with quadratic growth|f(x, u, p)| + c(1 + |p|2)

by approximation methods. Assuming some uniform control on approximatesolutions um of (3.3) in H1,2, Theorem 3.4 assures that

f(x, um,2um) $ f(x, u,2u) almost everywhere,

where u is the weak limit of a suitable sequence (um). Given some furtherstructure conditions on f , then there are various ways of passing to the limitm $ * in Equation (3.3); see, for instance, Frehse [2] or Evans [2].

3.5 Example. As a model problem consider the equation

A(u) = ,/u + u|2u|2 = h in %(3.4)u = 0 on &%(3.5)

on a smooth and bounded domain % & IRn, with h % L%(%). This is a specialcase of a problem studied by Bensoussan-Boccardo-Murat [1; Theorem 1.1,p. 350]. Note that the nonlinear term g(u, p) = u|p|2 satisfies the condition

(3.6) g(u, p)u ' 0 .

Approximate g by functions

g$(u, p) =g(u, p)

1 + !|g(u, p)| , ! > 0 ,

satisfying |g$| + 1$ and g$(u, p) · u ' 0 for all u, p.

Now, since g$ is uniformly bounded, the map H1,20 (%) < u 4$ g$(u,2u) %

H&1(%) is compact and bounded for any ! > 0. Denote A$(u) = ,/u +g$(u,2u) the perturbed operator A. By Schauder’s fixed point theorem, see, forinstance, Deimling [1; Theorem 8.8, p. 60], applied to the map u 4$ (,/)&1

%h,

g$(u,2u)&

on a su"ciently large ball in H1,20 (%), there is a solution u$ %

H1,20 (%) of the equation A$u$ = h for any ! > 0. In addition, since g$(u, p)·u '

0 we have


1u$12H1,2

0+ !u$, A$u$" = !u$, h" + 1u$1H1,2

01h1H!1 ,

and (u$) is uniformly bounded in H1,20 (%) for ! > 0. Moreover, since the

nonlinear term g$ satisfies

g$(u, p) =u|p|2

1 + !|u| |p|2 +%1 + |u|2

&|p|2

1 + !|u| |p|2

+ |p|2 + g$(u, p)u ,

we also deduce the uniform L1-bound

1g$(u$,2u$)1L1 + 1u$12H1,2

0+"

"u$g$(u$,2u$) dx

= !u$, A$u$" + c .

We may assume that a sequence (u$m) as !m $ 0 weakly converges in H1,20 (%)

to a limit u % H1,20 (%). By Theorem 3.4, moreover, we may assume that

um = u$m converges strongly in H1,q0 (%) and that um and 2um converge

pointwise almost everywhere. To show that u weakly solves (3.4), (3.5) wenow use the “Fatou lemma technique” of Frehse [2]. As a preliminary step weestablish a uniform L%-bound for the sequence (um).Multiply the approximate equations by um to obtain the di!erential inequality

,/% |um|2

2&+ ,/

% |um|2

2&

+ |2um|2 + umg$m(um,2um)

= hum + C(+) + +|um|2

2,

for any + > 0. Choosing + < *1, the first eigenvalue of ,/ on H1,20 (%), the

weak maximum principle implies a uniform bound for um in L%. (See TheoremB.7 and its application in Appendix B.)

Next, testing the approximate equations A$m(um)=h with ' = - exp(2um),where - % C%

0 (%) is non-negative, upon integrating by parts we obtain"

"

#2|2um|2 + g$m(um,2um)

$- exp(2um) dx

+"

"

%2um2- , h-

&exp(2um) dx = 0 .

Note that on account of the growth condition

|g$m(u, p)| + |u| |p|2

and the uniform bound 1um1L& + C0 derived above, for |2| ' C0 the term

2|2um|2 + g$m(um,2um)


has the same sign as 2. Moreover, this term converges pointwise almost ev-erywhere to the same expression involving u instead of um. Hence, by Fatou’slemma, upon passing to the limit m $ * we obtain

!- exp(2u), A(u), h" + 0

if 2 ' C0, respectively ' 0, if 2 + ,C0. This holds for all non-negative- % C%

0 (%) and hence also for - ' 0 belonging to H1,20 ( L%(%). Setting

- = -0 exp(,2u) we obtain

!-0, A(u) , h" + 0 and ' 0,

for any -0 ' 0, -0 % C%0 (%). Hence u is a weak solution of (3.4), (3.5), as

desired.More sophisticated variants and applications of Theorem 3.4 are given by

Bensoussan-Boccardo-Murat [1], Boccardo-Murat-Puel [1], and Frehse [1], [2].

Hardy Space Methods

The Hardy Hp-spaces play an important role in harmonic analysis. In thetheory of partial di!erential equations, the space H1 is of particular interest.For instance, the Calderon-Zygmund theorem, asserting that for 1 < p < *any function u % Lp with /u % Lp belongs to H2,p

loc , ceases to be valid inthe limit case p = 1. However, the theorem remains true if we substituteL1 by the slightly smaller space H1. While the harmonic analysts’ definitionof H1 is rather unwieldy and hardly lends itself to applications in the theoryof partial di!erential equations, recently an observation of Muller [2], [3] hasled to the discovery of simple criteria for composite functions to belong toH1. In particular, it was shown by Coifman-Lions-Meyer-Semmes [1] that theJacobian of a function u % H1,n(IRn; IRn) belongs to this space, and similarlyfor l # l-sub-determinants of 2u for u % H1,l(IRn; IRN ), for any l, n, N % IN.

As an application, we derive the assertion of Theorem 3.4 in the case thatfor a sequence um $ u weakly in H1,2(IRn; IRN ) the sequence ,/um = fm isbounded in H1. In fact, by the extension of the Calderon-Zygmund theorem toH1, the sequence (22um) is bounded in H1 & L1, and therefore, by Rellich’scompactness result, 2um $ 2u locally in L1 as m $ *. Boundedness of (fm)in H1 by the result of Coifman-Lions-Meyer-Semmes [1] is guaranteed if, forinstance, for each m % IN the function fm is a linear combination of 2# 2 sub-determinants of 2F (um), where F : IRN$ IRL is a smooth, bounded functionwith bounded derivative. For more information about Hardy spaces and theirapplications, see, for instance, Torchinsky [1] or Semmes [1].


4. The Concentration-Compactness Principle

As we have seen in our analysis of the Plateau problem, Section 2.7, a very seri-ous complication for the direct methods to be applicable arises in the presenceof non-compact group actions.

If, in a terminology borrowed from physics which we will try to makemore precise later, the action is a “manifest” symmetry – as in the case of theconformal group of the disk acting on Dirichlet’s integral for minimal surfaces– we may be able to eliminate the action by a suitable normalization. This wasthe reason for introducing the three-point condition on admissible functionsfor the Plateau problem in the proof of Theorem 2.8. However, if the actionis “hidden”, such a normalization is not possible and there is no hope thatall minimizing sequences converge to a minimizer. Even worse, the variationalproblem may not have a solution. For such problems, P.-L. Lions developedhis concentration-compactness principle. On the basis of this principle, formany constrained minimization problems it is possible to state necessary andsu"cient conditions for the convergence of all minimizing sequences satisfyingthe given constraint. These conditions involve a delicate comparison of thegiven functional in variation and a (family of) functionals “at infinity” (onwhich the group action is manifest).

Rather than dwell on abstract notions we prefer to give an example – a variantof problem (2.1), (2.3) – which will bring out the main ideas immediately.

4.1 Example. Let a: IRn $ IR be a continuous function a > 0 and supposethat

a(x) $ a% > 0 as |x| $* .

We look for positive solutions u of the equation

(4.1) ,/u + a(x)u = u|u|p&2 in IRn ,

decaying at infinity, that is

(4.2) u(x) $ 0 as |x| $* .

Here p > 2 may be an arbitrary number, if n = 1, 2. If n ' 3 we suppose thatp < 2n

n&2 . This guarantees that the imbedding

H1,2(%) $ Lp(%)

is compact for any % && IRn.Note that, as in the proof of Theorem 2.1, solutions of Equation (4.1)

correspond to critical points of the functional

E(u) =12

"

IRn

%|2u|2 + a(x)|u|2

&dx

on H1,2(IRn), restricted to the unit sphere

4. The Concentration-Compactness Principle 37

M = {u % H1,2(IRn) ;"

IRn

|u|p dx = 1}

in Lp(IRn). Moreover, if a(x) . a%, E is invariant under translations

u 4$ ux0(x) = u(x , x0) .

In general, for any u % H1,2(IRn), after a substitution of variables

E(ux0) =12

"

IRn

%|2u|2 + a(x + x0)|u|2

&dx $ 1

2

"

IRn

%|2u|2 + a%|u|2

&dx

as |xo| $* , whence it may seem appropriate to call

E%(u) :=12

"

IRn

%|2u|2 + a%|u|2

&dx

the functional at infinity associated with E. The following result is due toLions [2; Theorem I.2].

4.2 Theorem. Suppose

(4.3) I := infM

E < infM

E% =: I% ,

then there exists a positive solution u % H1,2(IRn) of Equation (4.1).Moreover, condition (4.3) is necessary and su"cient for the relative compact-ness of all minimizing sequences for E in M .

Proof. Clearly, (4.3) is necessary for the relative compactness of all minimizingsequences in M . Indeed, suppose I% + I and let (um) be a minimizing sequencefor E%. Then also (um), given by um = um(·+ xm), is a minimizing sequencefor E%, for any sequence (xm) in IRn. Choosing |xm| large enough such that

55E(um) , E%(um)55 + 1

m,

moreover, (um) is a minimizing sequence for E. In addition, we can achievethat

um $ 0 locally in L2 ,

whence (um) cannot be relatively compact.Note that this argument also proves that the weak inequality I + I%

always holds true, regardless of the particular choice of the function a.We now show that condition (4.3) is also su"cient for the relative com-

pactness of minimizing sequences. The existence of a positive solution to (4.1)then follows as in the proof of Theorem 2.1.

Let (um) be a minimizing sequence for E in M with E(um) $ I. We mayassume that um $ u weakly in Lp(IRn) . By continuity, a is uniformly positiveon IRn. Hence we also have


1um12H1,2 + c E(um) + C < * ,

and in addition we may assume that um $ u weakly in H1,2(IRn) and pointwisealmost everywhere. Let um = vm +u. Observe that the family (u|um,8u|p&1)is equi-integrable on IRn, uniformly in 0 + 8 + 1. Hence by Vitali’s theorem

(4.4)

"|um|p dx ,

"|um , u|p dx = ,

"" 1

0

d

d8|um , 8u|p d8 dx

= p

"" 1

0u(um , 8u)|um , 8u|p&2 d8 dx

$ p

"" 1

0u(u , 8u)|u , 8u|p&2 d8 dx =

"|u|p dx ,

where*

... dx denotes integration over IRn; that is,"

IRn

|u|p dx +"

IRn

|vm|p dx $ 1 .

Similarly, we have

(4.5)

E(um) = E(vm + u)

=12

"

IRn

+%|2u|2 + 22u2vm + |2vm|2

&

+ a(x)%|u|2 + 2uvm + |vm|2

&,dx

= E(u) + E(vm) +"

IRn

%2u2vm + a(x)uvm

&dx,

and the last term converges to zero by weak convergence vm = um , u $ 0 inH1,2(IRn).

Moreover, for any ! > 0, letting

%$ = {x % IRn ; |a(x) , a%| ' !} && IRn,

since vm $ 0 locally in L2, the integral"

IRn

(a(x) , a%)|vm|2 dx

+ !

"

IRn

|vm|2 + supIRn

|a(x)|"

""

|vm|2 dx

+ c! + o(1) .

Here and in the following, o(1) denotes error terms such that o(1) $ 0 asm $ *. Hence this integral can be made arbitrarily small if we first choose! > 0 su"ciently small and then let m ' m0(!) be su"ciently large. That is,we have


E(um) = E(u) + E%(vm) + o(1) .

By homogeneity, if we denote * =*IRn |u|p dx,

E(u) = *2/pE#*&1/pu

$' *2/pI, if * > 0 ,

E%(vm) = (1 , *)2/pE%#(1 , *)&1/pvm

$' (1 , *)2/pI% + o(1), if * < 1 .

Hence, for all * % [0, 1], we obtain the estimate

I = E(um) + o(1) = E(u) + E%(vm) + o(1)

' *2/pI + (1 , *)2/pI% + o(1)

'#*2/p + (1 , *)2/p

$I + o(1) .

Since p > 2 this implies that * % {0, 1}. But if * = 0, we obtain that

I ' I% + o(1) > I

for large m; a contradiction.Therefore * = 1; that is, um $ u in Lp, and u % M. By convexity of E,

moreover,E(u) + lim inf

m$%E(um) = I ,

and u minimizes E in M . Hence also E(um) $ E(u). Finally, by (4.5)

1um , u12H1,2 + cE(um , u)

= c (E(um) , E(u)) + o(1) $ 0 ,

and um $ u strongly in H1,2(IRn). The proof is complete.

Regarding |um|p dx as a measure on IRn, a systematic approach to suchproblems is possible via the following lemma (P.L. Lions [1; p. 115 !.]).

4.3 Concentration-Compactness Lemma I . Suppose µm is a sequence of prob-ability measures on IRn: µm ' 0,

*IRn dµm = 1. There is a subsequence (µm)

such that one of the following three conditions holds:(1#) (Compactness) There exists a sequence xm & IRn such that for any ! > 0there is a radius R > 0 with the property that

"

BR(xm)dµm ' 1 , !

for all m.(2#) (Vanishing) For all R > 0 there holds

limm$%

:sup

x"IRn

"

BR(x)dµm

;= 0 .


(3#) (Dichotomy) There exists a number *, 0 < *< 1, such that for any ! > 0there is a number R > 0 and a sequence (xm) with the following property:Given R( > R there are non-negative measures µ1

m, µ2m such that

0 + µ1m + µ2

m + µm ,

supp(µ1m) & BR(xm), supp(µ2

m) & IRn \ BR"(xm) ,

lim supm$%

:5555*,"

IRn

dµ1m

5555+5555(1 , *) ,

"

IRn

dµ2m

5555

;+ ! .

Proof. The proof is based on the notion of concentration function

Q(r) = supx"IRn

-"

Br(x)dµ

.

of a non-negative measure, introduced by Levy [1].Let Qm be the concentration functions associated with µm. Note that

(Qm) is a sequence of non-decreasing, non-negative bounded functions on [0,*[with limR$%Qm(R) = 1. Hence, (Qm) is locally bounded in BV on [0,*[ andthere exists a subsequence (µm) and a bounded, non-negative, non-decreasingfunction Q such that

Qm(R) $ Q(R) (m $ *) ,

for almost every R > 0. We normalize Q to be continuous from the left. SinceQm is non-decreasing, this then also implies that for every R > 0 we have

Q(R) + lim infm$%

Qm(R).

Let* = lim

R$%Q(R) .

Clearly 0 + * + 1. If * = 0, we have “vanishing”, case (2#). Suppose * = 1.Then for some R0 > 0 we have Q(R0) > 1

2 . For any m % IN let xm satisfy

Qm(R0) +"

BR0 (xm)dµm +

1m

.

Now for 0 < ! < 12 fix R such that Q(R) > 1 , ! > 1

2 and let ym satisfy

Qm(R) +"

BR(ym)dµm +

1m

.

Then, with error o(1) $ 0 as m $ *, we have"

BR(ym)dµm +

"

BR0 (xm)dµm ' Qm(R0) + Qm(R) + o(1)

' Q(R0) + Q(R) + o(1)


and the right-hand side is> 1 =

"

IRn

dµm

for su"ciently large m. It follows that for such m

BR(ym) ( BR0(xm) -= 7 .

That is, BR(ym) & B2R+R0(xm) and hence

1 , ! +"

B2R+R0 (xm)dµm

for large m. Choosing R even larger, if necessary, we can achieve that (1#)holds for all m.

If 0 < *< 1, given ! > 0 choose R and a sequence (xm) – depending on !and R – such that

Qm(R) '"

BR(xm)dµm > *, ! ,

if m ' m0(!). Enlarging m0(!), if necessary, we can also find a sequenceRm $ * such that

Qm(R) + Qm(Rm) < * + ! ,

if m ' m0(!). Moreover, given R( > R, we may assume that Rm ' R( for allm. Now let µ1

m = µm BR(xm), the restriction of µm to BR(xm). Similarly,define µ2

m = µm

%IRn \ BRm(xm)

&. Obviously

0 + µ1m + µ2

m + µm ,

and

supp(µ1m) & BR(xm), supp(µ2

m) & IRn \ BRm(xm) & IRn \ BR"(xm) .

Finally, for m ' m0(!) we can estimate5555*,

"

IRn

dµ1m

5555+55551 , *,

"

IRn

dµ2m

5555

=

55555*,"

BR(xm)dµm

55555+

55555

"

BRm (xm)dµm , *

55555 < 2! ,

which concludes the proof.

In the context of Theorem 4.2, Lemma 4.3 may be applied to µm = |um|p dx,m % IN. Dichotomy in this case is made explicit in (4.4). In view of thecompactness of the embedding H1,2(%) ($ Lp(%) on bounded domains % forall p < 2n

n&2 the situation dealt with in Example 4.1 is referred to as the locallycompact case.

Further complications arise in the presence of non-compact symmetrygroups acting locally; for instance, in the case of conformal invariance or in-variance under scaling.


Existence of Extremal Functions for Sobolev Embeddings

A typical example is the case of Sobolev’s embedding on a (possibly unbounded)domain % & IRn.

4.4 Sobolev embeddings. For u % C%0 (%), k ' 1, p ' 1, let

1u1pDk,p =

(

|!|=k

"

"|D!u|p dx ,

and let Dk,p(%) denote the completion of C%0 (%) in this norm. Suppose kp < n.

By Sobolev’s embedding, Dk,p(%) ($ Lq(%) where 1q = 1

p , kn , and there

exists a (maximal) constant S = S(%) = S(%, k, n, p) such that

(4.6) S1u1pLq + 1u1p

Dk,p , for all u % Dk,p(%) .

Using Schwarz-symmetrization (see for instance Polya-Szego [1; Note A.5,p. 189 !.]), for k = 1, best constants and extremal functions (on % = IRn)can be computed classically, see Talenti [1]; the earliest result in this regardseems to be due to Rodemich[1]. But for k > 1 this method can no longer beapplied. Using the concentration-compactness principle, however, the existenceof extremal functions for Sobolev’s embedding can be established in general;see Theorem 4.9 below.

First we note an important property of the embedding (4.6).

4.5 Scale invariance. By invariance of the norms in Dk,p(IRn), respectivelyLq(IRn), under translation and scaling

(4.7) u 4$ uR(x) = R&n/qu(x/R)

the Sobolev constant S is independent of %. Indeed, for any domain %, ex-tending a function u % C%

0 (%) by 0 outside %, we may regard C%0 (%) as a

subset of C%0 (IRn). Similarly, we may regard Dk,p(%) as a subset of Dk,p(IRn).

Hence we have

S(%) = inf+1u1p

Dk,p ; u % Dk,p(%), 1u1Lq = 1,

' S(IRn) .

Conversely, if um % Dk,p(IRn) is a minimizing sequence for S(IRn) with1um1Lq = 1, by density of C%

0 (IRn) in Dk,p(IRn) we may assume thatum % C%

0 (IRn). After translation, moreover, we have 0 % %. Scaling with(4.7), for su"ciently small Rm we can achieve that vm = (um)Rm % C%

0 (%).But by invariance of 1 ·1 Dk,p , 1 · 1Lq under (4.7) there now results

S(%) + lim infm$%

1vm1pDk,p = S(IRn) ,

and S(%) = S(IRn) = S, as claimed.


4.6 The case k = 1. For k = 1 the Sobolev inequality has an underlyinggeometric meaning which allows one to analyze this case completely. Considerfirst the case p = 1. For u % D1,1(IRn), or even u % BV (IRn), q1 = n

n&1 , weclaim

(4.8):"

IRn

|u|q1 dx

;1/q1

+ 1

n1/q191/nn&1

"

IRn

|2u| dx ,

where 9n&1 denotes the (n, 1)-dimensional measure of the unit sphere in IRn.Observe that equality holds if and only if u % BV (IRn) is a scalar multiple ofthe characteristic function of a ball in IRn. This reflects the fact that

K(n) =1

n1/q191/nn&1

= sup"++IRn

%Ln(%)

&1/q1

P (%; IRn)

equals the isoperimetric constant in IRn, and K(n) is achieved if and only if %is a ball in IRn. (The perimeter P (%; IRn) was defined in Theorem 1.4.)

The following proof of (4.8), based on Talenti [2; p. 404], reveals the deeprelation between isoperimetric inequalities and best constants for Sobolev em-beddings more clearly. (See also Cianchi [1; Lemma 1].)

Proof of (4.8). For u % C%0 (IRn), and t ' 0 let

%(t) = {x % IRn ; |u(x)| > t} .

Then|u(x)| =

" %

0)"(t)(x) dt

for almost every x % %, and hence by Minkowsky’s inequality

1u1Lq1 +" %

01)"(t)1Lq1 dt

=" %

0Ln%%(t)

&1/q1 dt

+ K(n)" %

0P%%(t); IRn& dt .

Finally, by the co-area formula (see for instance Federer [1; Theorem 3.2.11] orGiusti [1; Theorem 1.23])

"

IRn

|2u| dx ="

IRn

552|u|55 dx =

" %

0P%%(t); IRn& dt ,

and (4.8) follows.

From (4.8) the general case p ' 1 can be derived by applying Holder’sinequality. Denote q = np

n&p = sq1, where s = np&pn&p ' 1. Then for u % C%

0 (IRn)we can estimate


1u1Lq =DD|u|s

DD1/s

Lq1+%K(n)

&1/s:"

IRn

552|u|s55 dx

;1/s

+%sK(n)

&1/s:"

IRn

|2u| |u|s&1 dx

;1/s

+%sK(n)

&1/s1u11/sD1,p1u11&1/s

Lq

and

(4.9) 1u1Lq + sK(n)1u1D1,p .

(We do not claim that this constant is sharp.)

4.7 Bounded domains. In contrast to the case p = 1, for p > 1 the best constantin inequality (4.9) is never achieved on any domain % di!erent from IRn; inparticular, it is never achieved on a bounded domain. Indeed, if u % D1,p(%)achieves S = S(IRn), a multiple of u weakly solves the equation

(4.10) ,2%|2u|p&22u

&= u|u|q&2 in IRn

and vanishes on IRn \ %, which contradicts the strong maximum principle forEquation (4.10); see, for instance, Tolksdorf [1].

Of course, we suspect invariance under scaling (4.7) to be responsible forthis defect. Note that for any u % Dk,p(IRn) there holds

uR $ 0 weakly in Dk,p(IRn) as R $ 0 ,

while S is invariant under scaling. Hence, relative compactness of minimizingsequences cannot be expected. Observe, moreover, that for u % C%

0 (IRn) thesupport of uR lies in a fixed compact set for all R + 1. That is, we encounter anew type of loss of compactness compared to Example 4.1; we are dealing witha problem which is also locally non-compact.

This is the setting for the second concentration-compactness lemma fromP.L. Lions [3; Lemma I.1]. Denote

(

|!|=k

|D!u|p = |Dku|p ,

for convenience.

4.8 Concentration-Compactness Lemma II. Let k % IN, p ' 1, kp < n, 1q =

1p , k

n . Suppose um $ u weakly in Dk,p(IRn) and µm = |2kum|p dx $ µ ,6m = |um|q dx $ 6 weakly in the sense of measures where µ and 6 are boundednon-negative measures on IRn.Then we have:(1#) There exists some at most countable set J , a family {x(j) ; j % J} ofdistinct points in IRn, and a family {6(j) ; j % J} of positive numbers suchthat


6 = |u|q dx +(

j"J

6(j)+x(j) ,

where +x is the Dirac-mass of mass 1 concentrated at x % IRn.(2#) In addition we have

µ ' |2ku|p dx +(

j"J

µ(j)+x(j)

for some family {µ(j) ; j % J}, µ(j) > 0 satisfying

S#6(j)$p/q

+ µ(j) , for all j % J .

In particular,0

j"J

%6(j)&p/q

< *.

Proof. Let vm = um , u % Dk,p(IRn). Then vm $ 0 weakly in Dk,p, and by(4.4) we have

9m := 6m , |u|q dx =%|um|q , |u|q

&dx

= |um , u|q dx + o(1) = |vm|q dx + o(1) ,

where o(1) $ 0 as m $ *. Also let *m := |2kvm|p dx. We may assume that*m $ *, while 9m $ 9 = 6 , |u|q dx weakly in the sense of measures, where*,9 ' 0.

Choose - % C%0 (IRn). Then

"

IRn

|-|q d9 = limm$%

"

IRn

|-|q d9m = limm$%

"

IRn

|vm -|q dx

+ S&q/p lim infm$%

:"

IRn

|2k(vm-)|p dx

;q/p

= S&q/p lim infm$%

:"

IRn

|-|p|2kvm|p dx

;q/p

= S&q/p

:"

IRn

|-|p d*

;q/p

.

Observe that by Rellich’s theorem any lower order terms like |2l-||2k&lvm| $0 in Lp, as m $ *. That is, there holds

(4.11) S

:"

IRn

|-|q d9

;p/q

+"

IRn

|-|p d*

for all - % C%0 (IRn). Now let {x(j) ; j % J} be the atoms of the measure 9 and

decompose 9 = 90 +0

j"J 6(j)+x(j) , with 90 free of atoms. Since*IRn d9 < *,

J is an at most countable set. Moreover, 90 ' 0. Choosing - such that0 + - + 1, -

%x(j)&

= 1, from (4.11) we see that


* ' S%6(j)&p/q

+x(j) , for all j % J .

Since |2kum|p , |2kvm|p is of lower order than |2kvm|p at points of concen-tration, the latter estimate also holds for µ.

On the other hand, by weak lower semi-continuity we have

µ ' |2ku|p dx .

The latter measure and the measures +x(j) being relatively singular, (2#) follows.Now, for any open set % & IRn such that

*" d* + S, by (4.11) with

- = -k % C%0 (%) converging to the characteristic function of % as k $ *, we

have

(4.12)"

"d9 +

:"

"d9

;p/q

+ S&1

"

"d* + 1 .

That is, 9 is absolutely continuous with respect to * and by the Radon-Nikodym theorem there exists f % L1(IRn;*) such that d9 = f d*, *-almosteverywhere. Moreover, for *-almost every x % IRn we have

f(x) = lim,$0

-*B'(x) d9*

B'(x) d*

.

.

But then by (4.12), if x is not an atom of *,

S f(x)p/q = lim,$0

E

FGS#*

B'(x) d9$p/q

#*B'(x) d*

$p/q

H

IJ + lim,$0

-"

B'(x)d*

. q!pq

= 0 ,

*-almost everywhere. Since * has only countably many atoms and 90 has noatoms this implies that 90 = 0, that is, (1#).

Finally, we can state the following result; see P.L. Lions [3; Theorem I.1]:

4.9 Theorem. Let k % IN, p > 1, kp < n, 1q = 1

p , kn . Suppose (um) is a

minimizing sequence for S in Dk,p = Dk,p(IRn) with 1um1Lq = 1. Then (um)up to translation and dilation is relatively compact in Dk,p.

Proof. Choose xm % IRn, Rm > 0 such that for the rescaled sequence

vm(x) = R&n/qm um

:x , xm

Rm

;

there holds

(4.13) Qm(1) = supx"IRn

"

B1(x)|vm|q dx =

"

B1(0)|vm|q dx =

12

.


Since p > 1 we may assume that vm $ v weakly in Lq(IRn) and weakly inDk,p(IRn). Consider the families of measures

µm = |2kvm|p dx, 6m = |vm|q dx

and apply Lemma 4.3 to the sequence (6m). Vanishing is ruled out by our abovenormalization. If we have dichotomy, let * %]0, 1[ be as in Lemma 4.3.(3#) andfor ! > 0 determine R > 0, a sequence (xm), and measures 61

m, 62m as in that

lemma such that

0 + 61m + 62

m + 6m ,

supp(61m) & BR(xm), supp(62

m) & IRn \ B2R(xm) ,

lim supm$%

'5555

"

IRn

d61m , *

5555+5555

"

IRn

d62m , (1 , *)

5555

)+ ! .

Choosing a sequence !m $ 0 with corresponding Rm > 0 and xm, upon passingto a subsequence (6m) if necessary, we can achieve that

supp(61m) & BRm(xm), supp(62

m) & IRn \ B2Rm(xm)

andlim supm$%

'5555"

IRn

d61m , *

5555+5555"

IRn

d62m , (1 , *)

5555

)= 0 .

Moreover, in view of Lemma 4.3, we may suppose that Rm $ * (m $ *).Choose ' % C%

0 (B2(0)) with 0 + ' + 1 and such that ' . 1 in B1(0). Form % IN let 'm(x) = '

#x&xmRm

$. Since p ' 1 there holds

|2kvm|p ' |2kvm|p('pm + (1 , 'm)p).

By Minkowski’s inequality we have

1(2kvm)'m1Lp(IRn) ' 12k(vm'm)1Lp(IRn) , C(

l<k

12lvm2k&l'm1Lp(IRn),

and similarly for (1 , 'm) instead of 'm. Let Am denote the annulus Am =B2Rm(xm) \ BRm(xm). By Young’s inequality then for any + > 0 we obtain

12kvm1pLp(IRn) ' 12k(vm'm)1p

Lp(IRn) + 12k(vm(1 , 'm))1pLp(IRn) , "m,

where the error terms "m can be estimated

"m + +12kvm1pLp(IRn) + C(+)

(

l<k

12lvm2k&l'm1pLp(Am) .

Since |2k&l'm| + CRl&km , by interpolation (as in Adams [1; Theorem 4.14])

we can bound

(4.14)

DD|2lvm| |2k&l'm|DD

Lp(Am)+ C Rl&k

m 12lvm1Lp(Am)

+ C K 212kvm1Lp(Am) + C K R&km 2& l

k!l 1vm1Lp(Am) .


Here 2 can be chosen arbitrarily in ]0, 1], while the constant K depends onlyon k and n. (Note that estimate (4.14) is invariant under dilation.) Moreover,by Holder’s inequality

R&km 1vm1Lp(Am) + R&k

m

%Ln(Am)

& 1p& 1

q 1vm1Lq(Am) + C1vm1Lq(Am)

+ C

B"

IRn

d6m ,:"

IRn

d61m +

"

IRn

d62m

;C 1q

.

Hence this term tends to 0 as m $ *, while 12kvm1pLp(Am) + 1vm1p

Dk,p

remains uniformly bounded. Choosing a suitable sequence 2 = 2m $ 0, from(4.14) we thus obtain that "m + o(1), where o(1) $ 0 (m $ *). Also letting+ = +m $ 0 suitably, by Sobolev’s inequality we find

1vm1pDk,p ' 1vm'm1p

Dk,p + 1vm(1 , 'm)1pDk,p + o(1)

' S (1vm'm1pLq + 1vm(1 , 'm)1p

Lq) + o(1)

' S

K

L-"

BRm (xm)d6m

.p/q

+

-"

IRn\B2Rm (xm)d6m

.p/qM

N+ o(1)

' S

O:"

IRn

d61m

;p/q

+:"

IRn

d62m

;p/qP

+ o(1)

' S#*p/q + (1 , *)p/q

$, o(1) ,

where o(1) $ 0 (m $ *). But for 0 < *< 1 and p < q we have *p/q + (1 ,*)p/q > 1, contradicting the initial assumption that 1vm1p

Dk,p = 1um1pDk,p $ S.

There remains the case * = 1, that is, case (1#) of Lemma 4.3.Let xm be as in that lemma and for ! > 0 choose R = R(!) such that

"

BR(xm)d6m ' 1 , ! .

If ! < 12 our normalization condition (4.13) implies BR(xm)(B1(0) -= 7. Hence

the conclusion of Lemma 4.3.(1#) also holds with xm = 0, replacing R(!) by2R(!) + 1 if necessary. Thus, if 6m $ 6 weakly, it follows that

"

IRn

d6 = 1 .

By Lemma 4.8 we may assume that

µm $ µ ' |2kv|p dx +(

j"Jµ(j)+x(j)

6m $ 6 = |v|q dx +(

j"J6(j)+x(j)


for certain points x(j) % IRn, j % J , and positive numbers µ(j), 6(j) satisfying

S%6(j)&p/q + µ(j) , for all j % J .

By Sobolev’s inequality then

S + o(1) = 1vm1pDk,p =

"

IRn

dµm ' 1v1pDk,p +

(

j"Jµ(j) + o(1)

' S

E

G1v1p/qLq +

(

j"J

%6(j)&p/q

H

J+ o(1)

where o(1) $ 1 (m $ *). By strict concavity of the map * $ *p/q now thelatter will be

(4.15)' S

E

G1v1qLq +

(

j"J6(j)

H

Jp/q

+ o(1)

= S

:"

IRn

d6

;p/q

+ o(1) = S + o(1)

and equality holds if and only if at most one of the terms 1v1Lq , 6(j), j % J ,is di!erent from 0.

Note that our normalization (4.13) assures that

6(j) + 12

for all j % J .

Hence all 6(j) must vanish, 1v1Lq = 1, and vm $ v strongly in Lq(IRn). Butthen by Sobolev’s inequality 1v1p

Dk,p ' S and 1vm1Dk,p $ 1v1Dk,p as m $ *.It follows that vm $ v in Dk,p(IRn), as desired. The proof is complete.

As a consequence we obtain

4.10 Corollary. For any k % IN, any p > 1 such that kp < n there existsa function u % Dk,p(IRn) with 1u1Lq(IRn) = 1 and 1u1Dk,p(IRn) = S, where1q = 1

p , kn and where S = S(k, p, n) is the Sobolev constant.

Observe that, since Lemma 4.8 requires weak convergence um $ u in Dk,p(IRn)the above proof of Theorem 4.9 cannot be extended to the case p = 1. In fact,Corollary 4.10 is false in that case, as we have seen that the best constantfor Sobolev’s embedding D1,1(IRn) ($ L

nn!1 (IRn) is attained (precisely) on

characteristic functions of balls; that is, in BVloc(IRn).


4.11 Notes. (1#) The limiting case kp = n for Sobolev’s embedding behavesstrikingly di!erent from the case kp < n studied above. Consider k = 1 forsimplicity. By Sobolev’s embedding W k, n

k ($ W 1,n the results for k = 1 extendto the general case. However, for k > 1 the results that follow can be slightlyimproved, see Brezis-Waigner [1] or Ziemer [1].

By results of Trudinger [1] and Moser [3], for any % && IRn the spaceW 1,n

0 (%) embeds into the Orlicz space of functions u:% $ IR such thatexp%|u|

nn!1&% Lp(%) for any p < *, and there exists a limiting exponent

#n > 0 such that the map

(4.16) W 1,n0 (%) < u 4$ exp

%|u|

nn!1&% Lp(%)

is bounded on the unit ball

B1

%0; W 1,n

0 (%)&

='

u % W 1,n0 (%) ;

"

"|2u|n dx + 1

)

if and only if p + #n.Quite surprisingly, and in sharp contrast to the case kp < n considered

previously, Carleson-Chang [1] were able to establish that

supu"B1(0;W

1,n0 ("))

"

"exp%#n|u|

nn!1&dx

is attained if % is a ball. Struwe [17] then showed that also for domains % thatare close to a ball in measure the supremum is attained. Finally, Flucher [1]established the existence of an extremal function on any domain, if n = 2; seealso Bandle-Flucher [1].

Moreover, at least in the case that % is a 2-ball (Struwe [17]), even forsu"ciently small numbers p > #2 = 41 the functional

Ep(u) ="

"exp(p|u|2) dx

admits a relative maximizer in B1

%0; H1,2

0 (%)&. Thus, and since Ep(u) by the

result of Moser is unbounded on B1

%0; H1,2

0 (%)&, we are led to expect the exis-

tence of a further critical point of saddle type, for any p > 41 su"ciently closeto 41. In Struwe [17] such “unstable” critical points were, in fact, shown toexist for almost all such numbers p > 41 by minimax methods as we shall de-scribe in Chapter II. Note that for p > 41 we are dealing with a “supercritical”variational problem and quite delicate techniques are needed to overcome thepossible “loss of compactness”. It is an open problem whether a similar resultholds for all p > 41 su"ciently close to 41 and for any domain; moreover, theextension of this result to higher dimensions is open.(2#) Problems with exponential nonlinearities related to the embedding (4.16)above are studied, for instance, by Adimurthi-Yadava [1]. The embedding(4.16) is also relevant for the study of nonlinearities like

5. Ekeland’s Variational Principle 51

g(u) = e2u ,

arising in the 2-dimensional Kazdan-Warner problem (see Moser [5], Chang-Yang [1]) or in the uniformization problem, that is, the Yamabe problem forsurfaces; compare Section III.4.

5. Ekeland’s Variational Principle

In general it is not clear that a bounded and lower semi-continuous functionalE actually attains its infimum. The analytic function f(x) = arctanx, forexample, neither attains its infimum nor its supremum on the real line.

A variant due to Ekeland [1] of Dirichlet’s principle, however, permits oneto construct minimizing sequences for such functionals E whose elements um

each minimize a functional Em, for a sequence of functionals Em converginglocally uniformly to E.

5.1 Theorem. Let M be a complete metric space with metric d, and let E: M $IR ) +* be lower semi-continuous, bounded from below, and -. *.Then for any !, + > 0, any u % M with

E(u) + infM

E + !,

there is an element v % M strictly minimizing the functional

Ev(w) . E(w) +!

+d(v, w) .

Moreover, we haveE(v) + E(u), d(u, v) + + .

Fig. 5.1. Comparing E with Ev . v is a strict absolute minimizer of Ev if and only if the

downward cone of slope !/" with vertex at%v, E(v)

&lies entirely below the graph of E


Proof. Denote # = $# and define a partial ordering on M # IR by letting

(5.1) (v, ") + (v(, "() = ("( , ") + # d(v, v() + 0 .

This relation is easily seen to be reflexive, identitive, and transitive:

(v, ") + (v, ") ,

(v, ") + (v(, "() : (v(, "() + (v, ") = v = v(, " = "(,

(v, ") + (v(, "() : (v(, "() + (v((, "(() > (v, ") + (v((, "(() .

Moreover, if we denote

S = {(v, ") % M # IR ; E(v) + "} ,

by lower semi-continuity of E, S is closed in M # IR. To complete the proofwe need a lemma.

5.2 Lemma. S contains a maximal element (v, ") with respect to the partialordering + on M # IR such that (u, E(u)) + (v, ").

Proof. Let (v1, "1) = (u, E(u)) and define a sequence (vm, "m) inductively asfollows: Given (vm, "m), define

Sm = {(v, ") % S ; (vm, "m) + (v, ")}µm = inf{" ; (v, ") % Sm}

' inf{E(v) ; (v, ") % Sm} ' infM

E =: µ0 .

Note that µm + "m; moreover, Sm = {(vm, "m)}, if µm = "m. Now let(vm+1, "m+1) % Sm be chosen such that

(5.2) "m , "m+1 ' 12("m , µm) .

Note that by transitivity of + the sequence Sm is nested: S1 0 S2 0 . . . 0Sm 0 Sm+1 0 . . . . Hence also . . . + µm + µm+1 + . . . + "m+1 + "m + . . ..By induction, from (5.2) we obtain

"m+1 , µm+1 + "m+1 , µm

+ 12("m , µm) + ... +

:12

;m

("1 , µ1) .

Therefore, by definition of Sm, for any m % IN and any (v, ") % Sm we have

(5.3)|"m , "| = "m , " + "m , µm + C

:12

;m

,

d(vm, v) + #&1("m , ") + C#&1

:12

;m

.


In particular,%(vm, "m)

&m"IN

is a Cauchy sequence in M # IR. Thus, by com-pleteness of M ,

%(vm, "m)

&converges to some limit (v, ") %

!m"IN Sm. By

transitivity, clearly (u, E(u)) = (v1, "1) + (v, "). Moreover, (v, ") is maximal.In fact, if (v, ") + (v, ") for some (v, ") % M # IR, then also (vm, "m) + (v, ")for all m, and (v, ") % Sm for all m. Letting (v, ") = (v, ") in (5.3) we inferthat (vm, "m) $ (v, ") (m $ *), whence (v, ") = (v, "), as desired.

Proof of Theorem 5.1 (Completed). Let (v, ") be maximal in S with (u, E(u)) +(v, "). Comparing with (v, E(v)) % S at once yields " = E(v). By definition(5.1) the statement (u, E(u)) + (v, E(v)) translates into the estimate

E(v) , E(u) + # d(u, v) + 0 ;

in particular this impliesE(v) + E(u)

andd(u, v) + #&1 (E(u) , E(v)) + +

!

#infM

E + !, infM

E$

= + .

Finally, if w % M satisfies

Ev(w) = E(w) + # d(v, w) + E(v) = Ev(v) ,

by Definition 5.1 we have (v, E(v)) + (w, E(w)) . Hence w = v by maximalityof (v, E(v)); that is, v is a strict minimizer of Ev, as claimed.

5.3 Corollary. If V is a Banach space and E % C1(V ) is bounded from below,there exists a minimizing sequence (vm) for E in V such that

E(vm) $ infV

E, DE(vm) $ 0 in V ! as m $ * .

Proof. Choose a sequence (!m) of numbers !m > 0, !m $ 0 (m $ *). Form % IN choose um % V such that

E(um) + infV

E + !2m .

For ! = !2m, + = !m, u = um determine an element vm = v according to

Theorem 5.1, satisfying

E(vm) + E(vm + w) + !m 1w1V

for all w % V . Hence

1DE(vm)1V ' = lim sup0 ,=*w*V $0

E(vm) , E(vm + w)1w1V

+ !m $ 0 ,

as claimed.


In Chapter II we will re-encounter the special minimizing sequences of Corollary5.3 as “Palais-Smale sequences”. Compactness of such sequences by Corollary5.3 turns out to be a su"cient condition for the existence of a minimizer for adi!erentiable functional E which is bounded from below on a Banach space V .Moreover, we shall see that the compactness of Palais-Smale sequences (undersuitable assumptions on the topology of the level sets of E) will also guaranteethe existence of critical points of saddle type.

However, before turning our attention to critical points of general type wesketch another application of Ekeland’s variational principle.

Existence of Minimizers for Quasi-convex Functionals

Theorem 5.1 may be used to construct minimizing sequences for variationalintegrals enjoying better smoothness properties than can a-priori be expected.We present an example due to Marcellini-Sbordone [1].

5.4 Example. Let % be a bounded domain in IRn and let f :%#IRN#IRnN $ IRbe a Caratheodory function satisfying the growth and coercivity conditions

(5.4) |f(x, u, p)| + C%1 + |u|s + |p|s

&for some s > 1 , C % IR

and

(5.5) f(x, u, p) ' |p|s.

Moreover, suppose f is quasi-convex in the sense of Morrey [3]; that is, foralmost every x0 % %, u0 % IRN , po % IRnN , and any ' % H1,s

0 (%) there holds

(5.6)1

Ln(%)

"

"f%x0, u0, p0 + D'(x)

&dx ' f(x0, u0, p0).

Note that by Jensen’s inequality condition (5.6) is weaker than requiring f tobe convex in p. For u % H1,s(%; IRN ) now set

E(u) = E(u;%) ="

"f%x, u(x),2u(x)

&dx .

By a result of Fusco [1], conditions (5.4) and (5.6) alone already su"ce toensure weak lower semi-continuity of E in H1,t(%; IRN ) for any t > s; on theother hand, an example by Murat and Tartar shows that this is no longer truefor t = s; see for instance Marcellini-Sbordone [1; Section 2].

But if, in addition to (5.4) and (5.6), we also assume (5.5), Marcellini-Sbordone [1] succeed in finding a minimizing sequence for E for given boundarydata u0 which is locally bounded in H1,t for some t > s and therefore weaklyaccumulates at a minimizer of E. Their proof, which we shall presently explain,is based on Ekeland’s variational principle and Giaquinta-Modica’s [1] adaptionof a result of Gehring [1], Lemma 5.6 below.


Finally, it was shown by Acerbi-Fusco [1] that E, in fact, for any s ' 1 isweakly lower semi-continuous in H1,s(%; IRN ) whenever f only satisfies (5.4),(5.6) and the condition f(x, u, p) ' 0.

Regularity results for minimizers of (strictly) quasi-convex functionals havebeen obtained by Evans [1], Evans-Gariepy [1], and Giaquinta-Modica [2].

5.5 Theorem. Under the above hypotheses (5.4)–(5.6) on f , for any u0 %H1,s(%; IRN ) there is a minimizing sequence (um) for E on {u0}+H1,s

0 (%; IRN )which is locally bounded in H1,t for some t > s.

Proof. ChooseM = {u0} + H1,1

0 (%; IRN )

with metric d derived from the H1,10 -norm

d(u, v) ="

"|2u ,2v| dx .

Note that by Fatou’s lemma E: M $ IR ) +* is lower semi-continuous withrespect to d. Let um % {u0} + H1,s

0 (%; IRN ) & M be a minimizing sequence.By Theorem 5.1, if we let

!2m = E(um) , inf{E(u) ; u % M} ,

+m = !m, we can choose a new minimizing sequence (vm) in M such that eachvm minimizes the functional

Em(w) = E(w) + !m d(vm, w) .

In particular, for each %( && % and any w % {vm}+H1,s0 (%(; IRN ) there holds

(5.7) E(vm;%() + E(w;%() + !m

"

""|2w ,2vm| dx .

Choose x0 % % and for R < 12 dist(x0, &%), N % IN, 1 + 6 + N , choose

' = '/ % C%0

%B(1+//N)R(x0)

&& C%

0

%B2R(x0)

&satisfying 0 + ' + 1, ' . 1

on B(1+(/&1)/N)R(x0), |2'| + CN/R with C independent of N and R. Let

vm =1

Ln(B2R \ BR(x0))

"

B2R\BR(x0)vm dx

denote the mean value of u over the annulus B2R \ BR(x0). Define

w = wm,/ = (1 , ')vm + 'vm .

Then w = vm outside B(1+//N)R(x0) and by (5.5) and (5.7) we have

(5.8)

"

BR(x0)|2vm|s dx + E(vm; B(1+//N)R(x0)) + E

%w; B(1+//N)R(x0)

&

+ !m

#"

B2R(x0)'|2vm| dx +

"

B2R\BR(x0)|2'| |vm , vm| dx

$.


By Poincare’s inequality"

B2R\BR(x0)|vm , vm| dx + cR

"

B2R\BR(x0)|2vm| dx;

see Theorem A.10 of Appendix A. Hence, in particular, the last term in theprevious inequality is bounded by CN!m

*B2R(x0)

|2vm| dx.By choice of w and condition (5.4), moreover, we may estimate

(5.9)

E%w; B(1+//N)R(x0)

&+ C

"

B(1+(/N)R(x0)

%1 + |w|s + |2w|s

&dx

+ C

"

B(1+(/N)R(x0)

%1 + |vm|s + 's|vm , vm|s

&dx

+ C

"

B(1+(/N)R(x0)

%(1 , ')s|2vm|s + |2'|s|vm , vm|s

&dx

+ C

"

B2R(x0)

%1 + |vm|s + |2'|s|vm , vm|s

&dx

+ C

"

B(1+(/N)R\B(1+((!1)/N)R(x0)|2vm|s dx .

Note that we also used Holder’s inequality to estimate"

B2R(x0)|vm|s dx + CRn(1&s)

-"

B2R(x0)vm dx

.s

+ C

"

B2R(x0)|vm|s dx .

For u % L1loc(%), BR = BR(x0) & % let

—"

BR

u dx =1

Ln(BR)

"

BR

u dx

denote the mean value of u over BR, etc. By the Poincare-Sobolev inequality,interpolation, and Young’s inequality, for any + > 0 we can bound

—"

B2R(x0)|2'|s|vm , vm|s dx + CNsR&s —

"

B2R\BR(x0)|vm , vm|s dx

+ + —"

B2R(x0)|2vm|s dx + C(+, N)

-

—"

B2R(x0)|2vm| dx

.s

.

Summing the estimates (5.8) over 1 + 6 + N , we fill the annulus B2R \BR(x0)on the right of (5.9), and with a uniform constant C0 we obtain

N —"

BR(x0)|2vm|s dx + C(N) —

"

B2R(x0)

%1 + |vm|s + |2vm|

&dx

+ C(+, N)

-—"

B2R(x0)|2vm| dx

.s

+ C0(1 + +N) —"

B2R

|2vm|s dx .


Choosing N ' 2C0 + 1, + = N&1 > 0, upon dividing by N we find that

—"

BR(x0)|2vm|s dx + C —

"

B2R(x0)

%1 + |vm|s + |2vm|

&dx

+ C

-—"

B2R(x0)|2vm| dx

.s

+ : —"

B2R(x0)|2vm|s dx

with constants C and : = 2C02C0+1 < 1 independent of x0, R, and m. (This idea

in a related context first appears in Widman [1], Hildebrandt-Widman [1].)Letting g = |2vm|, h =

%1 + |vm|s + |2vm|

&1/s, p = s · min+

nn&s , s

,> s

and observing that by Sobolev’s embedding theorem |vm|s % Lp, Theorem 5.5now follows from the next lemma, due to Giaquinta-Modica [1].

5.6 Lemma. Suppose % is a domain in IRn and 0 + g % Ls(%), h % Lp(%)for some p > s > 1. Assume that for any x0 % % and 0 < R < 1

2 dist(x0, &%)there holds

—"

BR(x0)gs dx + b —

"

B2R(x0)hs dx + c

-—"

B2R(x0)g dx

.s

+ : —"

B2R(x0)gs dx

with uniform constants : < 1, b, and c independent of x0 and R. Then thereexists ! > 0, depending only on b, c, :, p, s, and n, such that g % Lt

loc(%) fors < t < s + ! + p, and for any x0 % %, 0 < R < 1

2 dist(x, &%), and any such tthere holds-

—"

BR(x0)gt dx

. 1t

+ C

<=

>

-—"

BR(x0)ht dx

. 1t

+

-—"

B2R(x0)gs dx

. 1s

?@

A ,

with C possibly depending also on t.

The proof of Lemma 5.6 goes beyond the scope of this book; a reference isGiaquinta [1; Proposition V.1.1, p. 122].

5.7 Note. Besides various other applications, Ekeland’s variational principlehas given rise to new interpretations of the mountain-pass lemma and its vari-ants that we discuss in Section II.6; see, for instance, De Figueiredo [1] orMawhin-Willem [1; Chapter 4.1] for an exposition.

The idea of choosing special minimizing sequences to ensure convergencetowards a minimizer already appears in the work of Hilbert [1] and Lebesgue[1]. In their solution of Dirichlet’s problem they use barriers and the “Courant-Lebesgue lemma”, that also inspired our proof of Lemma 2.8 above, to ensurethe equicontinuity and hence compactness of a suitably constructed minimizingsequence for Dirichlet’s integral. (The compactness criterion for families ofcontinuous functions on a compact domain was known from an earlier – thoughunsuccessful – attempt at solving Dirichlet’s problem by Arzela [1] in 1897.)


6. Duality

Let V be a Banach space and suppose G: V $ IR)+* is lower semi-continuousand convex. Geometrically, convexity of G is equivalent to convexity of theepigraph of G

epi(G) =+(v, ") % V # IR ; " ' G(v)

,,

while lower semi-continuity is equivalent to the closedness of epi(G). By theHahn-Banach separation theorem any closed convex set can be represented asthe intersection of the closed half-spaces which contain it, bounded by supporthyperplanes. Hence, for any lower semi-continuous, convex G: V $ IR ) +*there exists a set LG of continuous a"ne maps such that for any v % V thereholds

G(v) = sup {l(v) ; l % LG} ,

see, for instance, Ekeland-Temam [1; Proposition I.3.1]. Moreover, at any pointv % V where G is locally bounded there is a “support function” lv % LG suchthat lv(v) = G(v). The set of slopes of support functions

&G(v) = {Dl ; l % LG , l(v) = G(v)}

is called the subdi!erential of G at v.

Fig. 6.1.

6.1 Lemma. Suppose G: V $ IR ) +* is lower semi-continuous and convex.If G is (Gateaux) di!erentiable at v, then &G(v) = {DG(v)}. Conversely, ifG is locally bounded near v and if &G(v) = {v!} is single-valued, then G isGateaux-di!erentiable at v with DwG(v) = !w, v!" for all w % V .

(See, for instance, Ekeland-Temam [1; Proposition I.5.3] for a proof.)

6. Duality 59

6.2 The Legendre-Fenchel transform. For a function G: V $ IR ) +*, G -.+*, not necessarily convex, the function G!: V ! $ IR ) +*, given by

G!(v!) = sup {!v, v!" , G(v) ; v % V } ,

defines the Legendre-Fenchel transform of G.Note that G! is the pointwise supremum of a"ne maps, hence G! is lower

semi-continous and convex; moreover, the a"ne function l!v, given by

l!(v!) = !v, v!" , G(v) ,

is a support function of G! at v! if and only if

(6.1) G!(v!) + G(v) = !v, v!" .

For lower semi-continuous convex functions G the situation is, in fact, symmet-ric in G and G!. To see this, introduce G!! = (G!)!V . Note that the Legendre-Fenchel transform reverses order, that is, G + G implies G! ' G!. Hence,applying the transform twice preserves order: G + G implies G!! + G!!.Moreover, for a"ne functions l!! = l. Thus, for any a"ne map l + G we have

l(v) = l!!(v) + G!!(v) = supv'"V '

'!v, v!" , sup

w"V

+!w, v!" , G(w)

,)+ G(v) .

(Choose w = v inside {. . .}.)It follows that G!! is the largest lower semi-continuous convex function

below G, and G!! = G if and only if G is lower semi-continuous and convex.Our previous discussion implies:

6.3 Lemma. Suppose G: V $ IR)+*, G -. +*, is lower semi-continuous andconvex, and let G! be its Legendre-Fenchel transform. Then (6.1) is equivalentto either one of the relations v % &G!(v!) or v! % &G(v).

Proof. In fact, if v % &G!(v!), there is " % IR such that l!, given by

l!(w!) = !v, w!" , ",

belongs to LG' and satisfies l!(v!) = G!(v!).Since, by definition,

G!(v!) ' !v, v!" , G(v),

we conclude that G(v) ' ". On the other hand, since G = G!! + (l!)!, wehave

G(v) + supw'

'!v, w!" , l!(w!)

)= ",

and it follows that G(v) = "; that is, we obtain (6.1). The same reasoning showsthat the assumption v! % &G(v) implies (6.1). The converse is immediate fromthe definitions of G! and G = G!!.


In particular, if G % C1(V ) is strictly convex, which implies that

!v , w, DG(v) , DG(w)" > 0 if v -= w ,

then DG is injective, G! is finite on the range of DG, and &G!%DG(v)&

= {v}for any v % V .

If, in addition, DG is strongly monotone and coercive in the sense that forall v, w % V there holds

!v , w, DG(v) , DG(w)" ' #%1v , w1

&1v , w1

with a non-decreasing function #: [0,*[$ [0,*[ vanishing only at 0 and suchthat #(r) $ * as r $ *, then DG: V $ V ! is also surjective; see, for instance,Brezis [1; Corollary 2.4, p. 31]. Moreover, &G! – and hence G! – are locallybounded near any v! % V !. Thus, by Lemma 6.1, G! is Gateaux di!erentiablewith DG!(v!) = v for any v! = DG(v). Finally, from the estimate

1v! , w!1 ' !DG!(v!) , DG!(w!), v! , w!"1DG!(v!) , DG!(w!)1

' #%1DG!(v!) , DG!(w!)1

&

for all v!, w! % V ! it follows that DG! is continuous; that is, G! % C1(V !).We conclude that for any strictly convex function G % C1(V ) such that DGis strongly monotone, the di!erential DG is a homeomorphism of V onto itsdual V !. (In the following we apply these results only in a finite-dimensionalsetting. A convenient reference in this case is Rockafellar [1; Theorem 26.5,p. 258].)

Hamiltonian Systems

We now apply these concepts to the solution of Hamiltonian systems. For aHamiltonian H % C2(IR2n) and the standard symplectic form J on IR2n =IRn # IRn,

J =:

0 ,idid 0

;,

where id is the identity map on IRn, we consider the ordinary di!erential equa-tion

(6.2) x = J 2H(x) .

Note that by anti-symmetry of J we have

d

dtH%x(t)&

= 2H(x) · x(t) = 0 ,

that is, H%x(t)&. const. along any solution of (6.2) and any “energy-surface”

H = const. is invariant under the flow (6.2).

6. Duality 61

6.4 Periodic solutions. One would like to understand the global structure ofthe set of trajectories of (6.2) and their asymptotic behavior. This is motivatedof course by celestial mechanics, where questions of “stable and ramdon mo-tion” (see Moser [4]) also seem to be of practical importance. However, withexception of the – very particular – “completely integrable” case, this programis far too complex to be dealt with as a whole. Therefore, one is interested insub-systems of the flow (6.2) such as stationary points, periodic orbits, invari-ant tori, or quasi-periodic solutions. While stationary points in general do notreveal too much about the system, it turns out that – C1-generically at least– periodic orbits of (6.2) are dense on a compact energy surface H = const.;see Pugh-Robinson [1]. Such a result seems to have already been envisioned byPoincare [1; Tome 1, Article 36]. For particular systems, however, such resultsare much harder to obtain. In fact, the question whether any given energysurface carries a periodic solution of (6.2) has only recently been answered; seeSection II.9 for more details.

In this section we consider the special class of convex Hamiltonians. Thisclass includes as a simple model case the harmonic oscillator, described by theHamiltonian

H(p, q) =12%|p|2 + |q|2

&, x = (p, q) % IR2 ,

for which (6.2) possesses periodic solutions x of any given energy H(x(t)) =" > 0, all having the same period 21. In the general case, the followingresult of Rabinowitz [5] and Weinstein [2], extending earlier work of Seifert [1],holds.

6.5 Theorem. Suppose H % C1(IR2n) is strictly convex, non-negative andcoercive with H(0) = 0. Then for any # > 0 there is a periodic solutionx % C1

%IR ; IR2n& of (6.2) with H(x(t)) = # for all t. The period T is not

specified.

Remarks. Seifert and Weinstein essentially approached problem (6.2) usingdi!erential geometric methods, that is, by interpreting solutions of (6.2) asgeodesics in a suitable Riemannian or Finsler metric (the so-called Jacobi met-ric). Rabinowitz’ proof of Theorem 6.5 revolutionized the study of Hamiltoniansystems in the large as it introduced variational methods to this field.

Note that (6.2) may be interpreted as the Euler-Lagrange equation of thefunctional

(6.3) E(x) =12

" 1

0!x,J x" dt

on the class

C! = {x % C1(IR; IR2n) ; x(t + 1) = x(t)," 1

0H(x(t)) dt = #}.


Indeed, at a critical point x % C! of E, by the Lagrange multiplier rule thereexists T -= 0 such that

x = TJ2H(x) ,

and scaling time by a factor T we obtain a T -periodic solution of (6.2) on theenergy surface H = #.

However, the integral (6.3) is not bounded from above or below. In fact,E is a quadratic form given by an operator x 4$ J x with infinitely many posi-tive and negative eigenvalues. Owing to this complication, actually, for a longtime it was considered hopeless to approach the existence problem for periodicsolutions of (6.2) via the functional (6.3). Surprisingly, by methods that willbe presented in Chapter II below, and by using a delicate approximation pro-cedure, Rabinowitz was able to overcome these di"culties. In fact, his resultis somewhat more general than stated above as it applies to compact, strictlystar-shaped energy hypersurfaces.

On a compact, convex energy hypersurface – as was first observed byClarke [2], [4] – by duality methods his original proof can be considerablysimplified and the problem of finding a periodic solution of (6.2) can be recastin a way such that a solution again may be sought as a minimizer of a suitable“dual” variational problem. See also Clarke-Ekeland [1]. This is the proof wenow present.

Later we shall study the existence of periodic solutions of Hamiltoniansystems under much more general hypotheses; see Section II.8. Moreover, weshall study the existence of multiple periodic orbits; see Section II.5.

Proof of Theorem 6.5. In a first step we reformulate the problem in a way suchthat duality methods can be applied.

Note that by strict convexity and coerciveness of H the level surface S! =H&1({#}) bounds a strictly convex neighborhood of 0 in IR2n. Thus, for any- in the unit sphere S2n&1 & IR2n there exists a unique number r(-) > 0 suchthat x = r(-)- % S!. By the implicit function theorem r % C1(S2n&1). ReplaceH by the function

H(x) ='#r%x/|x|

&&q|x|q, if x -= 00, if x = 0

where q is a fixed number 1 < q < 2.Note that H % C1(IR2n) and is homogeneous of degree q on half-rays from

the origin. Moreover, if we let S! = {x % IR2n ; H(x) = #} we have S! = S!and hence 2H(x) is proportional to 2H(x). In fact, since H increases indirection 2H(x) at any point x % X!, there exists a function * > 0 such that

2H(x) = *(x)2H(x) , at any x % S! .

A periodic solution x on S! to (6.2) for H after a change of parameter thuswill yield a periodic solution

6. Duality 63

x(t) = x%s(t)&

to (6.2) on S! for the original function H, where s solves

s = *%x(s)

&.

This, incidentally, also shows that whether or not a level surface H = const.carries a periodic solution of (6.2) is a question concerning the surface and thesymplectic structure J – not the particular Hamiltonian H.

Finally, H is strictly convex. Indeed, consider any point x = 4- % IR2n\{0}with H(x) = " > 0. Note that

S% = {x % IR2n ; H(x) = "} =:"

#

;1/q

S! .

Thus, the hyperplane through (x, ") % IR2n+1, parallel to the hyperplanespanned by

TxS% ?= Tr()))S! & IR2n # {0} & IR2n+1

together with the vector

(x, x ·2H(x)) = (x, qH(x)) = (x, q") % IR2n+1 ,

is a support hyperplane for epi(H) which touches the graph of H precisely at(x, "). Similarly, since H ' 0 = H(0), the hyperplane IR2n # {0} is a supporthyperplane at (0, 0), and H is strictly convex.

Hence in the following we may assume that H = H.Let H! be the Legendre-Fenchel transform of H. Note that, since H is

homogeneous on rays of degree q > 1, the function H! is everywhere finite.Moreover, H!(0) = 0, H! ' 0. Also note that for a function H on IR2n whichis homogeneous of degree q > 1, strict convexity implies strong monotonicityof the gradient in the sense that, with a continuous function a = a(4) ' 0,vanishing only at 4 = 0, there holds

!x , y,2H(x),2H(y)" ' a

:|x , y||x| + |y|

;%|x| + |y|

&q&1|x , y|.

Hence, arguing as in the discussion following Lemma 6.3, we find that H! %C1(IR2n). Finally, letting p = q

q&1 > 2 be the conjugate exponent of q, we have

(6.4)

H!(y)|y|p = sup

'Qx

|y|p&1,

y

|y|

R, H(x)

|y|p ; x % IR2n

)

= sup'Q

x

|y|p&1,

y

|y|

R, H

:x

|y|p&1

;; x % IR2n

)

= H!:

y

|y|

;;

that is, H! is homogeneous on rays of degree p > 2.


(At this point we should remark that the components of the variable xabove include both position and momentum variables. Thus, although theseare certainly related, the conjugate H! of H di!ers from the usual Legendretransform of H which customarily only involves the momentum variables.)

Introduce the space

Lp0 =

'y % Lp

%[0, 1]; IR2n

&;" 1

0y dt = 0

).

Now, if x % C1%[0, 1]; IR2n& is a 1-periodic solution of (6.2), the function y =

,J x % Lp0 solves the system of equations

y = ,J x ,(6.5)y = 2H(x) .(6.6)

Equation (6.5) can be inverted (up to an integration constant x0 % IRn) byintroducing the integral operator

K: Lp0 $ H1,p

%[0, 1]; IR2n& , (Ky)(t) =

" t

0J y dt .

By Lemma 6.3 relation (6.6) is equivalent to the relation x = 2H!(y). Thatis, system (6.5), (6.6) is equivalent to the system

x = Ky + x0(6.5()x = 2H!(y)(6.6()

for some x0 % IR2n. The latter can be summarized in the single equation

(6.7)" 1

0

%2H!(y) , Ky

&· . dt = 0 , 3. % Lp

0 .

Indeed, if y % Lp0 satisfies (6.7), it follows that

2H!(y) , Ky = const. = x0 % IRn .

Hence y solves (6.5(), (6.6() for some x % H1,p%[0, 1]; IR2n

&. Transforming

back to (6.5), (6.6), from (6.6) we see that y % H1,p ($ C0, and thereforex % C1

%[0, 1]; IR2n

&is a 1-periodic solution of (6.2). Thus, (6.2) and its weak

“dual” form (6.7) are in fact equivalent.Now we can conclude the proof of Theorem 6.5: We recognize (6.7) as the

Euler-Lagrange equation of the functional E! on Lp0, given by

E!(y) =" 1

0

:H!(y) , 1

2!y, Ky"

;dt .

Note that by (6.4) the functional E! is Frechet-di!erentiable and coercive onLp

0. Moreover, E! is the sum of a continuous convex and a compact quadratic

6. Duality 65

term, hence weakly lower semi-continuous. Thus, by Theorem 1.2, a minimizery! % Lp

0 of E! exists, solving (6.7). By (6.4) the quadratic term ,* 10 !y, Ky" dt

in E! dominates near y = 0. Since, for instance, !y, Ky" > 0 for y(t) =ae2*Jt % Lp

0 with a % IR2n \ {0}, we have inf E! < 0, and y! -= 0. By theabove discussion there is a constant x0 such that the function x = Ky! + x0

solves (6.2). Since y! -= 0, also x is non-constant; hence H(x(t)) = " > 0.But H = H is homogeneous on rays. Thus a suitably rescaled multiple x of x,x = (!% )1/qx(T ·), will satisfy (6.2) with H(x) = #, as desired.

Periodic Solutions of Nonlinear Wave-Equations

As a second example we consider the problem of finding a non-constant, time-periodic solution u = u(x, t), 0 + x + 1, t % IR, of the problem

Au = utt , uxx = ,u|u|p&2 in ]0, 1[#IR(6.8)u(0, · ) = u(1, · ) = 0(6.9)u( · , t + T ) = u( · , t) for all t % IR ,(6.10)

where p > 2 and the period T are given.The following result again is due to Rabinowitz [6].

6.6 Theorem. Suppose T/1 % Q; then there exists a non-constant T -periodicweak solution u % Lp

%[0, 1]# IR

&of problem (6.8)–(6.10).

Remark. For simplicity, we consider only the case T = 21; the general caseT/1 % Q can be handled in a similar way. The situation, however, changescompletely if T is not a rational multiple of the spatial period, in this case, arational multiple of 1. Whether or not Theorem 6.7 holds true in this case is anopen problem which seems to call for techniques totally di!erent from those weare going to describe. (See Bobenko-Kuksin [1] or Poschel [1] for some recentresults in this regard.)

Proof. Problem (6.8)–(6.10) can be interpreted as the Euler-Lagrange equationsassociated with a constrained minimization problem for the functional

E(u) =12

" 2*

0

" *

0

%|ux|2 , |ut|2

&dx dt

on the space

H =+u % H1,2

loc

%[0, 1]# IR

&; u satisfies (6.9), (6.10)

,,

endowed with the H1,2-norm on % = [0, 1]# [0, 21], subject to the constraint

1u1Lp(") = 1 .


However, E is unbounded on this set. Moreover, the operator A = &2t , &2

x

related to the second variation of E has infinitely many positive and negativeeigenvalues and also possesses an infinite-dimensional kernel. Therefore – as inthe case of Hamiltonian systems considered above – the direct methods do notimmediately apply.

In order to convert (6.8)–(6.10) into a problem that we can handle, wewrite (6.8) as a system

v = Au(6.11),v = u|u|p&2 = 2G(u) ,(6.12)

where G(u) = 1p |u|

p. Since G is strictly convex, (6.12) may be inverted usingthe Legendre-Fenchel transform of G,

G!(v) = sup {uv , 1p|u|p ; u % IR} =

1q|v|q ,

where 1 < q < 2 is the exponent conjugate to p, satisfying 1p + 1

q = 1. ByLemma 6.3 then, (6.12) is equivalent to the equation

(6.13) u = 2G!(,v) = ,v|v|q&2 .

In order to invert (6.11) we need to collect some facts about the wave opera-tor A. In our exposition we basically follow Brezis-Coron-Nirenberg [1]. Therepresentation formula (6.14)–(6.16) is due to Lovicarowa [1].

6.7 Estimates for the wave operator A. For T = 21 the spectrum ;(A) andkernel N of A, acting on functions in L1(%) satisfying (6.9), (6.10), can becharacterized as follows:

;(A) = {j2 , k2 ; j % IN, k % IN0} ,

N ='

p(t + x) , p(t , x) ; p % L1loc(IR), p(s + 21) = p(s) for almost all s,

" 2*

0p dx = 0

).

The last condition appearing in the definition of N is a normalization condition.N is closed in L1(%); moreover, given f % L1(%) such that

*" f ' dx dt = 0

for all ' % N ( L%(%), there exists a unique function u % C(%), satisfying(6.9), (6.10), such that Au = f and

*" u' dx dt = 0 for all ' % N . In fact, u

is given explicitly as follows:

u(x, t) = 3(x, t) +%p(t + x) , p(t , x)

&,

where 3 is constructed from a 21-periodic extension of f to [0, 1]# IR by usingthe fundamental solution of the wave operator; that is,

6. Duality 67

3(x, t) = ,12

" *

x

-" t+()&x)

t&()&x)f(-, <) d<

.d- + c

1 , x

1,(6.14)

with

c =12

" *

0

-" t+)

t&)f(-, <) d<

.d- .(6.15)

Note that c is constant; here, the fact that f is orthogonal to N is used.The choice of c now guarantees that u satisfies the boundary condition (6.9);moreover, periodicity of f implies (6.10). Finally, choosing

(6.16) p(s) =121

" *

0

33(-, s + -) , 3(-, s, -)

4d-

ensures that u is L2-orthogonal to N , as desired.Formulas (6.14)–(6.16) determine an operator K = A&1 from the weak

orthogonal complement of N

N- ='

f % L1(%) ;"

"f ' dx dt = 0 for all ' % N ( L%(%)

)

into C(%) satisfying the condition

(6.17) 1Kf1L& + c 1f1L1 .

Moreover, for f % N- ( Lq(%), q > 1, we have Kf % C!(%), with # =1 , 1/q > 0 and

(6.18) 1Kf1C# + c 1f1Lq ;

in particular, K is a compact, selfadjoint linear operator of N- ( L2(%) intoitself, with eigenvalues 1/(j2 , k2), j % IN, k % IN0, j -= k. (6.17) and (6.18)are easy consequences of (6.14)–(6.16) and Holder’s inequality.

Proof of Theorem 6.6. Fix q = pp&1 and let V = N- ( Lq(%), endowed with

the Lq-norm. By (6.18) the operator K: V $ Lp(%) is compact. Define

E!(v) =12

"

"(Kv) v dx dt ;

clearly E! % C1(V ). Moreover, since K is compact, it follows that E! is weaklylower semi-continuous. Restrict E! to the unit sphere

M = {v % V ; 1v1Lq = 1}

in Lq and consider a minimizing sequence (vm) for E! in M . We may assumethat vm $ v! weakly in Lq, whence by weak lower semi-continuity

(6.19) E!(v!) + lim infm$%

E!(vm) = inf {E!(v) ; v % M} < 0 .


(To verify the last inequality recall that K possesses also negative eigenvalues.)In particular, v! -= 0 and v!/1v!1Lq % M . But then, since E!(4v) =

42E!(v) for all v, by (6.19) we must have 1v!1Lq = 1, and v! % M minimizesE! on M .

By the Lagrange multiplier rule, v! satisfies the equation

(6.20)"

"

%Kv! + µv!|v!|q&2

&' dx dt = 0 , for all ' % V ,

with a Lagrange parameter µ % IR. Choosing ' = v! in (6.20) we realize thatµ = ,2E!(v!) > 0. Scaling v! suitably, we obtain a non-constant functionv % V , satisfying (6.20) with µ = 1. But then v satisfies

Kv + v|v|q&2 % N ( Lp .

Letting u = ,v|v|q&2 % Lp , thus there exists 3 % N ( Lp such that

u = Kv + 3 ,

u = ,v|v|q&2 .

But this system of equations is equivalent to (6.11), (6.12), and we concludethat u is a non-constant solution of the equation

Au + u|u|p&2 = 0 ,

satisfying the boundary and periodicity conditions (6.9), (6.10), as desired.

6.8 Notes. (1#) Actually, the solution obtained above is of class L%, seeBrezis-Coron-Nirenberg [1; p. 672 f.].(2#) Theorem 6.6 remains valid for a large class of semilinear equations

(6.21) utt , xxx + g(u) = 0

involving functions g sharing the qualitative behavior of a superlinear monomialg(u) = u|u|p&2; see Rabinowitz [6], Brezis-Coron-Nirenberg [1]. It is not evenin general necessary that g is monotone (Coron [1]) – although clearly the proofgiven above cannot be extended to such a case.

7. Minimization Problems Depending on Parameters 69

In the case g is smooth (C%) and strictly increasing, it was shown byRabinowitz [6] and Brezis-Nirenberg [1] that any bounded solution u to (6.21),with boundary conditions (6.9), (6.10), is of class C%.(3#) Further aspects of problem (6.8)–(6.10) have been studied by Salvatore[1] and Tanaka [1]. See Chapter II, Remark 7.3 and Notes 10.6 for references.

Other applications of duality methods to semilinear wave equations andrelated problems have been given by Willem [1], [2].

7. Minimization Problems Depending on Parameters

Quite often in the calculus of variations a functional to be minimized maydepend on parameters. In particular, this is the case if we attempt to approx-imate a minimization problem with constraints by a family of unconstrainedvariational problems involving a “penalty term”.

7.1 Penalty method. Suppose that V is a Banach space and let E and G benon-negative functionals on V . We seek to minimize E on the set

M = {u % V ; G(u) = 0}

of admissible functions.Without any smoothness and non-degeneracy assumptions, the set M may

be very “rough”, and a direct approach to the constrained minimization prob-lem may be rather cumbersome. Thus, instead of tackling this constrainedoptimization problem directly, it is often more convenient to find approximatesolutions by minimizing, for = > 0, the functional

E0(u) = E(u) + =&1G(u), u % V.

That is, we relax the constraint G(u) = 0 and admit any u % V as admissible;however, as = $ 0, due to the “penalty term” =&1G(u), minimizers of E0 willcome closer and closer to M and, hopefully, will converge to a solution of theconstrained minimization problem.

Even if M = 7, minimizers u0 of E0 may still converge to a generalizedsolution of the Euler equations in some larger function space. This, however,requires controlling the penalty term G(u0). Sometimes this can be achievedvia the following lemma.

7.2 Lemma. Suppose that E0: V $ IR, 0 < = < 1, is a family of functionalswith the following properties:

(1#) For all = % ] 0, 1 [ there exists u0 % V such that E0(u0) = infV E0.

(2#) For each u % V the map = 4$ E0(u) is non-increasing.

Then for almost every =0 % ] 0, 1 [ the map


= 4$ 60 = infV

E0

is di!erentiable at =0 with di!erential 6(00 , and there holds5555&E0&=

|0=00 (u00)5555 + |6(00 |.

The same conclusion holds if, instead of (2#) , we assume that for each u % Vthe map = 4$ E0(u) is non-decreasing.

Proof. Since = 4$ E0(u) is monotone, so is the map = 4$ 60. By Rademacher’stheorem, the latter map therefore is di!erentiable at almost every =0 %] 0, 1 [.Finally, at any such =0, for = >= 0 by assumption (2#) we can estimate

0 + E00(u00) , E0(u00) + 600 , 60.

Dividing by =, =0, and passing to the limit = $ =0, we obtain the claim.

In the context of the penalty method described above, assumption (2#) ofthe lemma is always satisfied. Existence of a minimizer u0 for E0, that is,assumption (1#) of the lemma, can be verified under very general conditions onE and G; compare Section 1. Thus, Lemma 7.2 applies. As an illustration wefirst state an abstract result in this regard. Later we will also give a concreteapplication.

Theorem 7.3. Suppose V is reflexive, and let E, G: V $ IR be non-negativeand weakly lower semi-continuous. Also suppose that E is coercive. Then, forany = > 0 the functional E0 = E + =&1G attains its infimum 60 on V . Assumethat 60 = o(| ln =|) as = $ 0. Then there exists a sequence =n $ 0 such that=&1n G(u0n) $ 0 as n $ * for any sequence (u0n) of minimizers of E0n .

Proof. Existence of a minimizer u0 of E0 for = > 0 follows from Theorem 1.2.Moreover, for any fixed =0 > 0 and 0 < =1 < =0 by assumption on 60 we canestimate

(7.1) o(| ln =1|) ' 601 , 600 '" 00

01

|6(0|d= ' ess inf01<0<00

=|6(0| · | ln(=1/=0)|,

whence upon dividing by | ln =1|, and letting =1 $ 0 in (7.1), we infer that

lim inf0$0

=|6(0| = 0 .

Thus, by Lemma 7.2, for a suitable sequence =n $ 0 and any sequence (u0n)of minimizers of E0n there holds

=&1n G(u0n) = =n

5555&E0n&=

(u0n)5555 + =n|6(0n | $ 0 (n $ *),


as claimed.

Similarly, if M -= 7, by the same arguments as above we can show that for asequence =n $ 0 there are minimizers (u0n) of E0n such that, as n $ *, (u0n)converges weakly to a minimizer u0 of E in M and the penalty term decayslogarithmically as = $ 0; that is,

=&1n G(u0n) + C| ln =n|&1

with a uniform constant C.

Harmonic Maps with Singularities

Let % & IR2 be smooth and bounded and assume (for simplicity) that % issimply connected. We can identify &% ?= S1 via a parametrization of theboundary curve.

Given a smooth map u0: &% $ S1 & IR2, let

H1,2u0

(%; IR2) =+u % H1,2(%; IR2); u = u0 on &%

,.

We seek to find a minimizer u % H1,2u0

(%; IR2) of Dirichlet’s integral

E(u) =12

"

"|2u|2 dx

subject to the constraint|u| = 1 ,

that is, a harmonic map u:% $ S1 with prescribed Dirichlet data u = u0 on&%. However, if the topological degree d of u0, considered as a map u0: S1 $S1, is non-zero, there is no extension of u0 to a harmonic map u:% $ S1 ofclass H1,2. In fact, by a result of Helein [1] and its generalization to manifoldswith boundary by Qing [1] such a map would be smooth. But, if d -= 0, clearlythere cannot be a C1-map u: % $ S1 such that u = u0 on &%. That is, anyextension of u0 to a (weakly) harmonic map must have singularities in %.

7.4 The Ginzburg Landau model. Bethuel-Brezis-Helein proposed to study theresulting singular variational problem via the Ginzburg-Landau model. Thatis, for = > 0 they study the minimizers of the functional

E0(u) = E(u) + =&2G(u) , u % H1,2u0

(%; IR2) ,

whereG(u) =

14

"

"(1 , |u|2)2 dx

penalizes the violation of the target constraint.


(Choosing =&2 instead of =&1 gives = > 0 a meaning as a “characteristiclength” for the penalized problem; in our context, however, this is of no impor-tance.) In a series of papers, summarized in their monograph Bethuel-Brezis-Helein [1], they study the properties of minimizers of the Ginzburg-Landauenergy and their convergence to a harmonic map with singularities. Moreover,they derive a renormalized energy functional whose minimizers give the possiblelocations of the singularities of the limiting harmonic map.

Three estimates, in particular, are needed for the proof of the convergenceresult. Let u0 % H1,2

u0(%; IR2) be minimizers of E0 for = > 0. We may suppose

d > 0. Constants may depend on % and u0.It is rather straightforward to establish the following upper bound

(7.2) E0(u0) + 1d| ln =| + C .

Establishing the corresponding lower bound

(7.3) E0(u0) ' 1d| ln =|, C

is considerably more di"cult, while the final estimate

(7.4) lim sup0$0

=&2G(u0) + C

for the convergence proof seems to be the most delicate. Using “Pohozaev’sidentity”, Bethuel-Brezis-Helein proved this estimate for star-shaped domains;compare Lemma III.1.4. For general domains, the result is due to Struwe [22],[23].

For a sequence =n $ 0, estimate (7.4) is an immediate consequence of the“easy” upper bound (7.2) and Lemma 7.1; see Struwe [22].

7.5 Lemma. Suppose the estimate (7.2) holds true with a uniform constant C.Then we have

lim inf0$0

=&2G(u0) + 1d.

Proof. Let 60 = infH1,2u0 (";IR2) E0 ' 0. From (7.2) and Lemma 7.2, similar to

(7.1) we deduce that

1d| ln =| + C '" 1

0(=|6(0|)

d=

=

'" 1

0

5555=&E0&=

(u0)5555d=

=

= 2" 1

0

%=&2G(u0)

&d==

,

and the assertion follows.


In combination with the lower bound (7.3), the same ideas actually yield (7.4)for any sequence of minimizers. This observation is due to del Pino-Felmer [1].

7.6 Lemma. Suppose the estimates (7.2) and (7.3) hold true with a uniformconstant C. Then we have

lim sup0$0

=&2G(u0) + C.

Proof. Instead of di!erentiating, we now take finite di!erences. Taking aminimizer u0 of E0 as comparison function for E20 and using (7.2), (7.3), weobtain

34=&2G(u0) = E0(u0) , E20(u0)

+ 1d| ln =| + C , (1d| ln(2=)|, C)+ 1d ln 2 + C ,

as claimed.

Thus, the proof of estimate (7.4) is complete. The arguments of Bethuel-Brezis-Helein then finally yield the following convergence result.

7.7 Theorem. For any sequence =n $ 0 and any sequence of minimizers un

of E0n there exist exactly d points x1, . . . , xd % % such that, as n $ *, asubsequence un $ u weakly in H1,2

loc (% \ {x1, . . . , xd}; IR2) and in H1,p(%, IR2)for any p < 2.

The limit map u:% \ {x1, . . . , xd} $ S1 is harmonic and is given by

u(x) = ei((x)dS

j=1

x , xj

|x , xj |

with some harmonic function ,. (We identify IR2 ?= C.)The singularities x1, . . . , xd are minima of a renormalized energy func-

tional given explicitly in terms of u0 and the Green’s function on %.

7.8 Notes. The idea of using variations of parameters in order to obtain a-prioribounds on critical points (in fact, saddle points) of variational integrals depend-ing monotonically on a real parameter was introduced by Struwe [16], in thecontext of surfaces of prescribed mean curvature H % IR. Further applicationsare given in Struwe [17], [21], and Ambrosetti-Struwe[2]. Related ideas havebeen proposed by Schechter-Tintarev [1]. See the Preface to the Third Editionfor more recent developments and references.

In Section II.9 we return to the topic of parameter dependence in thecontext of saddle points and give applications of Hamiltonian systems.

Chapter II

Minimax Methods

In the preceding chapter we have seen that (weak sequential) lower semi-continuity and (weak sequential) compactness of the sub-level sets of a func-tional E on a Banach space V su!ce to guarantee the existence of a minimizerof E.

To prove the existence of saddle points we will now strengthen the regu-larity hypothesis on E and in general require E to be of class C1(V ), that iscontinuously Frechet di"erentiable. In this case, the notion of critical point isdefined and it makes sense to classify such points as relative minima or saddlepoints as we did in the introduction to Chapter I.

Moreover, we will impose a certain compactness assumption on E, to bestated in Section 2. First, however, we recall a classical result in finite dimen-sions.

1. The Finite Dimensional Case

In the finite dimensional case, the existence of saddle points can be obtainedfor instance as follows (see, for instance, Courant [1; p. 223 ".]):

1.1 Theorem. Suppose E ! C1(IRn) is coercive and suppose that E possessestwo distinct strict relative minima x1 and x2. Then E possesses a third criticalpoint x3 which is not a relative minimizer of E and hence distinct from x1, x2,characterized by the minimax principle

E(x3) = infp!P

maxx!p

E(x) =: ! ,

whereP = {p " IRn ; x1, x2 ! p, p is compact and connected}

is the class of “paths” connecting x1 and x2.

Proof. Let (pm) be a minimizing sequence in P , satisfying

maxx!pm

E(x) # ! (m #$) .

Since E is coercive, the sets pm are uniformly bounded. Therefore

1. The Finite Dimensional Case 75

p =!

m!IN

"

l"m

pl ,

the set of accumulation points of (pm), is the intersection of a decreasing se-quence of compact and connected sets, hence is compact and connected. More-over, by construction x1, x2 ! pm for every m. Hence x1,2 ! p, and p ! P .Thus

maxx!p

E(x) % infp!!P

maxx!p!

E(x) = ! .

By continuity, on the other hand,

maxx!p

E(x) & lim supm#$

(maxx!pm

E(x)) = ! ,

and maxx!p E(x) = !.Note that, in particular, since x1,2 are strict relative minima joined by p,

it now also follows that ! > max{E(x1), E(x2)}.To see that there is a critical point x3 ! p such that E(x3) = !, we

argue indirectly: Note that by continuity of E and compactness of p the setK = {x ! p ; E(x) = !} is compact. Suppose DE(x) '= 0 for every x ! K.Then there is a uniform number " > 0 such that |DE(x)| % 2" for all x ! K.

By continuity, there exists a neighborhood

U! = {x ! IRn ; (y ! K : |x) y| < #}

of K such that |DE(x)| % " in U!. Note that this implies x1, x2 '! U!. Let$ be a continuous cut-o" function with support in U! such that 0 & $ & 1and $ * 1 in a neighborhood of K. Let +E(x) denote the gradient of E at x,characterized by the condition

+E(x) · v = DE(x)v for all v ! IRn .

Define a continuous map %: IRn , IR # IRn by letting

%(x, t) = x) t$(x) +E(x) .

Note that % is continuously di"erentiable in t and

d

dtE (%(x, t))

t = 0= ) < $(x)+E(x), DE(x) >= )$(x)|+E(x)|2.

Moreover, |+E(x)|2 % "2 > 0 on supp($) " U!. By continuity then, thereexists T > 0 such that

d

dtE (%(x, t)) & )$(x)

2|+E(x)|2

for all t ! [0, T ], uniformly in x. Thus if we choose

pT = {%(x, T ) ; x ! p} ,

76 Chapter II. Minimax Methods

Fig. 1.1. A mountain pass in the E-landscape

for any point %(x, T ) ! pT we compute that

E (%(x, T )) = E(x) +# T

0

d

dtE (%(x, t)) dt

& E(x)) T

2$(x) |+E(x)|2 ,

and the latter is either & E(x) < !, if x '! K, or & ! ) T2 "

2 < !, if x ! K.Hence

maxx!pT

E(x) < ! .

But by continuity of % it follows that pT is compact and connected, whileby choice of U! and $ also xi = %(xi, T ) ! pT , i = 1, 2. Hence pT ! P ,contradicting the definition of !.

Finally, if all critical points u of E in p with E(u) = ! were relativeminima, the set K of such points would be open in p and (by continuity of Eand DE) also closed. Moreover, by the preceding argument K '= -. But p isconnected. Thus p = K, contradicting the fact that E(x1), E(x2) < !. Thisconcludes the proof.

1.2 Interpretation. It is useful to think of E(x) as measuring the elevationat a point x in a landscape. Our two minima x1, x2 then correspond to twovillages at the deepest points of two valleys, separated from each other by amountain ridge. If now we walk along a path p from x1 to x2 with the propertythat the maximal elevation E(x) at points x on p is minimal among all suchpaths we will cross the ridge at a mountain pass x3 which is a saddle point ofE. Because of this geometric interpretation Theorem 1.1 is sometimes calledthe finite dimensional “mountain pass theorem”.

2. The Palais-Smale Condition 77

2. The Palais-Smale Condition

From the experience in the preceding section we expect a functional to possesscritical points of saddle type whenever the set of points with energy less thana certain value is disconnected or has a non-trivial topology. However, even inthe finite dimensional setting of Theorem 1.1 and with suitable assumptionsabout the topology of the sub-level sets of E, saddle points in general need notexist unless a certain compactness property holds. This is illustrated by thefollowing simple example.

Example. Let E ! C1(IR2) be given by E(x, y) = exp()y) ) x2. ThenE0 := {(x, y) ; E(x, y) < 0} is disconnected, while there is no “mountain pass”of minimal height 0; see Figure 2.1.

Note, however, that in the example above there is a sequence of paths pm,given by pm(t) = (t, m), )1 & t & 1, connecting the two components of E0,such that E achieves its maximum on pm at points zm = (0, m), satisfyingE(zm) # 0, DE(zm) # 0 as m #$. Moreover, the points (zm) fail to have afinite accumulation point. It seems that this lack of compactness is responsiblefor the absence of saddle-type critical points.

Fig. 2.1. Searching in vain for an optimal mountain pass

As we have seen, one way of inducing the necessary compactness in thefinite dimensional case is by requiring E to be coercive, which generalizes tothe condition of bounded compactness in the infinite dimensional case.

However, as remarked earlier, in infinite dimensions the requirements ofbounded compactness and our regularity assumption E ! C1(V ) are incom-patible. Moreover, we would like to apply minimax methods to functionalswhich in general are neither bounded from above nor below. Thus, any of the


former conditions on E is too restrictive. Instead, we will require the so-calledPalais-Smale condition to be satisfied by E. Originally, in the work of Palaisand Smale this assumption is stated as follows:

(C)If S is a subset of V on which |E| is bounded but onwhich .DE. is not bounded away from zero, then thereis a critical point in the closure of S.

(See Palais [1], [2], Smale [2], Palais-Smale [1].)

We will replace condition (C) by a slightly stronger condition which iseasier to work with. It is convenient to introduce the following concept.

Definition. A sequence (um) in V is a Palais-Smale sequence for E if|E(um)| & c, uniformly in m, while .DE(um). # 0 as m #$.

In terms of this definition our compactness condition may be phrased as follows.

(P.-S.) Any Palais-Smale sequence has a (strongly) convergentsubsequence.

Condition (P.-S.) implies condition (C): From any set S as in condition (C),we may extract a Palais-Smale sequence, and, if the latter has a convergentsubsequence, the limit point of this sequence will be a critical point in theclosure of S. The converse, however, is not true: The functional E * 0 satisfies(C) but in general will not satisfy (P.-S.).

Note that (P.-S.) implies that any set of critical points of uniformlybounded energy is relatively compact, see Lemma 2.3.(1%). In fact, if we wereto strengthen condition (C) by this requirement, this new condition would beequivalent to (P.-S.).

In finite dimensions, a large class of functionals satisfying (P.-S.) can becharacterized as follows:

2.1 Proposition. Suppose E ! C1(IRn) and assume the function .DE. +|E|: IRn # IR is coercive. Then (P.-S.) holds for E.

Proof. If .DE. + |E| is coercive, clearly a Palais-Smale sequence will bebounded, hence will contain a convergent subsequence by the Bolzano-Weier-strass theorem.

Example. Suppose E: IRn # IR is a quadratic polynomial

E(x) =n$

i,j=1

aij xi xj +n$

i=1

bi xi + c

2. The Palais-Smale Condition 79

in x = (x1, . . . , xn) ! IRn, and that D2E(x) = (aij)1&i,j&n is non-degenerate inthe sense that the matrix (aij)1&i,j&n induces an invertible linear map IRn #IRn. Then E satisfies (P.-S.).

We may ask whether a similar non-degeneracy condition in general will guar-antee that a polynomial map satisfies (P.-S.). Suppose E is a polynomial ofdegree m in x = (x1, . . . , xn) ! IRn:

E(x) =$

|"|&m

a"x" ,

where & = (&1, . . . ,&n), x" = x"11 · · · x"n

n , |&| = &1 + . . . + &n. SupposeD2E(x): IRn, IRn # IR is non-degenerate for any x ! IRn. Does E satisfy (P.-S.)? (The answer seems to be unknown. In fact, this question seems relatedto a particular case of the Jacobi conjecture, a puzzling problem in algebraicgeometry; see, for instance, Bass-Connell-Wright [1].)

In general, we can say the following:

2.2 Proposition. Suppose that E has the following properties.(1%) Any Palais-Smale sequence for E is bounded in V .(2%) For any u ! V we can decompose

DE(u) = L + K(u) ,

where L: V # V ' is a fixed boundedly invertible linear map and the operatorK maps bounded sets in V to relatively compact sets in V '. Then E satisfies(P.-S.).

Proof. Any (P.-S.)-sequence (um) is bounded by assumption. Moreover

DE(um) = Lum + K(um) # 0

implies thatum = o(1)) L(1K(um) ,

where o(1) # 0 in V as m #$. By boundedness of (um) and compactness ofK the sequences (L(1K(um)) – and hence (um) – are relatively compact.

The Palais-Smale condition permits one to distinguish a certain family of neigh-borhoods of critical points of a functional E and thus o"ers a useful character-ization of regular values of E.

For ! ! IR, " > 0, ' > 0 let

E# = {u ! V ; E(u) < !},K# = {u ! V ; E(u) = !, DE(u) = 0} ,

N#,$ = {u ! V ; |E(u)) !| < ", .DE(u). < "} ,

U#,% ="

u!K!

{v ! V ; .v ) u. < '} .


That is, K# is the set of critical points of E having “energy” !, {U#,%}%>0 isthe family of norm-neighborhoods of K#.

2.3 Lemma. Suppose E satisfies (P.-S.). Then for any ! ! IR the followingholds:(1%) K# is compact.(2%) The family {U#,$}%>0 is a fundamental system of neighborhoods of K#.(3%) The family {N#,$}$>0 is a fundamental system of neighborhoods of K#.

Proof. (1%) Any sequence (um) in K# by (P.-S.) has a convergent subsequence.By continuity of E and DE any accumulation point of such a sequence alsolies in K#, and K# is compact.(2%) Any U#,%, ' > 0, is a neighborhood of K#. Conversely, let N be anyopen neighborhood of K#. Suppose by contradiction that for 'm # 0 there is asequence of points um ! U#,%m \N . Let vm ! K# be such that .um)vm. & 'm.Since K# is compact by (1%), we may assume that vm # v ! K#. But thenalso um # v, and um ! N for large m, contrary to assumption.(3%) Similarly, each N#,$, for " > 0, is a neighborhood of K# . Conversely,suppose that for some neighborhood N of K# and "m # 0 there is a sequenceum ! N#,$m \ N . By (P.-S.) the sequence (um) accumulates at a critical pointu ! K# " N . The contradiction proves the lemma.

2.4 Remarks. (1%) In particular, if K# = - for some ! ! IR there exists " > 0such that N#,$ = -; that is, the di"erential DE(u) is uniformly bounded innorm away from 0 for all u ! V with E(u) close to !.(2%) The conclusion of Lemma 2.3 remains valid at the level ! under the weakerassumption that (P.-S.)-sequences (um) for E such that E(um) # ! are rel-atively compact. This observation will be useful when dealing with limitingcases for (P.-S.) in Chapter III.

2.5 Cerami’s variant of (P.-S.). Cerami [1], [2; Teorema (*), p. 166] has pro-posed the following variant of (P.-S.):

(2.1)Any sequence (um) such that |E(um)| & c uniformlyand .DE(um).

%1 + .um.

&# 0 (m # $) has a

(strongly) convergent subsequence.

Condition (2.1) is slightly weaker than (P.-S.) while the most important im-plications of (P.-S.) are retained; see Cerami [1],[2] or Bartolo-Benci-Fortunato[1]. However, for most purposes it su!ces to use the standard (P.-S.) condition.Therefore and in order to achieve a coherent and simple exposition consistentwith the bulk of the literature in the field, in the following our presentationwill be based on (P.-S.) rather than (2.1).

Variants of the Palais-Smale condition for non-di"erentiable functionalswill be discussed in a later section. (See Section 10.)

3. A General Deformation Lemma 81

3. A General Deformation Lemma

Besides the compactness condition, the second main ingredient in the proof ofTheorem 1.1 is the (local) gradient-line deformation %. Following Palais [4], wewill now construct a similar deformation for a general C1-functional in a Banachspace. The construction may be carried out in the more general setting of C1-functionals on complete, regular C1,1-Banach manifolds with Finsler structures;see Palais [4]. However, for most of our purposes it su!ces to consider a Banachspace as the ambient space on which a functional is defined.

Pseudo-gradient Flows on Banach Spaces

In the following we will initially assume that E is a C1-functional on a Banachspace V . Moreover, we denote

V = {u ! V ; DE(u) '= 0}

the set of regular points of E. As a substitute for the notion of gradient (whichrequires an inner product to be defined) we introduce the following concept.

3.1 Definition. A pseudo-gradient vector field for E is a locally Lipschitzcontinuous vector field v: V # V such that the conditions(1%) .v(u). < 2 min{.DE(u)., 1} ,(2%) /v(u), DE(u)0 > min {.DE(u)., 1}.DE(u).hold for all u ! V .

Note that we require v to be locally Lipschitz. Hence, even in a Hilbert spacethe following result is somewhat remarkable.

3.2 Lemma. Any functional E ! C1(V ) admits a pseudo-gradient vector fieldv: V # V.

Proof. For u ! V choose w = w(u) such that

.w. < 2 min{.DE(u)., 1} ,(3.1)/w, DE(u)0 > min{.DE(u)., 1}.DE(u). .(3.2)

By continuity, for any u ! V there is a neighborhood W (u) such that (3.1) and(3.2) hold (with w = w(u)) for all u) ! W (u). Since V " V is metrizable andhence paracompact, there exists a locally finite refinement {W&}&!I of the cover{W (u)}u!V of V , consisting of neighborhoods W& " W (u&); see for instanceKelley [1; Corollary 5.35, p. 160].

Choose a Lipschitz continuous partition of unity {(&}&!I subordinate to{W&}&!I ; that is, choose Lipschitz continuous functions 0 & (& & 1 with supportin W& and such that

'&!I (& * 1 on V . For instance, following Palais [4; p. 205

f.], we may let


'&(u) = dist(u, V \ W&) = inf(.u) v. ; v '! W&

)

and define(&(u) =

'&(u)'&!!I '&!(u)

.

Clearly, 0 & (& & 1, (& = 0 outside W&, and'&!I (& * 1. Moreover, since

{W&}&!I is locally finite, for any u ! V there exists a neighborhood W of usuch that W 1W&! '= - for at most finitely many indices )) ! I, and Lipschitzcontinuity of (& on W is immediate from Lipschitz continuity of the family{'&}&!I .

Finally, we may let

v(u) =$

&!I

(&(u) w(u&) .

The relations (3.1) and (3.2) being satisfied by w(u&) on the support of (&,their convex linear combination v is a pseudo-gradient vector field for E, asrequired.

3.3 Remark. If E admits some compact group action G as symmetries, vmay be constructed to be G-equivariant. In particular, if E is even, that is,E(u) = E()u), with symmetry group {id,)id} 2= ZZ2, we may choose

v(u) =12

(v(u)) v()u)) ,

where v is any pseudo-gradient vector field for E, to obtain a ZZ2-equivariantpseudo-gradient vector field v for E, satisfying v()u) = )v(u).

More generally, suppose G is a compact Lie group acting on V ; that is,suppose there is a group homomorphism of G onto a subgroup – indiscrim-inately denoted by G – of the group of linear isometries of V such that theevaluation map

G, V # V ; (g, u) 3# gu

is continuous. Also suppose that G leaves E invariant

E(gu) = E(u), 4(g, u) ! G, V .

Then it su!ces to letv(u) =

#

Gg(1v(gu) dg

be the average of any pseudo-gradient vector field v for E with respect to aninvariant Haar’s measure dg on G in order to obtain a G-equivariant pseudo-gradient vector field v, satisfying

v(gu) = gv(u)

for all g and u.We are now ready to state the main theorem in this section.


3.4 Theorem (Deformation Lemma). Suppose E ! C1(V ) satisfies (P.-S.). Let! ! IR, # > 0 be given and let N be any neighborhood of K#. Then there exista number # !]0, #[ and a continuous 1-parameter family of homeomorphisms%(·, t) of V , 0 & t < $, with the properties(1%) %(u, t) = u, if t = 0, or DE(u) = 0, or |E(u)) !| % #;(2%) E (%(u, t)) is non-increasing in t for any u ! V ;(3%) % (E#+! \ N, 1) " E#(!, and % (E#+!, 1) " E#(! 5N .Moreover, %: V , [0,$[# V has the semi-group property; that is, %(·, t) 6%(·, s) = %(·, s + t) for all s, t % 0.

Proof. Lemma 2.3 permits one to choose numbers ", ' > 0 such that

N 7 U#,2% 7 U#,% 7 N#,$ .

We may suppose ", ' & 1. Let $ be a locally Lipschitz continuous functionon V such that 0 & $ & 1, $ * 1 outside N#,$, $ * 0 in N#,$/2. Also let( be a Lipschitz continuous function on IR such that 0 & ( & 1, ((s) * 0,if |! ) s| % min{#, "/4}, ((s) * 1, if |! ) s| & min{#/2, "/8}. Finally, letv: V # V be a pseudo-gradient vector field for E. Define

e(u) =*)$(u) ( (E(u)) v(u), if u ! V0, else

.

By choice of ( and $, the vector field e vanishes identically (and therefore isLipschitz continuous) near critical points u of E. Hence e is locally Lipschitzcontinuous throughout V . Moreover, since .v. < 2 uniformly, also .e. & 2 isuniformly bounded. Hence there exists a global solution %: V , IR # V of theinitial value problem

*

*t%(u, t) = e (%(u, t))

%(u, 0) = u .

% is continuous in u, di"erentiable in t and has the semi-group property %(·, s)6%(·, t) = %(·, s + t), for any s, t ! IR. In particular, for any t ! IR the map%(·, t) is a homeomorphism of V .

Properties (1%) and (2%) are trivially satisfied by construction and the prop-erties of v. Moreover, for # & min{#/2, "/8} and u ! E#+!, if E (%(u, 1)) % !)#it follows from (2%) that |E (%(u, t)))!| & # and hence that (

%E%%(u, t)

&&= 1

for all t ! [0, 1].Di"erentiating, by the chain rule we thus obtain

(3.3)

E%%(u, 1)

&= E(u) +

# 1

0

d

dtE (%(u, t)) dt

< ! + #)# 1

0$(%(u, t))/v(%(u, t)), DE(%(u, t))0 dt

& ! + #)#

{t;'(u,t)*!N!,"}/v(%(u, t)), DE(%(u, t))0 dt

& ! + #) L1%{t;%(u, t) '! N#,$}

&· "2 .


But if either u '! N or %(u, 1) '! N , by uniform boundedness .e. & 2 and sinceV \N and N#,$ are separated by the “annulus” U#,2%\U#,% of width ', certainly

(3.4) L1%{t ; %(u, t) '! N#,$}

&% '

2.

Hence, if we choose # & $2%4 , estimate (3.3) gives

E (%(u, 1)) < ! + #) '"2

2& ! ) # ,

and (3%) follows.

3.5 Remarks. (1%) Since the deformation %: V , [0,$[# V is obtained byintegrating a suitably truncated pseudo-gradient vector field, % will be calleda (local) pseudo-gradient flow.(2%) If K# = -, we may choose N = - and hence obtain a uniform reduction ofenergy near ! in this case.(3%) The Palais-Smale condition was only used to obtain estimate (3.4). Forthis it is enough to assume that (P.-S.) holds at the level !, see Remark 2.4(2%).In particular, if N = K# = - the conclusion of Theorem 3.4 remains valid ifcondition (P.-S.)is replaced by the assumption that N#,$ = - for some " > 0.(4%) If E is invariant under a compact group action G, as in Remark 3.3, wecan achieve that % is G-equivariant in the sense that there holds

%(gu, t) = g %(u, t) for all u ! V, g ! G, t % 0 .

3.6 Comparison with gradient flows. It may be of interest to consider thespecial case of a C2-functional E on a real Hilbert space H with scalar product(·, ·) and induced norm . ·. . In this case, a gradient vector field +E: H # H isdefined as in the finite dimensional case by letting +E(u) at any u ! H be theunique vector in H such that

%+E(u), v

&= DE(u)v for all v ! H, equivalently

characterized by

(3.5) .+E(u). = .DE(u)., /+E(u), DE(u)0 = .DE(u).2 .

Moreover, since

.+E(u))+E(v). = .DE(u))DE(v). ,

if E ! C2, +E is of class C1 and defines a local gradient flow % by letting

*

*t%(u, t) = )+E (%(u, t))

%(u, 0) = u .

To interpret % we identify E with its graph G(E) = {(u, E(u)) ! H,IR}. Thenin the picture outlined in Interpretation 1.2 the flow-lines of % become paths


of steepest descent, and the rest points of % are precisely the critical points ofE where G(E) has a horizontal tangent plane.

In a Banach space V as ambient space, note that in general by (3.5) a gradientvector need not be uniquely determined unless V is uniformly locally convex.Moreover, also in this case, the duality map j: V ' # V , which maps v ! V '

to w = j(v) ! V satisfying /w, v0 = .v.2 = .w.2, in general is only uniformlycontinuous on bounded sets but may fail to be Lipschitz.

Fortunately, it is not at all necessary to deform along lines of “steepest”descent to obtain existence results for saddle points, “steep enough” su!ces.The notions of pseudo-gradient vector field and pseudo-gradient flow allow forthe necessary flexibility.

Pseudo-Gradient Flows on Manifolds

The above constructions of pseudo-gradient vector fields and pseudo-gradientflows can easily be generalized to the setting of a C1-functional on a completeC1,1-Finsler manifold. We basically follow Palais [4].

3.7 Finsler manifolds. Let F be a Banach space bundle over a space M andlet . ·. be a continuous real valued function on F such that the restriction. · .u of . · . to each fiber Fu is an admissible norm for Fu. If we trivialize Fin a neighborhood of a point u0 ! M , using Fu0 as the standard fiber, thenfor each u near u0 the norm . ·. u becomes a norm on Fu0 . We say . · . is aFinsler structure for the bundle F if for any # > 0, each u0 ! M , and each suchtrivialization in an atlas defining the bundle structure of E

supv!Fu0\{0}

.v.u

.v.uo

, supv!F\{0}

.v.u0

.v.u< 1 + # ,

if u is su!ciently near u0.Recall that a topological space M is regular if for each point x ! M and

any neighborhood U of u there is a closed neighborhood U of u such thatU " U . Now, a Finsler manifold of class Cr, r % 1, is a regular Cr-Banachmanifold M , modeled on a Banach space V , together with a Finsler structure. ·. on the tangent bundle TM . Then also the co-tangent space T 'M carriesa natural Finsler structure, indiscriminately denoted by . ·. , characterized byletting

.v'.u = sup(|/v, v'0| ; v ! TuM, .v.u & 1

)

for any v' ! T 'uM . Finally, . ·. induces a metric

(3.5) d(u, v) = infp

# 1

0

++++d

dtp(t)++++

p(t)

dt ,

where the infimum is taken over all C1-paths p: [0, 1] # M joining p(0) =u, p(1) = v; see Palais [4; p. 208 ".]. We say that M is complete if M iscomplete with respect to this metric d.


As an example of a complete Cr-Finsler manifold we may consider any(norm-) complete Cr-submanifold M of a Banach space V , with TuM carryingthe norm induced by the inclusion TuM " TuV 2= V .

3.8 Definition. Let M be a C1,1 Finsler manifold modeled on a Banach spaceV , E ! C1(M), M = {u ! M ; DE(u) '= 0} the set of regular points of E. Apseudo-gradient vector field for E is a Lipschitz continuous vector field (section)v: M 3# TM in the tangent bundle TM with the property that v(u) ! TuM and(1%) .v(u).u < 2 min{.DE(u).u, 1},(2%) /v(u), DE(u)0 > min{.DE(u).u, 1}.DE(u).u,for all u ! M , where . · .u denotes the norm in the tangent space TuM 2= V toM at the point u.

As in the proof of Lemma 3.2, for any u ! M there exists a (constant) pseudo-gradient vector field in a local trivialization of the tangent bundle aroundu. Since a C1,1-Finsler manifold is paracompact, see Palais [4; p. 203], anda paracompact C1,1-Banach manifold always admits locally Lipschitz parti-tions of unity, see Palais [4; p. 205 f.], a family of local pseudo-gradient vectorfields as above may be patched together to yield a pseudo-gradient vector fieldv: M # TM just as in the “flat” case M = V . Thus we obtain (Palais [4; 3.3.p. 206]):

3.9 Lemma. Any functional E ! C1(M) on a C1,1-Finsler manifold M admitsa pseudo-gradient vector field v: M # TM .

3.10 Remark. Again, if G is a compact Lie group acting (smoothly) throughisometries on M and if E is G-invariant, v may be constructed to be G-equivariant in the sense that

v%g(u)&

= Dg(u) v(u) , 4u ! M, g ! G .

Indeed, we may let

v(u) =#

G(Dg(u))(1 v (g(u)) dg

with respect to an invariant measure dg on G. (Note that in the “flat” case M =V , with G acting through linear isometries, the tangent map Dg(u): TuM 2=V # Tg(u)M 2= V may be identified with the map g itself.)

For ! ! IR, ", ' > 0 define K# , N#,$, U#,$ as in the flat case, using theFinsler structure to define the norm .DE(u).u and the distance (3.5).

The Palais-Smale condition can now be stated as in Section 2, and Lemma2.3 remains true for E ! C1(M) on a C1,1-Finsler manifold M . Then theconstruction in the proof of Theorem 3.4 can be carried over to obtain a localpseudo-gradient flow %: D(%) " M , [0,$[# M for E. Note that for % to bedefined globally on M , [0,$[ we also need to assume that M is complete withrespect to the metric (3.5). This yields the following result.

4. The Minimax Principle 87

3.11 Theorem. Suppose M is a complete C1,1-Finsler manifold and E !C1(M) satisfies (P.-S.). Let ! ! IR, # > 0 be given and let N be any neighbor-hood of K#. Then there exist a number # !]0, #[ and a continuous 1-parameterfamily of homeomorphisms %(·, t) of M, 0 & t < $, with the properties(1%) %(u, t) = u, if t = 0, or DE(u) = 0, or |E(u)) !| % #;(2%) E (%(u, t)) is non-increasing in t for any u ! M ;(3%) %(E#+! \ N, 1) " E#(!, and %(E#+!, 1) " E#(! 5N .Moreover, % has the semi-group property %(·, s)6%(·, t) = %(·, s+t), 4s, t % 0.If M admits a compact group of symmetries G and if E is G-invariant, % canbe constructed to be G-equivariant, that is, such that %(g(u), t) = g (%(u, t)) forall g ! G, u ! M, t % 0.

3.12 Remarks. (1%) It su!ces to assume that (P.-S.) is satisfied at the level!, see Remark 2.4(2%). In particular, if N = K# = -, condition (P.-S.) may bereplaced by the assumption that N#,$ = - for some " > 0, see Remark 3.5(3%).(2%) Completeness of M is only needed to ensure that the trajectories of thepseudo-gradient flow % are complete in forward direction.

4. The Minimax Principle

The “deformation lemma” Theorem 3.4 is a powerful tool for proving the exis-tence of saddle points of functionals under suitable hypotheses on the topologyof the manifold M or the sub-level sets E" of E.

We proceed to state a very general result in this direction: the generalizedminimax principle of Palais [4; p. 210]. Later in this section we sketch someapplications of this result.

4.1 Definition. Let %: M , [0,$[# M be a semi-flow on a manifold M . Afamily F of subsets of M is called (positively) %-invariant if %(F, t) ! F forall F ! F , t % 0.

4.2 Theorem. Suppose M is a complete Finsler manifold of class C1,1 andE ! C1(M) satisfies (P.-S.). Also suppose F " P(M) is a collection of setswhich is invariant with respect to any continuous semi-flow %: M , [0,$[#Msuch that %(·, 0) = id, %(·, t) is a homeomorphism of M for any t % 0, andE (%(u, t)) is non-increasing in t for any u ! M . Then, if

! = infF!F

supu!F

E(u)

is finite, ! is a critical value of E.

Proof. Assume by contradiction that ! ! IR is a regular value of E. Choose# = 1, N = - and let # > 0, %: M , [0,$[# M be determined according toTheorem 3.11. By definition of ! there exists F ! F such that


supu!F

E(u) < ! + # ;

that is, F " E#+!. By property 3.11(3%) of % and invariance of F , if we letF1 = %(F, 1), we have F1 ! F and F1 " E#(!; that is,

supu!F1

E(u) & ! ) # ,

which contradicts the definition of !.

Of course, it would be su!cient to assume that condition (P.-S.) is satisfied atthe level ! and that F is forwardly invariant only with respect to the pseudo-gradient flow.

In the above form, the minimax principle can be most easily applied if Eis a functional on a manifold M with a rich topology. But also in the “flat”case E ! C1(V ), V a Banach space, such topological structure may be hiddenin the sub-level sets E( of E.

4.3 Examples. (Palais, [3; p. 190 f.]) Suppose M is a complete Finsler manifoldof class C1,1, and E ! C1(M) satisfies (P.-S.).(1%) Let F = {M}. Then F is invariant under any semi-flow %. Hence, if

! = infF!F

supu!F

E(u) = supu!M

E(u)

is finite, ! = maxu!M E(u) is attained at a critical point of E.(2%) Let F = {{u}; u ! M}. Then, if

! = infF!F

supu!F

E(u) = infu!M

E(u)

is finite, ! = minu!M E(u) is attained at a critical point of E.(3%) Let X be any topological space, and let [X, M ] denote the set of freehomotopy classes [f ] of continuous maps f : X # M . For given [f ] ! [X, M ] let

F = {g(X) ; g ! [f ]} .

Since [% 6 f ] = [f ] for any homeomorphism % of M homotopic to the identity,the family F is invariant under such mappings %. Hence, if

! = infF!F

supu!F

E(u)

is finite, ! is a critical value.(4%) Let Hk(M) denote the k-dimensional homology of M (with arbitrary co-e!cients). Given a non-trivial element f ! Hk(M), denote by F the collectionof all F " M such that f is in the image of

Hk(iF ): Hk(F ) # Hk(M) ,


where Hk(iF ) is the homomorphism induced by the inclusion iF : F +# M .Then F is invariant under any homeomorphism % homotopic to the identity,and by Theorem 4.2, if

! = infF!F

supu!F

E(u)

is finite, then ! is a critical value.There is a “dual version” of (4%):

(5%) If Hk is any k-dimensional cohomology functor, f a non-trivial element

f ! Hk(M), f '= 0 ,

let F denote the family of subsets F " M such that f is not annihilated bythe restriction map

Hk(iF ): Hk(M) # Hk(F ) .

Then, if! = inf

F!Fsupu!F

E(u)

is finite, ! is a critical value.In the more restricted setting of a functional E ! C1(V ), similar results

are valid if M is replaced by any sub-level set E(, , ! IR. We leave it tothe reader to find the analogous variants of (3%), (4%), and (5%). See alsoGhoussoub [1].

Closed Geodesics on Spheres

A di"erent construction of a flow-invariant family is at the basis of the nextfamous result. We assume the notion of geodesic to be familiar from di"erentialgeometry. Otherwise the reader may regard (4.2) below as a definition.

4.4 Theorem. (Birkho! [1]) On any compact surface S in IR3 which isC3-di!eomorphic to the standard sphere, there exists a non-constant closedgeodesic.

Proof. Denote u = ddtu. Define

H1,2%IR/2-; S

&=(u ! H1,2

loc

%IR; IR3& ; u(t) = u(t + 2-),

u(t) ! S for almost every t ! IR)

the space of closed curves u: IR/2- # S with finite energy

E(u) =12

# 2)

0|u|2 dt .

By Holder’s inequality

(4.1) |u(s)) u(t)| &# t

s|u| d. &

,|t) s|

# t

s|u|2 d.

-1/2

& (2|t) s|E(u))1/2 ,


functions u ! H1,2(IR/2-; S) with E(u) & , will be uniformly Holder contin-uous with Holder exponent 1/2 and Holder norm bounded by

82-,. Hence,

if S ! C3, H1,2(IR/2-; S) becomes a complete C2-submanifold of the Hilbertspace H1,2(IR/2-; IR3) with tangent space

TuH1,2(IR/2-; S)

= {( ! H1,2(IR/2-; IR3) ; ((t) ! Tu(t)S 2= IR2}

at any closed curve u ! H1,2(IR/2-; S); see, for instance, Klingenberg [1; The-orem 1.2.9].

By (4.1), if E(u) is su!ciently small so that the image of u is covered bya single coordinate chart, of course

TuH1,2(IR/2-; S) 2= H1,2(IR/2-; IR2) .

Moreover, E is analytic on H1,2(IR/2-; IR3), hence as smooth as H1,2(IR/2-; S)when restricted to that space. At a critical point u ! C2, upon integrating byparts,

(4.2)# 2)

0u ( dt =

# 2)

0)u( dt = 0 , 4( ! TuH1,2(IR/2-; S) ;

that is, u(t) 9 Tu(t)S for all t, which is equivalent to the assertion that u is ageodesic, parametrized by arc length.

More generally, at a critical point u ! H1,2(IR/2-; S), if n: S # IR3 denotesa (C2-) unit normal vector field on S, for any ( ! H1,2(IR/2-; IR3) we have

() n(u)%n(u) · (

&! TuH1,2(IR/2-; S) ,

where · is the inner product in IR3. Inserting this into (4.2) and observing that

(4.3) u · n(u) =%u · n(u)

&· ) u · Dn(u) u = )u · Dn(u) u

in the distribution sense, we obtain that

(4.4) u +%u · Dn(u) u

&n(u) = 0 .

From (4.4) we now obtain that u ! H2,1(IR/2-) +# C1(IR/2-). Hence, by (4.4)again, u ! C2(IR/2-), and our previous discussion shows that closed geodesicson S exactly corresponding to the critical points of E on H1,2(IR/2-; S).

Moreover, E satisfies the Palais-Smale condition on H1,2(IR/2-; S): If(um) is a sequence in H1,2(IR/2-; S) such that E(um) & c < $ and

.DE(um). = sup#"Tum H1,2(IR/2$;S)

###1,2$1

....# 2)

0um ( dt

....# 0 ,

then (um) contains a strongly convergent subsequence.


Proof of (P.-S.). Since E(um) & c uniformly, by (4.1) the sequence (um) isequi-continuous. But S is compact, in particular bounded; hence (um) is equi-bounded. Thus, by Arzela-Ascoli’s theorem we may assume that um # uuniformly and weakly in H1,2(IR/2-; IR3). It follows that u ! H1,2(IR/2-; S).

Via the unit normal vector field n we can define the projection-v: H1,2(IR/2-; IR3) # TvH1,2(IR/2-; S) by letting

( 3# -v((t) = ((t)) n (v(t))%n (v(t)) · ((t)

&,

as above. In particular, we have

(m := -um(um ) u) ! TumH1,2(IR/2-; S) .

Note that, since n ! C2 and since (um) is bounded in H1,2(IR/2-; S), thesequence ((m) is bounded in H1,2. Thus, /(m, DE(um)0 # 0. Moreover, sinceum / u weakly in H1,2 and uniformly, it follows that also (m / 0 weakly inH1,2 and uniformly. Consequently, we have

o(1) = /(m, DE(um)0 =# 2)

0um(m dt

=# 2)

0(um ) u)(m dt + o(1)

=# 2)

0|um ) u|2 ) (um ) u) · d

dt

/n(um)

%n(um) · (um ) u)

&0dt + o(1)

=# 2)

0|um ) u|2 )

%(um ) u) · n(um)

&%n(um) · (um ) u)

&dt + o(1) ,

where o(1) # 0 as m #$. But

n(u) · u = 0 = n(um) · um

almost everywhere. Hence# 2)

0|n(um)(um ) u)|2 dt =

# 2)

0| (n(um)) n(u)) u|2 dt

& 2.n(um)) n(u).2$E(u) # 0 (m #$) ,

and um # u strongly.

In order to construct a flow-invariant family F we now proceed as follows:By assumption, there exists a C3-di"eomorphism 0 : S # S2 from S

onto the standard sphere S2 " IR3. Let p: [))2 , )2 ] # H1,2(IR/2-; S) be a1-parameter family of closed curves u = p(1) on S, such that p(±)2 ) * p± areconstant “curves” in S. Via 0 we associate with p a map p ! C0(S2; S2) by rep-resenting S2 2= [))2 , )2 ], IR/2- in polar coordinates (1,2), with {))2 }, IR/2-,respectively {)2 }, IR/2-, collapsed to points. Then we let


p(1,2) = 0 (p(1)(2)) .

Consider now the collection

P = {p ! C01[)-

2,-

2]; H1,2(IR/2-; S)

2; p(±-

2) * const. ! S}

and let

F = {p ! P ; p is homotopic to id S2} .

Choosing for p(1) the pre-image under 0 of a family of equilateral circles cov-ering S2, we find that F '= -. Also note that the map P : p 3# p ! C0(S2; S2)is continuous. Hence F is %-invariant under any homeomorphism % ofH1,2(IR/2-; S) homotopic to the identity, and which maps constant maps toconstant maps. Note that, in particular, any % which does not increase E willhave this latter property.

S

p( )!

Fig. 4.1. An admissible comparison path p ! P

Finally, by Theorem 4.2

! = infp!F

supu!p

E(u)

is critical.This almost completes the proof of Theorem 4.4. However, it remains to

rule out the possibility that ! = 0: the energy of trivial (constant) “closedgeodesics”.


4.5 Lemma. ! > 0.

Proof. There exists " > 0 such that for any x at distance dist(x, S) & " fromS there is a unique nearest neighbor -(x) ! S, characterized by

|-(x)) x| = infy!S

|x) y| ,

and -(x) depends continuously on x. Moreover, - is C2 if S is of class C3. By(4.1) there exists , > 0 such that for u ! H1,2(IR/2-; S) with E(u) & , thereholds

(4.5) diam(u) = sup0&*,*!&2)

|u(2)) u(2))| < " .

Now suppose ! < ,, and let p ! F be such that E(u) & , for any curveu = p(1) ! p. By (4.5), if we fix 20 ! [0, 2-], we can continuously contract anysuch curve u to the constant curve u(20) in the "-neighborhood of S by letting

us(2) = (1) s)u(2) + s u(20), 0 & s & 1 .

Composing with -, we obtain a homotopy

ps(1,2) = -%(1) s)p(1)(2) + sp(1)(20)

&, 0 & s & 1,

between p = p0 and a path p1 ! P consisting entirely of constant loops p1(1) *p(1)(20) for all 1. Composing ps with 0 : S # S2, we also obtain a homotopyof p = p0 2 id S2 to the map p1, given by p1(1,2) = 0 (p(1)(20)). But, letting

p1,r(1,2) = p1(r1,2) = 0%p(r1)(20)

&, 0 & r & 1 ,

we see that p1 – and hence also p – is homotopic to a constant map, contraryto our choice of p ! F .

4.6 Notes. Birkho"’s result of 1917 and a later extension to spheres of arbi-trary dimension (Birkho" [2]) mark the beginning of the calculus of variationsin the large. A major advance then came with the celebrated work of Lusternik-Schnirelmann [1] in 1929 who – by variational techniques – established the exis-tence of three geometrically distinct closed geodesics free of self-intersections onany compact surface of genus 0. (Detailed proofs were published by Lusternik[1] in 1947.)

For recent developments in the theory of closed geodesics, see, for instance,Klingenberg [1], Bangert [1], and Franks [1].


5. Index Theory

In most cases the topology of the space M where a functional E is defined willbe rather poor. However, if E is invariant under a compact group G actingon M , this may change drastically if we can pass to the quotient M/G withrespect to the symmetry group. Often this space will have a richer topologicalstructure which we may hope to exploit in order to obtain multiple criticalpoints.

However, in general the group G will not act freely on M and the quotientspace will be singular, in particular, it will no longer be a manifold. Thereforethe results outlined in the preceding sections cannot be applied.

A nice way around this di!culty is to consider flow-invariant families F inTheorem 4.2 which are also invariant under the group action. Since by Remark3.5.(4%) we may choose our pseudo-gradient flows % to be equivariant if E is,this approach is promising. Moreover, at least for special kinds of group actionsG, the topological complexity of the elements of such equivariant families canbe easily measured or estimated in terms of an “index” which then may beused to distinguish di"erent critical points.

Krasnoselskii Genus

The concept of an index theory is most easily explained for an even functionalE on some Banach space V , with symmetry group G = ZZ2 = {id,)id}. Define

A = {A " V ; A closed, A = )A}

to be the class of closed symmetric subsets of V .

5.1 Definition. For A ! A, A '= -, following Co"man [1], let

,(A) =

3inf(m ; (h ! C0(A; IRm \ { 0}), h()u) = )h(u)

)

$, if {..} = -, in particular, if A : 0 ,

and define ,(-) = 0. Note that for any A ! A, by the Tietze extension theorem,any odd map h ! C0(A; IRm) may be extended to a map h ! C0(V ; IRm).Letting h(u) = 1

2

%h(u)) h()u)

&, the extension can be chosen to be symmetric.

,(A) is called the Krasnoselskii genus of A. (The equivalence of Co"-man’s definition above with Krasnoselskii’s [1] original definition – see alsoKrasnoselskii-Zabreiko [1; p. 385 ".] – was established by Rabinowitz [1;Lemma 3.6].) A notion of coindex with related properties was introduced byConnor-Floyd [1].

The notion of genus generalizes the notion of dimension of a linear space:

5. Index Theory 95

5.2 Proposition. For any bounded symmetric neighborhood 3 of the origin inIRm there holds: ,(*3) = m.

Proof. Trivially, ,(*3) & m. (Choose h = id.) Let ,(*3) = k and leth ! C0(IRm; IRk) be an odd map such that h(*3) ': 0. We may considerIRk " IRm. But then the topological degree of h: IRm # IRk " IRm on 3 withrespect to 0 is well-defined (see Deimling [1; Definition 1.2.3]). In fact, since his odd, by the Borsuk-Ulam theorem (see Deimling [1; Theorem 1.4.1]) we have

deg(h,3, 0) = 1 .

Hence by continuity of the degree also

deg(h,3, y) = 1 '= 0

for y ! IRm close to 0 and thus, by the solution property of the degree, h coversa neighborhood of the origin in IRm; see Deimling [1; Theorem 1.3.1]. But thenk = m, as claimed.

Proposition 5.2 has a converse:

5.3 Proposition. Suppose A " V is a compact symmetric subset of a Hilbertspace V with inner product (·, ·)V , and suppose ,(A) = m < $. Then Acontains at least m mutually orthogonal vectors uk, 1 & k & m, (uk, ul)V =0 (k '= l).

Proof. Let u1, . . . , ul be a maximal set of mutually orthogonal vectors in A, anddenote W = span {u1, . . . , ul} 2= IRl, -: V # W orthogonal projection onto W .Then -(A) ': 0, and - defines an odd continuous map h = -|A: A # IRl \ {0}.By definition of ,(A) = m we conclude that l % m, as claimed.

Moreover, the genus has the following properties:

5.4 Proposition. Let A, A1, A2 ! A, h ! C0(V ; V ) an odd map. Then thefollowing hold:(1%) ,(A) % 0, ,(A) = 0 ; A = -.(2%) A1 " A2 < ,(A1) & ,(A2).(3%) ,(A1 5 A2) & ,(A1) + ,(A2).(4%) ,(A) & ,

1h(A)2.

(5%) If A ! A is compact and 0 '! A, then ,(A) < $ and there is a neighborhoodN of A in V such that N ! A and ,(A) = ,(N).That is, , is a definite, monotone, sub-additive, supervariant and “continuous”map ,:A# IN0 5 {$}.

Proof. (1%) follows by definition.


(2%) If ,(A2) = $ we are done. Otherwise, suppose ,(A2) = m. By definitionthere exists h ! C0(A2; IRm \ {0}), h()u) = )h(u). Restricting h to A1 yieldsan odd map h|A1 ! C0(A1; IRm \ {0}), whence ,(A1) & ,(A2).(3%) Again we may suppose that both ,(A1) = m1, ,(A2) = m2 are finite, andwe may let h1, h2 be odd maps hi ! C0(Ai; IRmi \ {0}), i = 1, 2, as in thedefinition of the genus. As noted above, we may extend h1, h2 to odd mapshi ! C0(V ; IRmi), i = 1, 2. But then letting h(u) = (h1(u), h2(u)) defines anodd map h ! C0(V ; IRm1+m2) which does not vanish for u ! A1 5 A2, andclaim (3%) follows.(4%) Any odd map h ! C0

1h(A); IRm \ {0}

2induces an odd map h 6 h !

C0(A; IRm \ {0}), and (4%) is immediate.(5%) If A is compact and 0 '! A there is ' > 0 such that A 1 B%(0) = -.The cover

4B%(u) = B%(u) 5B%()u)

5

u!Aof A admits a finite sub-cover

{B%(u1), . . . , B%(um)}. Let {(j}1&j&m be a partition of unity on A sub-ordinate to {B%(uj)}1&j&m; that is, let (j ! C0

1B%(uj)

2with support in

B%(uj) satisfy 0 & (j & 1,'m

j=1 (j(u) = 1 for all u ! A. Replacing (j

by (j(u) = 12

%(j(u) + (j()u)

&, if necessary, we may assume that (j is even,

1 & j & m. By choice of ', for any j the neighborhoods B%(uj), B%()uj) aredisjoint.

Hence the map h: V # IRm with jth component

hj(u) =

67

8

(j(u), if u ! B%(uj) ,)(j(u), if u ! B%()uj) ,0, else ,

is continuous, odd, and does not vanish on A. This shows that ,(A) < $.Finally, assume that A is compact, 0 '! A, ,(A) = m < $ and let

h ! C0(A; IRm \ {0}) be as in the definition of ,(A). We may assumeh ! C0(V ; IRm). Moreover, A being compact, also h(A) is compact, and thereexists a symmetric open neighborhood N of h(A) whose closure is compactlycontained in IRm \ {0}. Choosing N = h(1(N), by construction h(N) ': 0 and,(N) & m. On the other hand, A " N . Hence ,(N) = ,(A) by monotonicityof ,, property (2%).

5.5 Observation. It is easy to see that if A is a finite collection of antipodalpairs ui,)ui (ui '= 0), then ,(A) = 1.

Minimax Principles for Even Functionals

Suppose E is a functional of class C1 on a closed symmetric C1,1-submanifoldM of a Banach space V and satisfies (P.-S.). Moreover, suppose that E iseven, that is, E(u) = E()u) for all u. Also let A be as above. Then for anyk & ,(M) & $ by Proposition 5.4(4%) the family

5. Index Theory 97

Fk = {A ! A ; A " M, ,(A) % k}

is invariant under any odd and continuous map and non-empty. Hence, analo-gous to Theorem 4.2, for any k & ,(M), if

!k = infA!Fk

supu!A

E(u)

is finite, then !k is a critical value of E; see Theorem 5.7 below.To see what happens when !k and !k+1 coincide for some k, it is instructive

to compare this result with the well-known Courant-Fischer minimax principlefor linear eigenvalue problems. Recall that on IRn with scalar product (·, ·) thek-th eigenvalue of a symmetric linear map K: IRn # IRn is given by

4k = minV !%V,

dim V !=k

maxu"V !#u#=1

(Ku, u) .

Translated into the above setting, we may likewise determine 4k by consideringthe functional

E(u) = (Ku, u)

on the unit sphere M = Sn(1 and computing !k as above. Trivially, E satisfies(P.-S.); moreover, it is easy to see that !k = 4k for all k. (By Proposition5.2 the inequality !k & 4k is immediate. The reverse inequality follows byProposition 5.3 and linearity of K.)

In the linear case, now, it is clear that if successive eigenvalues 4k =4k+1 = . . . = 4k+l(1 = 4 coincide, then K has an l-dimensional eigenspace ofeigenvectors u ! V satisfying Ku = 4u. Is there a similar result in the non-linear setting? Actually, there is. For this we again assume that E ! C1(M)is an even functional on a closed, symmetric C1,1-submanifold M " V \ {0},satisfying (P.-S.). Let !k, k & ,(M), be defined as above.

5.6 Lemma. Suppose for some k, l there holds

)$ < !k = !k+1 = . . . = !k+l(1 = ! < $ .

Then ,(K#) % l. By Observation 5.5, in particular, if l > 1, K# is infinite.

Proof. By (P.-S.) the set K# is compact and symmetric. Hence ,(K#) is well-defined and by Proposition 5.4(5%) there exists a symmetric neighborhood Nof K# in M such that ,(N) = ,(K#). For # = 1, N , and ! as above let # > 0and % be determined according to Theorem 3.11. We may assume % is odd.Choose A " M such that ,(A) % k + l ) 1 and E(u) < ! + # for u ! A.

Let %(A, 1) = A ! A. By property (3%) of % in Theorem 3.11

A " (E#(! 5N) .

Moreover, by definition of ! = !k it follows that

,%E#(!

&< k .


Thus, by Proposition 5.4(2%)–5.4(4%)we have

,(N) % ,%E#(! 5N

&) ,%E#(!

&

> ,(A)) k % ,(A)) k

% k + l ) 1) k = l ) 1 ;

that is, ,(N) = ,(K#) % l, as claimed.

In consequence, we note

5.7 Theorem. Suppose E ! C1(M) is an even functional on a completesymmetric C1,1-manifold M " V \ {0} in some Banach space V . Also sup-pose E satisfies (P.-S.) and is bounded from below on M . Let ,(M) =sup{,(K) ; K " M compact and symmetric}. Then the functional E pos-sesses at least ,(M) & $ pairs of critical points.

Remarks. Note that the definition of ,(M) assures that for k & ,(M) thenumbers !k are finite.

Completeness of M can be replaced by the assumption that the flow definedby any pseudo-gradient vector field on M exists for all positive time.

Applications to Semilinear Elliptic Problems

As a particular case, Theorem 5.7 includes the following classical result ofLusternik-Schnirelmann [2]: Any even function E ! C1(IRn) admits at least ndistinct pairs of critical points when restricted to Sn(1. In infinite dimensions,Theorem 5.7 and suitable variants of it have been applied to the solution ofnonlinear partial di"erential equations and nonlinear eigenvalue problems witha ZZ2-symmetry. See for instance Amann [1], Clark [1], Co"man [1], Hempel[1], Rabinowitz [7], Thews [1], and the surveys and lecture notes by Browder[2], Rabinowitz [7], [11].

Here we present only a simple example of this kind for which we return tothe setting of problem

)5u + 4u = |u|p(2u in 3 ,(I.2.1)u = 0 on *3 ,(I.2.3)

considered earlier. This time, however, we also admit solutions of varying sign.

5.8 Theorem. Let 3 be a bounded domain in IRn, and let p > 2; if n % 3we assume in addition p < 2n

n(2 . Then for any 4 % 0 problem (I.2.1), (I.2.3)admits infinitely many distinct pairs of solutions.

Proof. By Theorem 5.7 the even functional

5. Index Theory 99

E(u) =12

#

+

%|+u|2 + 4|u|2

&dx

admits infinitely many distinct pairs of critical points on the sphere S = {u !H1,2

0 ; .u.Lp = 1}, for any 4 % 0. Scaling suitably, we obtain infinitely manydistinct pairs of solutions for (I.2.1), (I.2.3).

General Index Theories

The concept of index can be generalized. Our presentation is based on Rabi-nowitz [11]. Related material can also be found in the recent monograph byBartsch [1]. Suppose M is a complete C1,1-Finsler manifold with a compactgroup action G. Let

A = {A " M ; A is closed, g(A) = A for all g ! G}

be the set of G-invariant subsets of M , and let

6 = {h ! C0(M ; M) ; h 6 g = g 6 h for all g ! G}

be the class of G-equivariant mappings of M . (Since our main objective is thatthe flow %( · , t) constructed in Theorem 3.11 be in 6 , we might also restrict6 to the class of G-equivariant homeomorphisms of M .) Finally, if G '= {id},denote

Fix G =(u ! M ; gu = u for all g ! G

)

the set of fixed points of G.

5.9 Definition. An index for (G,A,6 ) is a mapping i:A # IN0 5 {$} suchthat for all A, B ! A, h ! 6 there holds(1%) (definiteness:) i(A) % 0, i(A) = 0 =< A = -.(2%) (monotonicity:) A " B < i(A) & i(B).(3%) (sub-additivity:) i(A 5B) & i(A) + i(B).(4%) (supervariance:) i(A) & i

1h(A)2.

(5%) (continuity:) If A is compact and A 1 Fix G = -, then i(A) < $ andthere is a G-invariant neighborhood N of A such that i(N) = i(A).(6%) (normalization): If u '! Fix G, then i

19g!G gu

2= 1.

5.10 Remarks and examples. (1%) If A ! A and A 1 Fix G '= - then i(A) =supB!A i(B); indeed, by monotonicity, for u0 ! A1Fix G there holds i ({uo}) &i(A) & supB!A i(B). On the other hand, for any B ! A the map h: B # {u0},given by h(u) = u0 for all u ! B, is continuous and equivariant, whencei(B) & i ({u0}) by supervariance of the index. Hence, in general nothing willbe lost if we define i(A) = $ for A ! A such that A 1 Fix G '= -.


(2%) By Example 5.1 the Krasnoselskii genus , is an index for G = {id,)id},the class A of closed, symmetric subsets, and 6 the family of odd, continuousmaps.

Analogous to Theorem 5.7 we have the following general existence result forvariational problems that admit an index theory.

5.11 Theorem. Let E ! C1(M) be a functional on a complete C1,1-Finslermanifold M and suppose E is bounded from below and satisfies (P.-S.). Sup-pose G is a compact group acting on M without fixed points and let A be theset of closed G-invariant subsets of M , 6 be the group of G-equivariant home-omorphisms of M . Suppose i is an index for (G,A,6 ), and let

i(M) = sup{i(K) ; K " M is compact and G) invariant} & $ .

Then E admits at least i(M) critical points which are distinct modulo G.

The proof is the same as that of Theorem 5.7 and Lemma 5.6. Again note thatcompleteness of M can be replaced by the assumption that any pseudo-gradientflow on M is complete in forward time.

Lusternik-Schnirelman Category

The concept of category was introduced by Lusternik-Schnirelmann [2]. Thisnotion, in fact, is the first example of an index theory (in the above sense) inthe mathematical literature.

5.12 Definition. Let M be a topological space and consider a closed subsetA " M . We say that A has category k relative to M (catM (A) = k), if Ais covered by k closed sets Aj , 1 & j & k, which are contractible in M , andif k is minimal with this property. If no such finite covering exists, we letcatM (A) = $. Moreover, we define catM (-) = 0.

This notion fits in the frame of Definition 5.9 if we letG = {id}A = {A " M ; A closed },6 = {h ! C0(M ; M) ; h is a homeomorphism }.

Then we have

5.13 Proposition. catM is an index for (G,A,6 ).

Proof. (1%)–(3%) of Definition 5.9 are immediate. (4%) is also clear, since ahomeomorphism h preserves the topological properties of any sets Aj coveringA. (5%) Any open cover of a compact set A by open sets O whose closure iscontractible has a finite subcover {Oj , 1 & j & k}. Set N =

9Oj . (6%) is

obvious.

5. Index Theory 101

5.14 Categories of some standard sets. (1%) If M = Tm = IRm/ZZm is them-torus, then catT m(Tm) = m + 1, see Lusternik-Schnirelman [2] or Schwartz[2; Lemma 5.15, p. 161]. Thus, any functional E ! C1(Tm) possesses at leastm+1 distinct critical points. In particular, if m = 2, any C1-functional on thestandard torus, besides an absolute minimum and maximum, must possess atleast one additional critical point.(2%) For the m-sphere Sm " IRm+1 we have catSm(Sm) = 2. (Take A1, A2

slightly overlapping northern and southern hemispheres.)(3%) For the unit sphere S in an infinite dimensional Banach space we havecatS(S) = 1. (S is contractible in itself.)(4%) For real or complex m-dimensional projective space Pm we have catP m(Pm)= m + 1 (m & $).

Since real projective Pm = Sm/ZZ2, we may ask whether, in the presence of aZZ2-symmetry u # )u, the category and Krasnoselskii genus of symmetric setsare always related as in the above example 5.14(4%). This is indeed the case,see (Rabinowitz [1; Theorem 3.7]):

5.15 Proposition. Suppose A " IRm \ {0} is compact and symmetric, and letA = A/ZZ2 with antipodal points collapsed. Then ,(A) = catIRm\{0}/ZZ2(A).

Using the notion of category, results in the spirit of Theorem 5.8 have beenestablished by Browder [1], [3] and Schwartz [1], for example.

With index theories o"ering a very convenient means to characterize di"erentcritical points of functionals possessing certain symmetries, it is not surprisingthat, besides the classical examples treated above, a variety of other indextheories have been developed. See the papers by Fadell-Husseini [1], Fadell-Husseini-Rabinowitz [1], Fadell-Rabinowitz [1] on cohomological index theories– a very early paper in this regard is Yang [1]. Relative or pseudo-indices wereintroduced by Benci [3] and Bartolo-Benci-Fortunato [1].

For our final model problem in this section it will su!ce to consider theS1-index of Benci [2] as another particular case.

A Geometrical S1-Index

If M is a complete C1,1-Finsler manifold with an S1-action (in particular, if Mis a complex Hilbert space with S1 = {ei* ; 0 & 2 & 2-} acting through scalarmultiplication) we may define an index for this action as follows; see Benci [2].

5.16 Definition. Let A be the family of closed, S1-invariant subsets of M , and6 the family of S1-equivariant maps (or homeomorphisms). For A '= -, define

.(A) =

67

8

inf(m ; (h ! C0

%A; Cm \ {0}

&, l ! IN :

h 6 g = gl 6 h for all g ! S1), if {. . .} '= -,

$, if {. . .} = -,and let .(-) = 0. (Note the similarity with the Krasnoselskii index ,.) Here,S1 acts on Cm by component-wise complex multiplication.


5.17 Proposition. . is an index for (S1,A,6 ).

Proof. It is easy to see that . satisfies properties (1%), (2%), and (4%) of Def-inition 5.9. To see (3%) we may assume that .(Ai) = mi < $, i = 1, 2, andwe may choose hi, li as in the definition of . such that hi ! C0(Ai; Cmi \ {0})satisfies hi 6 g = gli 6 hi for all g ! S1, i = 1, 2.

Extending hi to M and averaging

hi(u) =#

S1g(lihi(gu) dg

with respect to an invariant measure (arc-length) on S1, we may assume thathi ! C0(M ; Cm), i = 1, 2. But then the map

h(u) =,%

h1(u)&l2 ,%h2(u)

&l1-

,

where for (z1, . . . , zm) ! Cm, l ! IN we let (z1, . . . , zm)l := (zl1, . . . , z

lm), defines

a maph ! C0

%A1 5 A2 ; Cm1+m2 \ {0}

&

such that h6g =,%

gl1 6h1

&l2 ,%gl2 6h2

&l1-

= gl 6h for all g ! S1 with l = l1l2.

To see (6%), for an element u0 '! Fix (S1) let

G0 = {g ! S1 ; gu0 = u0}

be the subgroup of S1 fixing u0. Since u0 '! Fix (S1), G0 is discrete, hencerepresented by

G0 =(e2)ik/l ; 0 & k < l

)

for some l ! IN.For u = gu0 now let h(u) = gl ! S1 " C \ {0}. Then h is well-defined

and continuous along the S1-orbit S1u0 = {gu0 ; g ! S1} of u0. Extending hequivariantly, we see that .(S1u0) = 1, which proves (6%).

Finally, to see (5%), suppose A is S1-invariant, compact, and A1Fix(S1) =-. For any u0 ! A let h be constructed as in the proof of (6%) and let O(u0)be an S1-invariant neighborhood of S1u0 such that h

%O(u0)

&': 0.

By compactness of A, finitely many such neighborhoods {O(ui)}1&i&m

cover A, whence .(A) < $ by sub-additivity. The remainder of the proof of(5%) is the same as in Proposition 5.4.

As in the case of the Krasnoselskii genus, Proposition 5.2, the S1-index may beinterpreted as a generalized dimension of a closed S1-invariant set; see Benci[2; Proposition 2.6]. However, we will not pursue this.

Instead, we observe that in the case of a free S1-action on a manifold M ,a simpler variant of Benci’s S1-index can be defined as follows. (Recall thata group G acts freely on a manifold M if only the identity element in G fixespoints in M .)

5. Index Theory 103

5.16) Definition. Suppose S1 acts freely on a manifold M . Let A be the familyof closed, S1-invariant subsets of M , and 6 the family of S1-equivariant maps(or homeomorphisms). For A '= -, define

.(A) =

67

8

inf(m ; (h ! C0

%A; Cm \ {0}

&:

h 6 g = g 6 h for all g ! S1),

$, if {. . .} = -,

and let .(-) = 0.

The proof of Proposition 5.17 may be carried over easily to see that . is anindex for (S1,A,6 ).

Multiple Periodic Orbits of Hamiltonian Systems

As an application, we present the following theorem on the existence of “many”periodic solutions of Hamiltonian systems, due to Ekeland and Lasry [1]:

5.18 Theorem. Suppose H ! C1(IR2n; IR), and for some ! > 0 assume thatC = {x ! IR2n ; H(x) & !} is strictly convex, with boundary S = {x !IR2n ; H(x) = !} satisfying x · +H(x) > 0 for x ! S. Suppose that fornumbers r, R > 0 with

r < R <8

2 r

we haveBr(0) " C " BR(0) .

Then there exist at least n distinct periodic solutions of the equation

(5.1) x = J+H(x)

on S. (J is defined on p. 60.)

Theorem 5.18 provides a “global” analogue of a result by Weinstein [1] on theexistence of periodic orbits of Hamiltonian systems near an equilibrium. Fur-ther extensions and generalizations of Theorem 5.18 were given by Ambrosetti-Mancini [2] and Berestycki-Lasry-Mancini-Ruf [1] who allow for energy surfaces“pinched” between ellipsoids rather than spheres. Moreover, Ekeland-Lassoued[1] have been able to show that any strictly convex energy surface carries atleast two distinct periodic orbits of (5.1). It is conjectured that a result likeTheorem 5.18 holds true in general on such surfaces; the proof of this conjec-ture, however, remains open.

Proof of Theorem 5.18. We follow Ambrosetti and Mancini [2]. As observed inSection I.6 we may assume that

(5.2) H(sx) = sqH(x)


is homogeneous of degree q, 1 < q < 2, and strictly convex. Moreover, dividingH by !q we may assume that ! = 1

q . By our assumption on S and (5.2), finally,we have

(5.3)1

qRq|x|q & H(x) & 1

qrq|x|q .

Let H' be the Legendre-Fenchel transform of H as in I.6. Recall that H' ! C1,and H'(sy) = spH'(y) with p = q

q(1 > 2; moreover, (5.3) translates into

(5.4)1prp|y|p & H'(y) & 1

pRp|y|p .

Also let K be the integral operator

(Ky)(t) =# t

0J y dt

on

Lp0 =(y ! Lp

loc

%IR; IR2n

&; y(t + 2-) = y(t),

# 2)

0y dt = 0

).

Then – as described in detail in Section I.6, following Equations (I.6.5)), (I.6.6))– a function y ! Lp

0 \ {0} solves the Equation (I.6.7), that is, the equation

(5.5) +H'(y))Ky = x0

for some x0 ! IR2n, if and only if the function x = Ky + x0 ! C1([0, 2-]; IR2n)solves (5.1) with H

%x(t)&

=: h/q > 0, which in turn implies that x(t) =h(1/qx

%h2/q(1t

&solves (5.1) with H

%x(t)&* ! = 1

q .Suppose we can exhibit n distinct solutions yk of (5.5) with minimal period

2- corresponding to distinct solutions xk of (5.1) with energies H%xk(t)&

= hk.Then either hj = hk and the corresponding xj '= xk (since the solutions xk aredistinct). Or hj '= hk and xj, xk will have di"erent minimal period and hencebe distinct. In any event we will have achieved the proof of the theorem. Thisis the strategy that we now follow.

Denote E: Lp0 # IR the dual variational integral corresponding to (5.5),

given by

E(y) =# 2)

0

,H'(y)) 1

2/y, Ky0

-dt .

Note that we have an S1-action on Lp0, via

(., y) 3# y, (t) = y(t + .) , for all , = ei, ! S1 .

This action leaves E invariant. Moreover, y has minimal period 2- if and onlyif y, = y precisely for . ! 2-ZZ. Denote

m = inf(E(y) ; y ! Lp

0

)

and let

5. Index Theory 105

m' = inf(E(y) ; y ! Lp

0, (k ! IN, k % 2 : y, = y for . = 2-/k)

.

Observe that, since p > 2 and since the spectrum of K contains positive eigen-values, we have m & m' < 0. Also note that the S1-action will be free on theset

M = {y ! Lp0 ; m & E(y) < m'} .

In particular, any y ! M will have minimal period 2-.Finally, E satisfies condition (P.-S.). Indeed, since E is coercive on Lp

0,any (P.-S.)-sequence (ym) for E is bounded in Lp

0. Thus we may assume thatym / y weakly in Lp

0 and Kym # Ky strongly in Lq%[0, 2-]; IR2n&. Recall from

Section I.6 that by strict convexity and homogeneity of H' the di"erential+H'

is strongly monotone. That is, we have

o(1).ym ) y.Lp = /ym ) y, DE(ym))DE(y)0

=# 2)

0/ym ) y,+H'(ym))+H'(y)0 ) /ym ) y, Kym )Ky0 dt

% &%.ym ) y.Lp

&.ym ) y.Lp ) o(1) ,

where o(1) # 0 as m # $, with a non-negative, non-decreasing function &that vanishes only at 0. If follows that ym # y strongly in Lp

0, as claimed.Thus, by Theorem 5.11, the proof will be complete if we can show that the

(simplified) S1-index of a suitable compact S1-invariant subset of M is % n.(From the definition of M it is clear that any pseudo-gradient flow on M willbe complete in forward time.)

Note that since E is weakly lower semi-continuous and coercive on Lp0 and

since in both cases the set of comparison functions is weakly closed, m and m'

will be attained in their corresponding classes. Let y' ! Lp0 satisfy

E(y') = m', y', = y' for some . '! 2-ZZ .

By minimality of y'

/y', DE(y')0 = p

# 2)

0H'(y') dt)

# 2)

0/y', Ky'0 dt = 0 ,

whence in particular

m' =,

1p) 1

2

-# 2)

0/y', Ky'0 dt .

We may assume that . = 2)k for some k ! IN, k > 1. Hence we obtain as

comparison function

y(t) = y',

t

k

-! Lp

0 ,

and


m & infs>0

E(sy) = infs>0

,sp

# 2)

0H'(y) dt) s2

2

# 2)

0/y, Ky0 dt

-.

But # 2)

0H'(y) dt =

# 2)

0H'(y') dt =

1p

# 2)

0/y', Ky'0 dt ,

while

(5.6)# 2)

0/y, Ky0 dt = k

# 2)

0/y', Ky'0 dt .

Hence

(5.7)

m & infs>0

,sp

p) s2

2k

-# 2)

0/y', Ky'0 dt

= kp

p&2

,1p) 1

2

-# 2)

0/y', Ky'0 dt

& 2p

p&2 m' < 0 .

To obtain a lower bound on m, let y ! Lp0 satisfy

E(y) = m .

Then

(5.8) /y, DE(y)0 = p

# 2)

0H'(y) dt)

# 2)

0/y, Ky0 dt = 0 ,

and hence

(5.9) m =,

1p) 1

2

-# 2)

0/y, Ky0 dt =

11) p

2

2# 2)

0H'(y) dt .

Note that by (5.4) for z ! Lp0 we have

(5.10)12-

# 2)

0H'(z) dt % rp

2-p

# 2)

0|z|p dx % rp

p

,12-

# 2)

0|z|2 dx

-p/2

.

Let 7 = {z ! Lp0 ; 1

2)

: 2)0 |z|2 dt = 1} and denote

b = sup*# 2)

0/z, Kz0 dt ; z ! 7

;> 0 .

Then, if we let y = 4z, z ! 7, by (5.8), (5.10) we have

(5.11)42(pb % 42(p

# 2)

0/z, Kz0 dt = p

# 2)

0H'(z) dt

% 2-rp .

5. Index Theory 107

Since p > 2 this implies a bound from above for 4. By estimating

(5.12)m =,

1p) 1

2

-42

# 2)

0/z, Kz0 dt %

,1p) 1

2

-42b

%,

1p) 1

2

-(2-)(2/(p(2)bp/(p(2)r(2p/(p(2) =: c0r

(2p/(p(2)

with a constant c0 < 0, the latter translates into a lower bound for m, andhence for m', by (5.7). Now let

7n =*

z(t) = eJ t(8, $) = (8 cos t) $ sin t, 8 sin t + $ cos t) ;

(8, $) ! IR2n with |8|2 + |$|2 = 1;" 7 .

Clearly 7n is S1-invariant. Moreover, note that for z ! 7n we have H'%z(t)&&

Rp/p for all t, whence

(5.13)12-

# 2)

0H'(z) dt & Rp/p .

Now, we have

Lemma 5.19.: 2)0 /z, Kz0 dt = b for any z ! 7n.

Postponing the proof of Lemma 5.19, we conclude the proof of Theorem 5.18as follows. For z ! 7n let 4 = 4(z) > 0 satisfy

(5.14) p4p

# 2)

0H'(z) dt) 42

# 2)

0/z, Kz0 dt = 0 .

By (5.10) the map z 3# 4(z)z is an S1-equivariant C1-embedding of 7n intoLp

0, mapping 7n onto an S1-invariant set 7n di"eomorphic to 7n by radialprojection; in particular, the (simplified) S1-index .(7n) % .(7n).

From (5.13), (5.14) and Lemma 5.19, as in (5.11) we now obtain that forz ! 7n, 4 = 4(z) there holds

42(pb = 42(p

# 2)

0/z, Kz0 dt = p

# 2)

0H'(z) dt & 2-Rp .

Hence analogous to (5.12) we obtain

supy!-n

E(y) & c0R(2p/p(2 < 0 .

Since by assumption R <8

2 r, and in view of (5.7), this implies that

supy!-n

E(y) <1

2p/(p(2)c0r

(2p/(p(2) & 12p/p(2

m & m' ;


that is, 7n " M . Hence the proof of Theorem 5.18 is complete if we show that.(7n) % n.

But any S1-equivariant map h:7n # Cm\{0} with h6g = g6h induces anodd map of S2n(1 =

((8, $) ! IR2n ; |8|2+ |$|2 = 1

)into Cm\{0} 2= IR2m\{0},

given by(8, $) 3# h

%eJ t(8, $)

&.

By Proposition 5.2 we conclude that 2m % 2n, whence .(7n) % n. (Since themap

z = z(t) = eJ t(8, $) 3# z(0) = (8, $) ! IR2n \ {0} 2= Cn \ {0}is S1-equivariant, we actually have equality .(7n) = n.)

Proof of Lemma 5.19. Since K is compact there exists z ! 7 satisfying# 2)

0/z, Kz0 dt = b .

By (5.6), z must have minimal period 2-.Moreover, z satisfies

Kz + x0 = 4z

for some 4 ! IR, where x0 = ) 12)

: 2)0 K(z) dt. Setting

x = Kz + x0,

equivalently, x solves4x = J x .

Hence, |x| * const. Moreover, the properties of z imply that x '= 0 and that xhas minimal period 2-. A scalar multiple of x (and hence also a scalar multipleof z) thus belong to 7n.

By S1-invariance of I(z) =: 2)0 /z, Kz0 dt and invariance of I under ro-

tations in IR2n, if I(z) = b for some z ! 7n, it follows that I(z) = b for allz ! 7n. The proof is complete.

6. The Mountain Pass Lemma and its Variants

The minimax principle and its variants essentially cover all possibilities howexistence results for saddle points can be drawn from information about thetopology of the sub-level sets of a functional E.

However, unless the domain of E itself has a rich topology, finding the rightnotion of flow-invariant family may be quite tiresome. Fortunately, there areexistence results for saddle points tailor-made for applications. These are thefamous (infinite dimensional) mountain pass lemma and its variants, due toAmbrosetti and Rabinowitz [1]. The simplest form of these results reads asfollows.

6. The Mountain Pass Lemma and its Variants 109

6.1 Theorem. Suppose E ! C1(V ) satisfies (P.-S.). Assume that(1%) E(0) = 0 ;(2%) (' > 0, & > 0 : .u. = '< E(u) % &;(3%) (u1 ! V : .u1. % ' and E(u1) < &.Define

P =(p ! C0 ([0, 1]; V ) ; p(0) = 0, p(1) = u1

).

Then! = inf

p!Psupu!p

E(u) % &

is a critical value.E

M

u!

"

!

Fig. 6.1. On the mountain pass lemma of Ambrosetti and Rabinowitz

Proof. Suppose by contradiction that K# = -. For # = min{&,&)E(u1)} andN = - determine # > 0 and a deformation % as in Theorem 3.4. By definitionof !, there exists p ! P such that

supu!p

E(u) < ! + # .

Consider p1 = %(p, 1). Note that by choice of # the deformation %(·, 1) leavesu0 = 0 and u1 fixed. Hence p1 ! P . Moreover, %(E#+!, 1) " E#(!; therefore

supu!p1

E(u) = supu!p

E (%(u, 1)) < ! ) # ,

by choice of p. But this contradicts the definition of !, and the proof is com-plete.

Remark. Observe that by Remark 3.5(3%) the proof, and hence the assertionof Theorem 6.1, remain true if we only require (P.-S.) at the level !.


Applications to Semilinear Elliptic Boundary Value Problems

Theorem 6.1 permits an alternative proof of Theorem I.2.1. However, Theorem6.1 can also be applied to more general problems of the type

)5u = g(·, u) in 3 ,(6.1)u = 0 on *3 ,(6.2)

which cannot be solved by a constrained minimization method.

6.2 Theorem. Let 3 be a smooth, bounded domain in IRn, n % 3, and let g:3,IR # IR be a Caratheodory function with primitive G(x, u) =

: u0 g(x, v) dv.

Suppose that the following conditions hold:(1%) g(x, 0) = 0 and lim supu#0

g(x,u)u & 0, uniformly in x ! 3;

(2%) (p < 2' = 2nn(2 , C: |g(x, u)| & C

%1+|u|p(1

&, for almost every x!3, u!IR;

(3%) (q > 2, R0 : 0 < q G(x, u) & g(x, u) u, for almost every x ! 3, if |u| % R0.Then problem (6.1, (6.2) admits non-trivial solutions u+ % 0 % u(. If, inaddition, g is Holder continuous in both variables, then u+ > 0 > u( in 3.

Remark. Here again, G(x, u) =: u0 g(x, v) dv denotes a primitive of g. An

analogous result is valid for n = 2. However, notation is simpler if we consideronly n % 3. Wang, Z.Q. [1], under (essentially) the assumptions of Theorem6.2, recently has established the existence of a third non-trivial solution. (Notethat by (1%) problem (6.1), (6.2) always admits the solution u = 0.)

Proof. Problem (6.1), (6.2) corresponds to the Euler-Lagrange equation of thefunctional

E(u) =12

#

+|+u|2 dx)

#

+G(x, u) dx

on the space H1,20 (3). Assumption (2%) implies that E is of class C1.

To see that E satisfies (P.-S.) , first note that by (2%) the map u 3# g(·, u)takes bounded sets in Lp(3) into bounded sets in Lp/(p(1)(3) " H(1(3).Therefore, and since for p < 2n

n(2 by Rellich’s theorem the space H1,20 (3)

embeds into Lp(3) compactly, the map K: H1,20 (3) # H(1(3), given by

K(u) = g(·, u), is compact. Since DE is of the form DE(u) = )5u ) g(·, u),by Proposition 2.2 it su!ces to show that any (P.-S.)-sequence (um) for E isbounded in H1,2

0 (3).Let (um) be a (P.-S.)-sequence. Then we obtain

(6.3)

C + o(1).um.H1,20% q E(um)) /um, DE(um)0

=q ) 2

2

#

+|+um|2 dx +

#

+

%g(x, um)um ) qG(x, um)

&dx

% q ) 22

.um.2H1,20

+ Ln(3) · ess infx!+,v!IR

%g(x, v)v) qG(x, v)

&,

where o(1) # 0 (m #$).


But by (2%) and (3%) the last term is finite and the desired conclusionfollows.

Next observe that E(0) = 0; moreover, by (1%), (2%) for any # > 0 thereexists C(#) such that

G(x, u) & #|u|2 + C(#)|u|p

for all u ! IR and almost every x ! 3. It follows that

E(u) % 12

#

+|+u|2 dx) #

#

+|u|2 dx) C(#)

#

+|u|p dx

%,

12) #

41) C(#).u.p(2

H1,20

-.u.2

H1,20% & > 0 ,

provided .u.H1,20

= ' is su!ciently small. Here 41 denotes the first eigenvalueof )5 in 3 with homogeneous Dirichlet boundary conditions, given by theRayleigh-Ritz quotient (I.2.4). Moreover, we have used the Sobolev embeddingH1,2

0 (3) +# Lp(3); see Theorem A.5 of Appendix A.Finally, condition (3%) can be restated as a di"erential inequality for the

function G, of the form

u|u|q d

du

%|u|(qG(x, u)

&% 0 , for |u| % R0 .

Upon integration we infer that for |u| % R0 we have

(6.4) G(x, u) % ,0(x)|u|q

with ,0(x) = R(q0 min

(G(x, R0), G(x,)R0)

)> 0. Hence, if u ! H1,2

0 (3) doesnot vanish identically, we obtain that with constants C(u), c(u) > 0 there holds

(6.5)

E(4u) =42

2

#

+|+u|2 dx)

#

+G(x,4u) dx

& C(u)42 ) c(u)4q + Ln(3) ess supx!+,|v|&R0

|G(x, v)| ,

# )$ as 4#$ .

But then, for fixed u '= 0 and su!ciently large 4 > 0 we may let u1 = 4u, andfrom Theorem 6.1 we obtain the existence of a non-trivial solution u to (6.1),(6.2).

In order to obtain solutions u+ % 0 % u(, we may truncate g above orbelow u = 0, replacing g by 0, if u & 0, to obtain a positive solution, respectivelyreplacing g by 0, if u % 0, in order to obtain a solution u( & 0. Denote thetruncated functions by g±(x, u), with primitive G±(x, u) =

: u0 g±(x, v) dv.

Note that (1%), (2%) remain valid for g± while (3%) will hold for ±u % R0,almost everywhere in 3. Moreover, for ±u & 0 all terms in (3%) vanish.

DenoteE±(u) =

12

#

+|+u|2 dx)

#

+G±(x, u) dx


the functional related to g±. Then as above E± ! C1(H1,20 (3)) satisfies (P.-

S.), E±(0) = 0, and condition (2%) of Theorem 6.1 holds. Moreover, choosing acomparison function u > 0, and letting u1 = 4u for large positive, respectivelynegative 4, by (6.4), (6.5), also condition (3%) of Theorem 6.1 is satisfied.Our former reasoning then yields a non-trivial critical point u± of E±, weaklysolving the equation

)5u± = g±(·, u±) in 3 .

Since g±(u) = 0 for ±u & 0, by the weak maximum principle ±u± % 0; seeTheorem B.5 of Appendix B. Hence the functions u± in fact are weak solutionsof the original Equation (6.1).

Finally, if g is Holder continuous, by (2%) and Lemma B.3 of Appendix Band the remarks following it, u± ! C2,"(3) for some & > 0. Hence the strongmaximum principle gives u+ > 0 > u(, as desired. (For a related problem asimilar reasoning was used by Ambrosetti-Lupo [1].)

The Symmetric Mountain Pass Lemma

More generally, for problems which are invariant under the involution u 3#)u, we expect the existence of infinitely many solutions, as in the case ofproblem (I.2.1), (I.2.3); see Theorem 5.8. However, if a problem does notexhibit the particular homogeneity of problem (I.2.1), (I.2.3), in general itcannot be reduced to a variational problem on the unit sphere in Lp, and aglobal method is needed. Fortunately, there is a “higher-dimensional” versionof Theorem 6.1, especially adapted to functionals with a ZZ2-symmetry, thesymmetric mountain pass lemma – again due to Ambrosetti-Rabinowitz [1]:

6.3 Theorem. Suppose E ! C1(V ) is even, that is E(u) = E()u), andsatisfies (P.-S.). Let V +, V ( " V be closed subspaces of V with codim V + &dim V ( < $ and suppose V = V ( + V +. Also suppose there holds(1%) E(0) = 0 ,(2%) (& > 0, ' > 0 4u ! V + : .u. = ' < E(u) % & ,(3%) (R > 0 4u ! V ( : .u. % R < E(u) & 0 .Then for each j, 1 & j & k = dim V ( ) codim V + the numbers

!j = infh!.

supu!Vj

E (h(u))

are critical, where

6 = {h ! C0(V ; V ); h is odd, h(u) = u if u ! V ( and .u. % R} ,

and where V1 " V2 " . . . Vk = V ( are fixed subspaces of dimension

dim Vj = codim V + + j .

Moreover, !k % !k(1 % . . . % !1 % &.

For the proof of Theorem 6.3 we need a topological lemma.


6.4 Intersection Lemma. Let V, V +, V (, 6, Vj , R be as in Theorem 6.3.Then for any ' > 0, and any h ! 6 there holds

,%h(Vj) 1 S% 1 V +

&= j ,

where S% denotes the '-sphere S% = {u ! V ; .u. = '}, and , denotes theKrasnoselskii genus introduced in Section 5.1; in particular h(Vj)1S%1V + '= -.

Proof. Denote S+% = S% 1 V + . For any h ! 6 the set A = h(Vj) 1 S+

% issymmetric, compact, and 0 '! A. Thus by Proposition 5.4(5%) there exists aneighborhood U of A such that ,(U) = ,(A). Then

,%h(Vj) 1 S+

%

&% ,%h(Vj) 1 S% 1 U

&

% , (h(Vj) 1 S%)) , (h(Vj) 1 S% \ U) .

Let Z " V + be a direct complement of V ( and denote by -: V # V (

the continuous linear projection onto V ( associated with the decompositionV = V ( > Z. Then, since U is a neighborhood of h(Vj) 1 S+

% , we have- (h(Vj) 1 S% \ U) ': 0, and hence

, (h(Vj) 1 S% \ U) & dim V ( < $ .

On the other hand, by Proposition 5.4(2%), (4%)

, (h(Vj) 1 S%) % ,%h(1(S%) 1 Vj

&.

But h(0) = 0, h = id on Vj \ BR(0); hence h(1(S%) 1 Vj bounds a symmetricneighborhood of the origin in Vj . Thus, by Proposition 5.2 we conclude that

, (h(Vj) 1 S%) % dim Vj = dimV ( + j ,

which implies the lemma.

Proof of Theorem 6.3. By Lemma 6.4 we have !j % & for all j ! {1, . . . , k}.Moreover, clearly !j+1 % !j for all j. Suppose by contradiction that !j isregular for some j ! {1, . . . , k}. For ! = !j , N = -, and # = & let # > 0 and% be as constructed in Theorem 3.4. By Remark 3.5(4%) we may assume that%(·, t) is odd for any t % 0.

Note that by choice of # we have %( ·, t)6h ! 6 for any h ! 6 . Choose h ! 6such that there holds E (h(u)) < !+# for all u ! Vj . Then h1 = %(·, 1)6h ! 6 ,and for all u ! Vj there holds

E (h1(u)) = E (% (h(u), 1)) < ! ) # ,

contradicting the definition of !.

Note that in contrast to Theorem 5.7, our derivation of Theorem 6.3 doesnot allow us to obtain optimal multiplicity results in the case of degeneratecritical values !j = !j+1 = . . . = !k. However, for most applications this defectdoes not matter. (Actually, by using a notion of pseudo-index, Bartolo-Benci-Fortunato [1; Theorem 2.4] have been able to prove optimal multiplicity resultsalso in the case of degenerate critical values.) The following result is typical.


6.5 Theorem. Suppose V is an infinite dimensional Banach space and supposeE ! C1(V ) satisfies (P.-S.), E(u) = E()u) for all u, and E(0) = 0. SupposeV = V ( > V +, where V ( is finite dimensional, and assume the followingconditions:(1%) (& > 0, ' > 0 : .u. = ', u ! V + < E(u) % & .(2%) For any finite dimensional subspace W " V there is R = R(W ) such that

E(u) & 0 for u ! W, .u. % R.Then E possesses an unbounded sequence of critical values.

Proof. Choose a basis {(1, . . .} for V + and for k ! IN let Wk = V ( >span {(1, . . . ,(k}, with Rk = R(Wk). Since Wk " Wk+1, we may assumethat Rk & Rk+1 for all k.

Define classes

6k =(h ! C0(V ; V ) ; h is odd,

4j & k, u ! Wj : .u. % Rj < h(u) = u)

and let!k = inf

h!.k

supu!Wk

E (h(u)) .

Then, by the argument of Theorem 6.3, each of these numbers defines a criticalvalue !k % & of E. Indeed, for each k we let Wk take the role of V ( in Theorem6.3 and consider the case j = k with Vk = Wk. Observe that for the pseudo-gradient flow % constructed in the proof of Theorem 6.3 there holds %(·, t) ! 6k

for all t. Moreover,

6k " 6 =(h ! C0(V ; V ); h is odd, h(u) = u

if u ! Wk and .u. % Rk

).

Hence also Lemma 6.4 remains true with 6k instead of 6 , and !k is critical, asclaimed.

The proof therefore will be complete when we show the following:

Assertion. The sequence (!k) is unbounded.

First observe that, since 6k 7 6k+1 while Wk " Wk+1 for all k, the sequence(!k) is non-decreasing. Suppose by contradiction that supk !k = ! < $. Notethat K = K# is compact and symmetric. Moreover, since ! % & > 0 wealso have 0 '! K. By Proposition 5.4(5%) then ,(K) < $, and there exists asymmetric neighborhood N of K such that ,(N) = ,(K#).

Choose # = & > 0, and let # !]0, #[ and % be determined according toTheorem 3.4 corresponding to #, !, and N . Observe that % may be chosento be odd, see Remark 3.5(4%), and that by choice of # for any j ! IN wehave %(u, t) = u for all t % 0 and any u ! Wj such that .u. % Rj; that is,%(·, t) ! 6k for any k.


Let ! = !(0), # = #(0), % = %#(0) . We iterate the above procedure.For each number !) ! [&, ! ) #] also the set K ) = K#! is compact, has finitegenus ,(K#!), and possesses a neighborhood N ) = N#! with ,(N

)) = ,(K )).

Moreover, for each such !), with # = & and N = N#! we may let ##! !]0, #[, %#!

be determined according to Theorem 3.4. Finitely many neighborhoods ]!(l))#(l), !(l) +#(l)[, l = 1, . . . , L, where !(l) ! [&, !)#], #(l) = ##(l) , cover [&, !)#].Clearly, we may assume

! = !(0) % !(1) % . . . % !(l) % !(l+1) % . . . % !(L) % &

and also that!(l(1) > !(l) + #(l) > !(l(1) ) #(l(1) > !(l)

for all l, 1 & l & L.For any k ! IN, with # = #(0) > 0 as above, now choose h ! 6k satisfying

supu!Wk

E (h(u)) < !k + # & ! + # .

Composing h with % = %#(0) yields a map %(·, 1) 6 h =: h) ! 6k with

h)(Wk) " E#(! 5N .

Consider the compositions

H(m,m) = id, H(m,m(1) = %#(m)(·, 1) ,

H(m,l) = %#(m)(·, 1) 6 . . . 6 %#(l+1)(·, 1), 0 & l < m) 1 < L

and let h(0) = %#(0)(·, 1)6h = h), h(l) = %#(l)(·, 1)6h(l(1) = H(l,0)6h(0), for l =1, . . . , L. By Theorem 3.4, each H(m,l) is the composition of homeomorphisms,hence a homeomorphism itself.

By induction, letting N (0) = N, N (l) = N#(l) , l = 1, . . . , L, we have

h(m)(Wk) " E#(m)(!(m)

"

0&l&m

H(m,l)1N (l)2

for all m, 0 & m & L. Moreover, h(m) " 6k for all m, 0 & m & L. Inparticular, h(L) ! 6k and

h(L)(Wk) " E""

0&l&L

H(L,l)1N (l)2

.

Let S+% = *B%(0; V +). Then by assumption (1%) we have

h(L)(Wk) 1 S+% "

"

0&l&L

1H(L,l)

1N (l)21 S+

%

2""

0&l&L

H(L,l)1N (l)2

.

By monotonicity and sub-additivity of the genus ,, see Proposition 5.4(2%),(3%), and since each map H(L,l) is a homeomorphism, this implies that


(6.6)

,1h(L)(Wk) 1 S+

%

2&$

0&l&L

,1H(L,l)

1N (l)22

=$

0&l&L

,1(N (l)

2=$

0&l&L

,%K#(l)

&=: k0 < $ ,

with k0 independent of k.

On the other hand, by the Intersection Lemma 6.4, for any k ! IN we have

(6.7) ,%h(Wk) 1 S+

%

&% k .

This holds for any h ! 6k, in particular, for h = h(L) .Since k is arbitrary, this contradicts (6.6); hence the proof of Theorem 6.5

is complete.

Application to Semilinear Equations with Symmetry

As an application we state the following existence result for problems of thetype (6.1), (6.2) involving odd nonlinearities g. Results of this kind are wellknown; see the references in Section 5. Note that the symmetry assumptionallows removal of any conditions on g near u = 0.

6.6 Theorem. Let 3 be a smoothly bounded domain in IRn, n % 3, and letg:3,IR # IR be a Caratheodory function with primitive G(·, u) =

: u0 g(·, v) dv.

Suppose:(1%) g is odd: g(x,)u) = )g(x, u),and conditions (2%),(3%) of Theorem 6.2 are satisfied, that is(2%) (p < 2' = 2n

n(2 , C : |g(x, u)| & C%1 + |u|p(1

&almost everywhere,

(3%) (q > 2, R0 : 0 < q G(x, u) & g(x, u)u for almost every x, |u| % R0.Then problem (6.1), (6.2) admits an unbounded sequence (uk) of solutions uk !H1,2

0 (3).

Proof. As in the proof of Theorem 6.2, define

E(u) =12

#

+|+u|2 dx)

#

+G(x, u) dx .

Hypothesis (2%) implies that E is Frechet-di"erentiable on H1,20 (3). A compu-

tation similar to (6.3) shows that the assertion of the theorem is equivalent tothe assertion that E admits an unbounded sequence of critical values.

To prove the latter we invoke Theorem 6.5. Note that by the proof ofTheorem 6.2 the functional E satisfies (P.-S.); see (6.3). Moreover, since g isodd, E is even: E(u) = E()u). Finally, E(0) = 0.

Denote 0 < 41 < 42 & 43 & . . . the eigenvalues of )5 on 3 with homoge-neous Dirichlet data, as usual, and let (j be the corresponding eigenfunctions.

We claim that for k0 su!ciently large there exist ' > 0, & > 0 suchthat for all u ! V + := span{(k ; k % k0} with .u.H1,2

0= ' there holds


E(u) % &. Indeed, by (2%), Sobolev’s embedding H1,20 (3) +# L2'

(3), andHolder’s inequality, for u ! V + we have (with constants C1, C2 independent ofu)

(6.8)

E(u) % 12

#

+|+u|2 dx) C

#

+|u|p dx) C

% 12.u.2

H1,20) C .u.r

L2.u.p(rL2' ) C

%,

12) C14

(r/2k0

.u.p(2

H1,20

-.u.2

H1,20) C2

where r2 + p(r

2' = 1. In particular, r = n(1 ) p2' ) > 0, and we may let ' =

2<

(C2 + 1) and choose k0 ! IN such that

C14(r/2k0

'p(2 & 14

to achieve

E(u) % 1 =: & , for all u ! V + with .u.H1,20

= ' .

Now fix V + as above and denote V ( = span{(j ; j < k0} its orthogonalcomplement.

Finally, on any finite dimensional subspace W " H1,20 , by (3%) and (6.4),

(6.5) there exist constants Ci = Ci(W ) > 0 such that

(6.9) supu!/BR(0;W )

E(u) & C1R2 ) C2R

q + C3 # )$

as R #$.Hence, Theorem 6.5 guarantees the existence of an unbounded sequence

of critical values!k = inf

h!.k

supu!Wk

E (h(u)) , k % k0 ,

where Wk = span{(j ; j & k} and 6k is defined as in the proof of Theorem6.5. The proof is complete.

6.7 Remark. We can estimate the rate at which the sequence (!k) diverges.Indeed, by the Intersection Lemma 6.4 , letting Vk = span{(j ; j % k},Wk = span{(1 . . . ,(k} as in the proof above, we have

h(Wk) 1 S% 1 Vk '= -

for any ' > 0, any k ! IN. Hence, by (6.8), with r = n%1) p

2'

&= p ) n(p(2)

2we obtain


!k % sup%>0

infu!S%+Vk

E(u)

% sup%>0

,12) C14

(r/2k 'p(2

-'2 ) C2

% c4kr

p&2 = c4k( p

p&2(n2 ) ,

for k large. Finally, by the asymptotic formula (see Weyl [1], or Edmunds-Moscatelli [1] )

4kk(2/n # c > 0 (k #$) ,

and it follows that with a constant c > 0 we have

(6.10) !k % ck2p

n(p&2)(1

for large k. Estimate (6.10) will prove useful in the next section.

7. Perturbation Theory

A natural question, which even today is not adequately settled, is whether thesymmetry of the functional is important for results like Theorem 6.5 to hold.

A partial answer for problems of the kind studied in Theorem 6.6 wasindependently obtained by Bahri-Berestycki [1] and Struwe [1] in 1980.

In abstract form, the variational principle underlying these results laterwas phrased by Rabinowitz [10]. It will be convenient to write E for the non-symmetric functional and to reserve the notation E for its even symmetrization.

7.1 Theorem. Suppose E ! C1(V ) satisfies (P.-S.). Let W " V be a finitedimensional subspace of V , w' ! V \W , and let W ' = W > span{w'}; also let

W '+ = {w + tw' ; w ! W, t % 0}

denote the “upper half-space” in W '. Suppose(1%) E(0) = 0 ,(2%) (R > 0 4u ! W : .u. % R < E(u) & 0 ,(3%) (R' % R 4u ! W ' : .u. % R' < E(u) & 0 ,and let

6 = {h !C0(V, V ) ; h is odd, h(u) = u if max{E(u), E()u)} & 0 ,

in particular, if u ! W, .u. % R, or if u ! W ', .u. % R'} .

Then, if

!' = infh!.

supu!W '

+

E(h'(u)) > ! = infh!.

supu!W

E (h(u)) % 0 ,

the functional E possesses a critical value % !'.

Proof. For , !]!, !'[ let

7. Perturbation Theory 119

9 = {h ! 6 ; E (h(u)) & , for u ! W} .

By definition of !, clearly 9 '= -. Hence

,' = infh!0

supu!W '

+

E (h(u)) % !'

is well-defined. We contend that ,' is critical.Assume by contradiction that ,' is regular and choose # > 0, % according

to Theorem 3.4, corresponding to ,', # = ,' ) , > 0, N = -. Also selecth ! 9 with

supu!W '

+

E (h(u)) < ,' + # .

Define an odd map h): W ' # V

h)(u) =*

% (h(u), 1) , if u ! W '+

)% (h()u), 1) , if )u ! W '+ .

Note that by choice of # and since h ! 9 we have

% (h()u), 1) = h()u) = )h(u) = )% (h(u), 1)

for u ! W . Hence h) is well-defined, odd, and continuous. Moreover, since0 & ! < , < ,' ) #, the map %(·, 1) fixes any point u that satisfies E(u) & 0and E()u) & 0. By definition of 9 so does h, and hence the composition%(·, 1) 6 h. It follows that h) may be extended to a map h) ! 9. But now theestimate

supu!W '

+

E (h)(u)) = supu!W '

+

E (% (h(u), 1)) < ,' ) #

yields the desired contradiction.

Theorem 7.1 suggests comparing a non-symmetric functional E satisfying theremaining hypotheses (1%), (2%) of Theorem 6.5 with its symmetrization

E(u) =12

1E(u) + E()u)

2.

Then Theorem 6.5 applies to E and we obtain an unbounded sequence (!k)of critical values of E. Playing the speed of divergence !k # $ o" againstthe perturbation from symmetry E ) E, from Theorem 7.1 we can glean theexistence of infinitely many critical points for functionals perturbed from sym-metry. Stating a general theorem of this kind, however, seems to involve somany technical conditions that any reader would ask himself if these conditionscan ever be met in practice. Therefore we prefer to present such an applicationimmediately.


Applications to Semilinear Elliptic Equations

We return to the setting of problem (6.1), (6.2) under (essentially) the assump-tions (1%)–(3%) of Theorem 6.6 studied previously. However, in order to makeestimate (6.4) uniform in x ! 3, we assume that g is continuous in all itsvariables. For convenience we restate also the remaining conditions.

7.2 Theorem. Let 3 be a smoothly bounded domain in IRn, n % 3. Suppose(1%) g:3,IR # IR is continuous and odd with primitive G(x, u) =

: u0 g(x, v) dv;

(2%) (p < 2' = 2nn(2 , C : |g(x, u)| & C

%1 + |u|p(1

&almost everywhere;

(3%) (q > 2, R0 : 0 < q G(x, u) & g(x, u)u for almost every x, |u| % R0.Moreover, suppose that

2p

n(p) 2)) 1 >

q

q ) 1.

Then for any f ! L2(3) the problem

)5u = g(·, u) + f

u = 0in 3 ,

on *3 ,

has an unbounded sequence of solutions uk ! H1,20 (3), k ! IN.

Proof. The assertion of the theorem is equivalent to the claim that the func-tional

E(u) =12

#

+|+u|2 dx)

#

+G(·, u) dx)

#

+fu dx

on H1,20 (3) possesses an unbounded sequence of critical values.Clearly, E(0) = 0, and on any finite dimensional subspace W " H1,2

0 (3)by (3%) and (6.4), (6.5) the estimate

(7.1) supu!/B%(0;W )

E(u) & C1'2 ) C2'

q + C3' + C4 # )$ ('#$)

holds with constants Ci = Ci(W ) > 0. (The term:+ fu dx was estimated by

using Holder’s inequality.) To see (P.-S.), as in (6.3) in the proof of Theorem6.2, for a (P.-S.) sequence (um) consider the estimate

C + o(1) .um.H1,20% qE(um)) /um, DE(um)0

= q E(um)) /um, DE(um)0 ) (q ) 1)#

+fu dx

% q ) 22

.um.2H1,20) C.um.H1,2

0) C ,

from which boundedness of (um) in H1,20 (3) follows at once. (Note that the

symmetrized functional E(u) = 12

%E(u) + E()u)

&equals the functional E

considered in Theorem 6.6.) Using Proposition 2.2, the existence of a strongly


convergent subsequence of (um) now is established exactly as in the proof ofTheorem 6.2.

Let 4j , (j , V + = span{(k ; k % k0}, Wk = span{(j ; j & k} as in theproof of Theorem 6.6. By (7.1) we may choose a non-decreasing sequence (Rk)such that E(u) & 0 for all u ! Wk with .u.H1,2

0% Rk; hence also E(u) & 0 for

all such u. With this sequence (Rk) let (6k) be defined as in Theorem 6.5 andfor k % k0 let

(7.2) !k = infh!.k

supu!Wk

E (h(u)) % c k2p

n(p&2)(1 ) !0

be the sequence of critical values of the symmetrized functional E constructedin Theorem 6.6, the lower estimate following from (6.10). Suppose that E doesnot have any critical values larger than some number ! ) 1. Fix k % k0 suchthat !k % ! for k % k. We will use Theorem 7.1 to derive a uniform bound

!k & C, for all k % k ,

which will yield the desired contradiction to (7.2).For k % k let W = Wk, w' = (k+1, W ' = Wk+1, and let 6 be defined as

in Theorem 7.1, with R = Rk, R' = Rk+1. Note that 6 " 6k+1 " 6k. Sinceby assumption there are no critical values ! > ! ) 1 for E, from Theorem 7.1we conclude that

!' := !'k = inf

h'!.sup

u!W '+

E (h'(u))

= ! := !k = infh!.

supu!W

E (h(u)) .

From oddness of any map h ! 6 and the estimate

E(u) =12

1E(u) + E()u)

2& max{E(u), E()u)} ,

valid for all u, we also deduce that

(7.3) !k % infh!.

supu!W

E%h(u)&% !k % ! , for all k % k .

Given # > 0 (say # = 1) choose a map h' " 6 such that

supu!W

E%h'(u)

&& sup

u!W '+

E%h'(u)

&< !'

k + # = !k + # .

Let . ·. denote the norm in H1,20 (3), respectively H(1(3). Consider a pseudo-

gradient vector field v for E such that

.v. & 2, /v(u), DE(u)0 % min(1, .DE(u).

).DE(u). .

Since E is even, we may assume v to be odd. Note that


/v(u), DE(u)0 = /v(u), DE(u)0+ /v(u), DE(u))DE(u)0% min{1, .DE(u).} .DE(u). ) 2.DE(u))DE(u).% min{1, .DE(u).} .DE(u). ) 4 .DE(u))DE(u). .

That is, v also is a pseudo-gradient vector field for E o" N$ for any " % 8,where for " > 0 the set N$ is given by

N$ =(u ! H1,2

0 (3) ; .DE(u))DE(u). > "(1 min(1, .DE(u).

).DE(u).

).

Note that – unless f * 0, which is trivial – N$ is a neighborhood of the setof critical points of E and E, for any " > 1. Denote N$ = N$ 5

1)N$2

the

symmetrized sets N$. Let 0 & $ & 1 be an even, locally Lipschitz functionsatisfying $(u) = 0 in N10 and $(u) = 1 o" N = N20. Also let ( be Lipschitz,0 & ( & 1, ((s) = 0 for s & 0, ((s) = 1 for s % 1, and let

e(u) = )(%max{E(u), E()u)}

&$(u)v(u)

denote the truncated pseudo-gradient vector field on H1,20 (3).

Then, if % denotes the (odd) pseudo-gradient flow for E induced by e, %will also be a pseudo-gradient flow for E and will strictly decrease E o" N with“speed”

d

dtE%%(u, t)

&|t=0 & )1

2min(1, .DE(u).2

), if E(u) % 1 .

Moreover, %(·, t) ! 6 for all t % 0.Composing h' with %(·, t), unless E achieves its supremum on %

%h'(W '), t

&

at a point u ! N , thus we obtain that

d

dtsup

u!W 'E%%%h'(u), t

&&& )c < 0 ,

uniformly in t % 0. (Here, condition (P.-S.) was used; the constant c maydepend on our initial choice for the map h.) Replacing h' by %(·, t) 6 h' andchoosing t large, if necessary, we hence may assume that E achieves its supre-mum on h'(W ') in N . That is, we may estimate

(7.4)

!k + # = !'k + # % sup

u!W 'E (h'(u))) sup

u"N,

E(u)$!k+&

|E(u)) E()u)|

= supu!W '

E (h'(u))) supu"N,

E(u)$!k+&

|2#

+fu dx|

% !k+1 ) c supu"N

E(u)$!k+&

.u.L2 .

But for u ! N with E(u) & !k + #, with constants c > 0 from (3%) and (6.4),(6.5) we obtain


.u.qL2 & c .u.q

Lq & c

,q ) 2

2

-#

+G(u) dx + C

& c

#

+

,12g(u)u)G(u)

-dx + C

= c

,E(u)) 1

2/u, DE(u)0+

12

#

+fu dx

-+ C

& c E(u) + c%1 + .DE(u).

&.u.H1,2

0+ C

& c E(u) + c%1 + .DE(u))DE(u).

&.u.H1,2

0+ C

& c !k + c .u.H1,20

+ C .

From this there follows

.u.2H1,2

0& 2 E(u) + 2

#

+G(u) dx + C.u.L2

& c !k + c .u.H1,20

+ C .

Together with the former estimate this shows that for such u there holds

.u.L2 & c !1/qk + C & c !1/q

k .

Inserting this estimate in (7.4) above yields the inequality

!k+1 & !k + c !1/qk = !k

,1 + c !

1&qq

k

-,

for all k % k0, with a uniform constant c. By iteration therefore

!k0+l & !k0

k0+l(1=

k=k0

,1 + c !

1&qq

k

-

& !k0 exp

>k0+l(1$

k=k0

ln

,1 + c !

1&qq

k

-?

& !k0 exp

>

ck0+l(1$

k=k0

!1&q

q

k

?

.

But by (7.2), (7.3) we have

!k % !k % c k2p

n(p&2)(1 .

Since we assume that

µ =1) q

q·,

2p

n(p) 2)) 1-

< )1 ,

we can uniformly estimate


!k0+l & !k0 exp

>c

k0+l(1$

k=k0

!1&q

q

k

?

& !k0 exp

>c

$$

k=k0

kµ

?< $ ,

for all l ! IN, which yields the desired contradiction. Thus, E possesses anunbounded sequence of critical values and the proof is complete.

Remark 7.3. (1%) In the special case g(x, u) = u|u|p(2, by Theorem 7.2 forany f ! L2 the problem

)5u = u|u|p(2 + f in 3 ,(7.5)u = 0 on *3 ,(7.6)

admits infinitely many solutions, provided 2 < p < p', where p' > 2 is thelargest root of the equation

(7.7)2p'

n(p' ) 2)= 2 +

1p' ) 1

.

Observe that p' < 2'.(2%) Recently, Bahri-Lions [1] have been able to improve the estimate for !k

as follows!k % c k

2pn(p&2) ,

which yields the improved boundsq

q ) 1<

2p

n(p) 2)for p and q for which Theorem 7.2 is valid. In particular, for the model problem(7.5), (7.6) above, one obtains infinitely many solutions, provided 2 < p < p,where 2 < p < 2' solves the equation

2p

n(p) 2)= 1 +

1p) 1

.

(3%) Bahri [1] has shown that for any p !]2, 2'[ there is a dense open set off ! H(1(3) for which problem (7.5), (7.6) possesses infinitely many solutions.

Generally, it is expected that this should be true for all f (for instance inL2(3)), also for more general problems of type (6.1), (6.2) considered earlier.Indeed, this is true in the case of ordinary di"erential equtions – see for exampleNehari [1], Struwe [2], or Turner [1] – and in the radially symmetric case, wheredi"erent solutions of (6.1), (6.2) can be characterized by the nodal propertiesthey possess; see Struwe [3], [4]. However, the proof of this conjecture in generalremains open.(4%) Applications of perturbation theory to Hamiltonian systems are given byBahri-Berestycki [2], Rabinowitz [10], and – more recently – by Long [1].(5%) Finally, Tanaka [1] has obtained similar perturbation results for nonlinearwave equations, as in Section I.6.

8. Linking 125

8. Linking

In the preceding chapter we have seen that symmetry is not essential for avariational problem to have “many” solutions. In this chapter we will describeanother method for dealing with non-symmetric functionals. This method wasintroduced by Benci [1], Ni [1], and Rabinowitz [8]. Subsequently, it was gen-eralized to a setting possibly involving also “indefinite” functionals by Benci-Rabinowitz [1]. It is based on the topological notion of “linking”.

8.1 Definition. Let S be a closed subset of a Banach space V , Q a submanifoldof V with relative boundary *Q. We say S and *Q link if(1%) S 1 *Q = -, and(2%) for any map h ! C0(V ; V ) such that h|/Q = id there holds h(Q) 1 S '= -.More generally, if S and Q are as above and if 6 is a subset of C0(V ; V ), thenS and *Q will be said to link with respect to 6 , if (1%) holds and if (2%) issatisfied for any h ! 6 .

8.2 Example. Let V = V1 > V2 be decomposed into closed subspaces V1, V2,where dimV2 < $. Let S = V1, Q = BR(0; V2) with relative boundary*Q = {u ! V2 ; .u. = R}. Then S and *Q link.

Fig. 8.1.

Proof. Let -: V # V2 be the (continuous) projection of V onto V2, andlet h be any continuous map such that h|/Q = id. We have to show that-%h(Q)&: 0.

For t ! [0, 1], u ! V2 define

ht(u) = t -%h(u)&

+ (1) t)u .


Note that ht ! C0(V2; V2) defines a homotopy of h0 = id with h1 = - 6 h.Moreover, ht|/Q = id for all t. Hence the topological degree deg(ht, Q, 0) iswell-defined for all t. By homotopy invariance and normalization of the degree(see for instance Deimling [1; Theorem 1.3.1]), we have

deg(- 6 h, Q, 0) = deg(id, Q, 0) = 1 .

Hence 0 ! - 6 h(Q), as was to be shown.

8.3 Example. Let V = V1 > V2 be decomposed into closed subspaces withdim V2 < $, and let u ! V1 with .u. = 1 be given. Suppose 0 < ' < R1, 0 <R2 and let

S = {u ! V1 ; .u. = '} ,

Q = {su + u2 ; 0 & s & R1, u2 ! V2, .u2. & R2} ,

with relative boundary *Q = {su+u2 ! Q ; s ! {0, R1} or .u2. = R2}. ThenS and *Q link.

Fig. 8.2.

Proof. Let -: V # V2 denote the projection onto V2, and let h ! C0(V ; V )satisfy h|/Q = id. We must show that there exists u ! Q such that the relations.h(u). = ' and -

%h(u)&

= 0 simultaneously hold. For t ! [0, 1], s ! IR, u2 ! V2

let

ht(s, u2) =%t.h(u)) -

%h(u)&.+ (1) t)s) ', t-

%h(u)&

+ (1) t)u2

&,

where u = su + u2. This defines a family of maps ht: IR , V2 # IR , V2

depending continuously on t ! [0, 1]. Moreover, if u = su + u2 ! *Q, we have

ht(s, u2) =%t.u) u2.+ (1) t)s) ', u2

&= (s) ', u2) '= 0

8. Linking 127

for all t ! [0, 1]. Hence, if we identify Q with a subset of IR , V2 via thedecomposition u = su+u2, the topological degree deg(ht, Q, 0) is well-definedand by homotopy invariance

deg(h1, Q, 0) = deg(h0, Q, 0) = 1 ,

where h0(s, u2) = (s ) ', u2). Thus, there exists u = su + u2 ! Q such thath1(u) = 0, which is equivalent to

-%h(u)&

= 0 and .h(u). = ' ,

as desired.

8.4 Theorem. Suppose E ! C1(V ) satisfies (P.-S.). Consider a closed subsetS " V and a submanifold Q " V with relative boundary *Q. Suppose(1%) S and *Q link,(2%) & = infu!S E(u) > supu!/Q E(u) = &0 .Let

6 = {h ! C0(V ; V ) ; h|/Q = id} .

Then the number

! = infh!.

supu!Q

E%h(u)&

defines a critical value ! % & of E.

Proof. Suppose by contradiction that K# = -. For # = & ) &0 > 0, N = -let # > 0 and %: V , [0, 1] # V be the pseudo-gradient flow constructed inTheorem 3.4. Note that by choice of # there holds %(·, t)|/Q = id for all t. Leth ! 6 such that E

%h(u)&

< ! + # for all u ! Q. Define h) = %(·, t) 6 h. Thenh) ! 6 and

supu!Q

E%h(u)&

< ! ) #

by Theorem 3.4(3%), contradicting the definition of !.

Remark. By Example 8.3, letting S = *B%(0; V ), Q = {tu ; 0 & t & 1}with relative boundary *Q = {0, u}, 0 < '< .u., Theorem 8.4 contains themountain pass lemma Theorem 6.1 as a special case. In contrast to Theorem6.1, the above Theorem 8.4 also allows higher-dimensional comparison sets Q,similar to Theorem 6.3 in the symmetric case.


Applications to Semilinear Elliptic Equations

We demonstrate this with yet another variant of Theorem 6.2, allowing forlinear growth of g near u = 0. Conceivably, condition (1%) below may befurther weakened. However, we will not pursue this.

8.5 Theorem. Let 3 be a smooth, bounded domain in IRn, n % 3, and letg:3, IR # IR be measurable in x ! 3, di!erentiable in u ! IR with derivativegu, and with primitive G. Moreover, suppose that(1%) g(x, 0) = 0, and

g(x, u)u

% gu(x, 0), for almost every x ! 3, u ! IR ;

(2%) (p < 2nn(2 , C :

..gu(x, u).. & C

%1 + |u|p(2

&, for almost every x ! 3, u ! IR;

(3%) (q > 2, R0 : 0 < q G(x, u) & g(x, u)u , for almost every x ! 3, if |u| % R0.Then the problem

)5u = g(·, u) in 3(8.1)u = 0 on *3(8.2)

admits a solution u '* 0.

Proof. For u ! H1,20 (3) let

E(u) =12

#

+|+u|2 dx)

#

+G(x, u) dx .

By assumptions (2%), (3%) the functional E is of class C2 and as in the proof ofTheorem 6.2 satisfies (P.-S.) on H1,2

0 (3). It su!ces to show that E admits acritical point u '* 0. Note that by assumption (1%) problem (8.1), (8.2) admitsu * 0 as a trivial solution.

Let (k denote the eigenfunctions of the linearized equation

)5(k = gu(x, 0)(k + 4k(k

(k = 0in 3

on *3

with eigenvalues 41 < 42 & 43 & . . . . Denote k0 = min{k ; 4k > 0} and letV + = span{(k ; k % k0}, V ( = span{(1, . . . ,(k0(1}. Note that by (1%) wehave

G(x, u) =# u

0g(x, v) dv % 1

2gu(x, 0)u2 .

Hence there holds

(8.3) E(u) & 12D2E(0)(u, u) & 0, for u ! V ( ,

while by definition of V + clearly we have

8. Linking 129

D2E(0)(u, u) % 4ko.u.2L2 , for u ! V + .

To strengthen the latter inequality note that by (2%) the function gu(x, 0) isessentially bounded. Thus for su!ciently large k1 we have

D2E(0)(u, u) =#

0|+u|2 dx)

#

+gu(x, 0)u2 dx % 1

2.u.2

H1,20

uniformly for u ! span{(k ; k % k1}. Since the complement of this space inV + has finite dimension, we conclude that there exists 4 > 0 such that

D2E(0)(u, u) % 4.u.2H1,2

0,

uniformly for u ! V +. But E ! C2%H1,2

0 (3)&; it follows that for su!ciently

small ' > 0 we have

(8.4) infu!S+

%

E(u) % 12D2E(0)(u, u)) o

%.u.2

H1,20

&% 4'2

4> 0 ,

where S+% = {u ! V + ; .u.H1,2

0= '} and where o(s)/s # 0 (s # 0) .

By (3%) finally, as in the proof of Theorem 6.2 (see (6.4), (6.5)), for anyfinite dimensional subspace W we have

E(u) # )$, as .u. # $, u ! W .

Recalling (8.3), (8.4), we see that the assumptions of Theorem 8.4 are satisfiedwith S = S+

% and

Q = {u( + s(k0 ; u( ! V (, .u(.H1,20& R, 0 & s & R} ,

for su!ciently large R > 0. Thus E admits a critical point u with E(u) % 1%2

4 .The proof is complete.

Further applications of Theorem 8.4 to semilinear elliptic boundary value prob-lems are given in the survey notes by De Figueireido [1] or Rabinowitz [11; p. 25".]. In particular, Theorem 8.4 o"ers a simple and unified approach to asymp-totically linear equations, possibly “resonant” at u = 0 or at infinity, as inAhmad-Lazer-Paul [1], Amann [4], Amann-Zehnder [1]. (See for instance Ra-binowitz [11; Theorem 4.12], or Bartolo-Benci-Fortunato [1] and the referencescited therein. See also Chang [1; p. 708].)


Applications to Hamiltonian Systems

A more refined application of linking is given in the next theorem due to Hoferand Zehnder [1], prompted by the work of Viterbo [1]. Once again we deal withHamiltonian systems

(8.5) x = J+H(x) ,

where H is a given smooth Hamiltonian and J is the skew-symmetric matrix

J =,

0 )idid 0

-

on IR2n = IRn , IRn.

8.6 Theorem. Let H ! C2(IR2n; IR). Suppose that 1 is a regular value of Hand S = S1 = H(1(1) is compact and connected. Then for any " > 0 there isa number ! !]1 ) ", 1 + "[ such that S# = H(1(!) carries a periodic solutionof the Hamiltonian system (8.5).

(Note that by the implicit function theorem and compactness of S there existsa number "0 > 0 such that for any ! !]1) "0, 1 + "0[ the hypersurface S# is ofclass C2, compact, and di"eomorphic to S.)

Remarks. By Theorem 8.6, in order to obtain a periodic solution of (8.5) onthe fixed surface S = S1, it would su!ce to obtain a-priori bounds (in termsof the action integral) for periodic solutions to (8.5) on surfaces S# near S.Benci-Hofer-Rabinowitz [1] have shown that this is indeed possible, providedcertain geometric conditions are satisfied, including, for instance, the conditionthat S be strictly star-shaped with respect to the origin. However, also energysurfaces of Hamiltonians that are only convex either in the position or in themomentum variables are allowed.

The existence of periodic solutions to (8.5) on S likewise can be derivedif S is of “contact type”. This notion, introduced by Weinstein [3], allows oneto give an intrinsic interpretation of the existence results by Rabinowitz [5]and Weinstein [2] for closed trajectories of Hamiltonian systems on convex orstrictly star-shaped energy hypersurfaces. See Hofer-Zehnder [1], or Zehnder[2] for details.

Proof of Theorem 8.6. Observe that, as remarked earlier in the proof of Theo-rem I.6.5, the property that a level hypersurface S# = H(1(!) carries a periodicsolution of (8.5) is independent of the particular Hamiltonian H having S# asa level surface. We now use this freedom to redefine H suitably. For 0 < " < "0let U = H(1

%[1) ", 1 + "]

&? S , [)1, 1]. Then IR2n \ U has two components.

Indeed, by Alexander duality

8. Linking 131

H0(IR2n \ U ; ZZ) ? H2n(1(U ; ZZ) ? H

2n(1(S ; ZZ) ? ZZ ,

in Spanier’s [1] notation. Denote by A the unbounded component of IR2n \ Uand by B the bounded component. We may assume 0 ! B. Also let , =diam U > 0. Fix numbers r, b > 0 such that

, <r < 2,32-r2 <b < 2-r2

and choose a smooth function f : ] ) "0, "0[# IR such that f |]($0,($] = 0 ,f |[$,$0[ = b, and such that f )(s) > 0 for )" < s < ". Also let g: IR # IR be asmooth function satisfying

(8.6)g(s) = b for s & r, g(s) % 3

2-s2 for s > r, g(s) =

32-s2 for large s ,

and 0 < g)(s) & 3-s for s > r .

Then define

H(x) =

6@7

@8

0, if x ! B,f(s), if H(x) = 1 + s, )" & s & ",b, if x ! A, |x| & r,g(|x|), if |x| > r .

Note that for x ! IR2n we can estimate

(8.7) )b +32-|x|2 & H(x) & 3

2-|x|2 + b .

Now we look for solutions of (8.5) of period 1, with H replacing H. Suchsolutions uniquely correspond to the critical points of the functional

E(x) =12

# 1

0/x,J x0 dt)

# 1

0H(x) dt

on the space of 1-periodic (C1))maps x: IR # IR2n.By the following lemma we are able to distinguish a periodic solution on

a level hypersurface S#.

8.7 Lemma. Suppose x is a 1-periodic C1-solution of (8.5), with H replacingH, and which satisfies E(x) > 0. Then x(t) ! S# for some ! ! [1) ", 1 + "],for all t.

Proof. Since H % 0, any constant solution x of (8.5) satisfies E(x) & 0. Assumex is non-constant with |x(t0)| > r. Then g

%|x(t0)|

&= H%x(t0)&

= H%x(t)&

=g%|x(t)|

&also for t close to t0 and by (8.6) it follows that |x(t)| *| x(t0)| = s0.

In particular, x satisfies


x = J g)(s0) x

s0

and we compute

E(x) =12g)(s0) s0 ) H

%x(t0)&

& 32-s2

0 )32-s2

0 = 0 ,

by definition of H. The claim now follows.

In order to exhibit a 1-periodic solution x of (8.5) with E(x) > 0, a variationalargument involving linking will be applied.

Denote by V = H1/2,2(S1 ; IR2n) the Hilbert space of all 1-periodic func-tions

(8.8) x(t) =$

k!ZZ

exp (2-ktJ ) xk ! L2%[0, 1] ; IR2n& ,

with Fourier coe!cients xk ! IR2n satisfying

.x.2 = 2-

>$

k!ZZ

|k| |xk|2?

+ |x0|2 < $ .

The inner product on V is given by

(x, y)V := 2-

>$

k!ZZ

|k|/xk, yk0?

+ /x0, y00

inducing the norm . ·. above.Split V = V ( > V 0 > V + orthogonally, where

V ( = {x ! V ; xk = 0 for k % 0} ,

V 0 = {x ! V ; xk = 0 for k '= 0} ,

V + = {x ! V ; xk = 0 for k & 0}

with reference to the Fourier decomposition (8.8) of an element x ! V . Alsodenote P(, P 0 and P+ the corresponding projections. Thus, any x ! V maybe uniquely expressed x = x( + x0 + x+, where x( = P(x, etc.

Note that the self-adjoint operator

Lx = )J x

has eigenspacesVk = {exp(2-ktJ )xk ; xk ! IR2n}

with eigenvalues 2-k, k ! ZZ. Moreover, dimVk = 2n, for all k. Hence therelated quadratic form

8. Linking 133

A(x) =12

# 1

0/x,J x0 dt

67

8

< 0, for x ! V (,= 0, for x ! V 0,> 0, for x ! V +.

In fact, using the projections P±, A can be written

A(x) =12%()P( + P+)x, x

&V

.

Note that since H is of class C2 and behaves like |x|2 for large |x|, its mean

G(x) =# 1

0H(x) dt

is of class C2 on L2(S1; IR2n). Hence also E ! C2(H1/2,2(S1)). RestrictingG to V , from compactness of the embedding H1/2,2(S1) +# L2(S1) we alsodeduce that DG is a completely continuous map from V into its dual.

8.8 Lemma. There exist numbers & > 0, ' !]0, 1[ such that E(x) % & forx ! V +, .x. = '.

Proof. +H(0) = 0, +2H(0) = 0 implies that DG(0) = 0, D2G(0) = 0. HenceD2E(0) = )P( + P+. Since E ! C2, E(0) = 0, DE(0) = 0, the lemmafollows.

Now define

Q = {x = x( + x0 + s e ! V ; .x( + x0. & R, and 0 & s & R} ,

with R > 1 to be determined, and with a unit vector

e =182-

exp (2-tJ )a ! V +, |a| = 1 .

Denote *Q the relative boundary of Q in V ( > V 0 > IR · e; that is,

*Q =(x( + x0 + s e ! Q ; .x( + x0. = R or s ! {0, R}

).

8.9 Lemma. If R is su"ciently large, then E|/Q & 0.

Proof. Note that (8.7) implies that

G(x) =# 1

0H(x) dt % )b +

32-.x.2L2 .

Therefore, for x = x(+x0+se with s = R or .x(+x0.2 = .x(.2+.x0.2L2 =R, we obtain


E(x) =12%()P( + P+)x, x

&V)G(x)

& )12.x(.2 +

12s2 + b) 3

2-%.x(.2L2 + .x0.2L2 +

s2

2-&

& )12.x(.2 ) 1

4s2 ) 3

2-.x0.2L2 + b

& 0 ,

if R > 0 is su!ciently large. Moreover, H % 0 implies G % 0 and thereforeE & 0 on V ( > V 0, that is at s = 0, which concludes the proof.

Fix 0 < ' < R as in Lemmas 8.8, 8.9. Denote S+% = {x ! V + ; .x. = '}.

Define a class 6 of maps V # V as follows:

6 is the class of maps h ! C0(V ; V ) such that h is homotopic to the identitythrough a family of maps ht = Lt + Kt, 0 & t & T , where L0 = id, K0 = 0,and where for each t there holds:

Lt = (L(t , L0

t , L+t ): V ( > V 0 > V + # V ( > V 0 > V +

is a linear isomorphism preserving sub-spaces, Kt is compact and ht(*Q)1S+% =

-, for each t.

Following Benci-Rabinowitz [1], we now establish

8.10 Lemma. *Q and S+% link with respect to 6 .

Proof. Choose h ! 6 . We must show that h(Q)1 S+% '= -, or equivalently that

the equations

(8.9)%(P( + P 0) 6 h

&(x) = 0, .h(x). = '

are satisfied for some x ! Q.Using a degree argument as in Example 8.3, we establish (8.9) for every

ht in the family defining h ! 6 . Consider Q " V ( > V 0 > IR · e =: W andrepresent x = x( + x0 + se. Applying L(1

t to (8.9), these equations become

x( + x0+(P( + P 0)L(1t Kt(x) = 0 ,

.ht(x). = ' .

Note that kt := (P( + P 0)L(1t Kt: W # V ( > V 0 is compact. Now define a

map Tt: W # W by letting

Tt(x) = Tt(x( + x0 + se) = x( + x0 + kt(x) + .ht(x).e= x + kt(x) +

%.ht(x). ) s

&e .

8. Linking 135

Observe that Tt is of the form id+ compact. Moreover, the condition ht(*Q)1S+% = - translates into the condition Tt(x) '= 'e for x ! *Q. Hence the Leray-

Schauder degree (see for instance Deimling [1; 2.8.3, 2.8.4]) of Tt on Q withrespect to 'e is well-defined and, in fact,

deg(Tt, Q, 'e) = deg(T0, Q, 'e) = 1 ,

as desired.

Let +E denote the gradient of E with respect to the scalar product in V . Since+E has linear growth, the gradient flow %: V , [0,$[# V given by

*

*t%(x, t) = )+E

%%(x, t)

&

%(x, 0) = x

exists globally. Note that E is non-increasing along flow-lines with

(8.10)d

dtE%%(x, t)

&= )+++++E%%(x, t)

&++++2

.

8.11 Lemma. %(·, T ) ! 6 for any T % 0.

Proof. Clearly ht = %(·, t) for 0 & t & T is a homotopy of %(·, T ) to theidentity; moreover by (8.10)

E%%(x, t)

&& E(x) & 0 for x ! *Q .

By Lemmata 8.8 and 8.9 therefore ht(*Q) 1 S+% = - for all t. Finally, observe

that

(8.11) +E(x) = )x( + x+ ++G(x) ,

where +G is compact. Thus the desired form of %(x, t) may be read o" thevariation of constant formula

%(x, t) = etx( + x0 + e(tx+ +# t

0

%et(sP( + P 0 + e((t(s)P+

&+G%%(x, s)

&ds

=: L(t x( + L0

tx0 + L+

t x+ + Kt(x) .

It remains to verify the Palais-Smale condition for E.


8.12 Lemma. E satisfies (P.-S.) on V .

Proof. Let (xm) be a (P.-S.)sequence for E. By (8.11) and Proposition 2.2it su!ces to show that (xm) is bounded in V . Suppose by contradiction that.xm. # $. Normalize zm = xm

,xm, and note that

(8.12) ()P( + P+)zm ) +G(xm).xm.

# 0 .

We claim that the sequence ym := -G(xm),xm, is precompact. Indeed, by (8.12),

the sequence (ym) is bounded and we may assume that ym # y weakly inH1/2,2 and strongly in L2. Moreover, by (8.6), for 2 ! H1/2,2(S1) we mayestimate....

,+G(xm).xm.

,2

-

V

.... =

.....

# 1

0

/+H(xm),20.xm.

dt

..... & c

,1 +++++

xm

.xm.

++++L2

-.2.L2 .

Choosing 2 = 2m = ym ) y above, it then follows that ym # y strongly inH1/2,2 as m # $. Hence, from (8.12) it follows that zm # z in H1/2,2 andalmost everywhere in [0, 1]. Since +H grows linearly, passing to the limit inthe expression

A2

.xm., DE(xm)

B=# 1

0/)J zm,20 )

C+H%.xm.zm

&,2D

.xm.dt

is allowed by Vitali’s convergence theorem. From (8.6) we thus infer that zsatisfies the equation

z = 3-J z .

However, 3- does not belong to the spectrum of L = )J ddt . Hence z * 0.

But since zm # z, and .zm. = 1, this is impossible. Therefore the originalsequence (xm) must be bounded.

In order to conclude the proof of Theorem 8.6 we can now repeat the argumentof Theorem 8.4. Define

! = inft"0

supx!Q

E%%(x, t)

&.

From Lemmata 8.10 and 8.11 it follows that %(Q, t) 1 S+% '= - for all t % 0.

Hence ! % & > 0. Suppose by contradiction that ! is a regular value for E.By (P.-S.) there exists " > 0 such that

.+E(x). % "

for all x such that |E(x)) !| < ". Choose t0 > 0 such that

supx!Q

E%%(x, t0)

&< ! + " .

9. Parameter Dependence 137

Then by definition of !, choice of ", and (8.10), for t % t0 the number

supx!Q

E%%(x, t)

&

is achieved only at points where+++++E%%(x, t)

&++++ % ".

Hence by (8.10)

d

dt

,supx!Q

E%%(x, t)

&-& )"2 ,

which gives a contradiction to the definition of ! after time T % t0 + 1$ . The

proof is complete.

In the next section we will see that by a slight refinement of the above linkingargument one can obtain periodic solutions of (8.5) not only with energy arbi-trarily close to a given value !0, but, in fact, for almost every energy level in asuitable neighborhood of !0 – a vast abundance of periodic solutions!

9. Parameter Dependence

The existence result for periodic solutions of Hamiltonian systems given inthe previous section can be considerably improved, if combined with the ideaspresented in Section I.7.

In fact, we have the following result of Struwe [21].

9.1 Theorem. Let H ! C2(IR2n, IR) and suppose that 1 is a regular value of Hand the energy level surface S = S1 = {x; H(x) = 1} is compact and connected.Then there is a number "0 > 0 such that for almost every ! !] 1) "0, 1 + "0 [there exists a periodic solution x of (8.5) with H(x(t)) * !.

In fact, "0 > 0 may be any number as determined in the remark followingTheorem 8.6 such that for ! !] 1) "0, 1 + "0 [ the level surface S# = H(1(!)is C2-di"eomorphic to S.

For the proof of Theorem 9.1 we may modify the Hamiltonian H as in thepreceding section while preserving the level surfaces S# , |1)!| < "0. However,now we shift H, as follows. Fix any number !0 !] 1 ) "0, 1 + "0 [ and let0 < " <

%"0 ) |1) !0|

&/3. If su!ces to show that (8.5) has a periodic solution

with energy ! for almost every ! !] !0 ) " , !0 + " [=: I0.Let U = H(1

%[1 ) "0 + ", 1 + "0 ) "]

& 2= S , [)1, 1], IR2n \ U = A 5 B,, = diam U , , < r < 2,, 3

2-r2 < b < 2-r2 as in Section 8 above and choosesmooth functions f and g as in that section such that f(s) = 0 for s & )",f(s) = b for s % ", f )(s) > 0 for |s| < ", and where g satisfies (8.6). For ! ! I0,m ! IN, then define


H#,m(x) =

6@7

@8

0 , if x ! B ,f%m(H(x)) !)

&, if x ! U ,

b , if x ! A, |x| & r ,g%|x|&

, if |x| > r ,

satisfying (8.7), and for x ! V = H1/2,2(S1; IR2n) let

E#,m(x) =12

# 1

0/x, Jx0 dt)

# 1

0H#,m(x) dt

= A(x))G#,m(x) .

Then, as in Section 8, the functional E#,m is of class C2 and satisfies (P.-S.)onV . Moreover, critical points x ! V of E#,m with E#,m(x) > 0 correspond toperiodic solutions y of (8.5) with energy |H(y) ) !| < "/m; compare Lemma8.7. More precisely, we have

9.2 Lemma. Let x ! V be a critical point of E#,m with E#,m(x) > 0. ThenH%x(t)&* h ! IR with |h) !| < "/m, and letting

T (x) = mf ),

m%H(x)) !

&-> 0 ,

the function y(t) = x%t/T (x)

&is a T (x)-periodic solution of (8.5).

Proof. The first assertion follows just as in the proof of Lemma 8.9. The secondassertion is immediate from the equation

(9.1)x = J+H#,m(x) = mf )

,m%H(x)) !

&-J+H(x)

= T (x)J+H(x) .

Moreover, we have (compare Lemma 8.8):

9.3 Lemma. There exist numbers & > 0, ' > 0 such that E#,m(x) % & forx ! S+

% =(x ! V +; .x. = '

), uniformly in m ! IN, ! ! I0.

Proof. Let r0 > 0 be such that Br0(0; IR2n) " B. By Tchebychev’s inequality,we have

L1

,(t; |x(t)| > r0

)-& r(2

0

# 1

0|x(t)|2 dt & C.x.2

for x ! V . There exists a constant C0 such that

H#,m(x) & C0|x|2 for all x ,


uniformly in m ! IN, ! ! I0. Since V +# L4%[0, 1]; IR2n&, moreover, from

Holders inequality we obtain

G#,m(x) =# 1

0H#,m

%x(t)&

dt & C0

#

{t;|x(t)|"r0

) |x(t)|2 dt

& C0L1

,(t; |x(t)| % r0

)-1/2%# 1

0|x(t)|4 dt

&1/2 & C.x.3 .

On the other hand, for x ! V + there holds

A(x) =12.x.2 ,

and the claim follows.

In addition, there exists a uniform number R > ' such that

E#,m|/Q & 0

for all ! ! I0, m ! IN, where Q is defined as in Section 8; compare Lemma 8.9.Finally, the gradient flow %#,m(·, t) for E#,m for any t % 0 is a member of

6 , for each ! ! I0, m ! IN.Hence, as in Section 8, for each ! ! I0, m ! IN there exists a periodic

solution x#,m of (9.1) with

E#,m(x#,m) = infh!.

supx!Q

E#,m

%h(x)&

=: :m(!).

We now use variations of ! to obtain a-priori bounds on T (x#,m) that allowpassing to the limit m # $ in (9.1) for a suitable sub-sequence and suitablevalues of !.

Observe that, since f ) % 0, for any fixed x ! V , m ! IN the map ! 3#E#,m(x) is non-decreasing in ! ! I0; in fact, for any x ! V we have

*

*!E#,m(x) = m

# 1

0f ),

m%H(x(t))) !

&-dt .

In particular, if x ! V is critical for E#,m with E#,m(x) > 0, by Lemma 9.2 thisyields

*

*!E#,m(x) = T (x) .

As a consequence of the monotonicity of E#,m, clearly also the map ! 3#:m(!) is non-decreasing for any m ! IN. Hence this map is almost everywheredi"erentiable for any m with /

/# :m % 0, and there holds

# $

($

*

*!:md! & sup

m,#:m(!)) & & sup

m,#

*supx!Q

E#,m(x);) & = C < $ ,


uniformly in m ! IN, with & as determined in Lemma 9.3.But then also

lim infm#$

% **!

:m

&! L1%[)", "]

&,

and by Fatou’s lemma we obtain# $

($lim infm#$

% **!

:m

&d! & lim inf

m#$

# $

($

*

*!:m & C .

In particular, lim infm#$//# :m(!) < $ for almost every ! ! I0. Fix such a !

and let 9 " IN be a subsequence such that

*

*!:m(!) # lim inf

m#$

*

*!:m(!) = C# < $

as m #$, m ! 9. We may assume //# :m(!) & C# + 1 for all m ! 9.

We claim:

9.4 Lemma. For any m ! 9 there exists a critical point xm of E#,m such thatE#,m(xm) = :m(!) and T (xm) = /

/#E#,m(xm) & C# + 4.

Proof: Choose a sequence !k @ !. We claim there exists a (P.-S.)-sequence(xk

m) for E#,m such that +E#,m(xkm) # 0 as k #$ and such that

(9.2):m(!)) 2(!k ) !) & E#,m(xk

m) & E#k,m(xkm)

& :m(!k) + (!k ) !) & :m(!) + (C# + 2)(!k ) !)

for large k. This will imply the assertion of the lemma: By (P.-S.), a sub-sequence (xk

m) as k # $ will accumulate at a critical point xm of E#,m,satisfying E#,m(xm) = :m(!) and

C# + 4 % lim infk#$

E#k,m(xkm))E#,m(xk

m)!k ) !

= lim infk#$

# 1

0

# #k

#

mf )%m(H(xkm)) !))

&

!k ) !d!) dt

=# 1

0mf )%m(H(xm)) !)

&dt = T (xm) .

Negating the above claim, suppose there is no (P.-S.)-sequence for E#,m sat-isfying (9.2). Then there exists # > 0 and k0 ! IN such that for all x ! Vsatisfying (9.2) for some k % k0 there holds

(9.3) .+E#,m(x).2 % 2# .

We may assume k0 = 1. Let 0 & ( & 1 be a Lipschitz continuous function suchthat ((s) = 0 for s & 0, ((s) = 1 for s % 1 and define a family of vector fieldsek by letting


ek(x) = )1%

1) (k(x)&+E#k,m(x) + (k(x)+E#,m(x)

2

= x( ) x+ +1%

1) (k(x)&+G#k,m(x) + (k(x)+G#,m(x)

2,

where

(k(x) = (

>E#,m(x))

%:m(!)) 2(!k ) !)

&

!k ) !

?

.

Let %k: V , [ 0,$[ # V be the corresponding flows, satisfying

*

*t%k(x, t) = ek

%%k(x, t)

&,

%k(x, 0) = x

for all x ! V, t % 0. We claim that %k(·, t) ! 6 for any t % 0.Indeed, we may assume :m(!) % 2(!k)!) for all k. Thus, if E#k,m(x) & 0

for some x ! V , it follows that (k(x) = 0, ek(x) = )+E#k,m(x), and henceE#k,m

%%k(x, t)

&& 0 for all t % 0. Since this applies, in particular, to any point

x ! *Q, it follows that%k(*Q, t) 1 S+

% = -

for all t % 0. The remaining properties defining the class 6 are verified as inthe proof of Lemma 8.11.

Choose h ! 6 such that

(9.4) supx!Q

E#k,m

%h(x)&& :m(!k) + (!k ) !)

and consider a point x ! h(Q). If

E#,m(x) < :m(!)) 2(!k ) !) ,

by definition of ek we have ek(x) = )+E#k,m(x) and hence%ek(x),+E#k,m(x)

&V& 0

for such x. If, on the other hand,

E#,m(x) % :m(!)) 2(!k ) !) ,

by assumption (9.4) estimate (9.2) is verified for x.Observe that

%ek(x),+E#k,m(x)

&V& )(k(x)

%+E#,m(x),+E#k,m(x)

&V

.

But, by (9.3), for points x satisfying (9.2) we can estimate


%+E#,m(x),+E#k,m(x)

&V

= .+E#,m(x).2 )%+E#,m(x),+G#k,m(x))+G#,m(x)

&V

% 12.+E#,m(x).2 ) 1

2.+G#k,m(x))+G#,m(x).2

% #) C

#

{t;|x(t)|&r}|+H#k,m(x))+H#,m(x)|2 dt

% #) o(1) ,

where o(1) # 0 as k # $ by dominated convergence, uniformly in x. Hence,for large k, for any x ! h(Q) we have

d

dtE#k,m

%%k(x, t)

&|t=0 =

%ek(x),+E#k,m(x)

&V& 0 .

In particular, letting ht = %k(·, t) 6 h for t % 0, estimate (9.4) holds for anyht, t % 0. Moreover, ht ! 6 for any t % 0. Hence, by definition of :m(!), forany t % 0 we have

(9.5) M(t) = supx!Q

E#,m

%ht(x)

&% :m(!) .

Together with (9.4), this implies that M(t) is achieved only at points x ! ht(Q)satisfying (9.2). Moreover, (k(x) = 1 at such points and hence

%ek(x),+E#,m(x)

&V

= ).E#,m(x).2 & )2#

by (9.3).Thus, M(t) is strictly decreasing with

d

dtM(t) & )2# < 0 ,

contradicting (9.5) for su!ciently large t > 0. This completes the proof.

Proof of Theorem 9.1: For ! ! I0 with

lim infm#$

*

*!:m(!) = C# < $

let 9 and (xm)m!0 be as in Lemma 9.4. For any m the function xm solves(9.1) with T (xm) & C# + 4 and satisfies

|H(xm(t))) !| <"

m.

Hence the sequences (xm), (xm) are equi-bounded and equi-continuous;By the theorem of Arzela-Ascoli, therefore, a subsequence converges C1-

uniformly to a 1-periodic solution x of

x = TJ+H(x)

10. Critical Points of Mountain Pass Type 143

for some T % 0, with H%x(t)&* !. Moreover, A(xm) % E#,m(xm) % &; hence

A(x) % & > 0. In particular, x is non-constant, T > 0, and y(t) = x(t/T ) is aT -periodic solution of (8.5) on S#.

9.5 Notes. (1%) Recently, Ginzburg [1] and Herman [1] have given an exampleof a smooth Hamiltonian H, possessing a smooth, compact energy level surfacecarrying no periodic orbit. Theorem 9.1 therefore is best possible. A relatedresult is due to Kuperberg [1].(2%) Further applications of the above method are given in Struwe [16], [17],Ambrosetti-Struwe [2]. However, an abstract statement would be quite cum-bersome and in each instance, features that are particular to the given problemcome into play. Common to all the above applications is a family of functionalsE# ! C1(V ) depending monotonically on ! ! IR. A key technical point is thata pseudo-gradient flow for E# should be a pseudo-gradient flow also for E#k

near points x ! V such that |E#k(x))E#(x)| & C|!k ) !|, when |!k ) !|A 1.

10. Critical Points of Mountain Pass Type

Di"erent critical points of functionals E sometimes can be distinguished bythe topological type of their neighborhoods in the sub-level sets of E. Thisis the original idea of M. Morse which led to the development of what is nowcalled Morse theory; see the Introduction. Hofer [2] has observed that suchinformation about the topological type in some cases is available already fromthe minimax characterization of the corresponding critical value.

10.1 Definition. Let E ! C1(V ), and suppose u is a critical point of E withE(u) = !. We say u is of mountain pass type if for any neighborhood N of uthe set N 1 E# is disconnected.

With this notion available we can strengthen the assertion of Theorem 6.1 asfollows. For convenience we recall

6.1 Theorem. Suppose E ! C1(V ) satisfies (P.-S.). Suppose

(1!) E(0) = 0 .

(2!) "! > 0, " > 0 : #u# = ! $ E(u) % ".

(3!) "u1 ! V : #u1# % ! and E(u1) < ".

Define

# =(

p ! C0 ([0, 1]; V ) ; p(0) = 0, p(1) = u1

).

Then$ = inf

p"!supu"p

E(u)

is a critical value.

Now we assert:


10.2 Theorem. Under the hypotheses of Theorem 6.1 the following holds:(1%) either E admits a relative minimizer u '= 0 with E(u) = !, or(2%) E admits a critical point u of mountain pass type with E(u) = !.

Remark. Simple examples on IR show that in general also case (1%) occurs;see Figure 10.1.

Fig. 10.1. A function possessing no critical point of mountain pass type

Another variant of Theorem 6.1 is related to results of Chang [4] andPucci-Serrin [1], [2].

10.3 Theorem. Suppose E ! C1(V ) satisfies (P.-S.). Suppose 0 is a relativeminimizer of E with E(0) = 0, and suppose that E admits a second relativeminimizer u1 '= 0. Let 6 and ! be defined as in Theorem 6.1. Then,(1%) either there exists a critical point u ! K# which is not of minimum type,or(2%) the origin and u1 can be connected in any neighborhood of the set of relativeminimizers u of E with E(u) = 0. Necessarily then, ! = E(u1) = E(0) = 0.

The proofs of these results are quite similar in spirit.

Proof of Theorem 10.3. Let 6 and ! be defined as in Theorem 6.1. Supposethat K# consists entirely of relative minimizers of E. Then for each u ! K#there exists a neighborhood N(u) such that

E(u) = ! & E(v) for all v ! N(u) .

Let N0 =9

u!K!N(u), and for any neighborhood N of K# let # > 0 and % be

determined according to Theorem 3.4 for # = 1 and N = N0 1 N . Choosingp ! 6 such that p([0, 1]) " E#+!, by (1%),(3%) of Theorem 3.4 the path p) =%(·, 1) 6 p ! 6 satisfies

p)([0, 1]) " %(E#+!, 1) " N 5E#(! " N0 5 E#(! .


But N0 and E#(! by construction are disjoint, hence disconnected. Thus eitherp)([0, 1]) " N or p)([0, 1]) " E#(!. Since the latter contradicts the definition of! we conclude that p)([0, 1]) " N " N , whence p(0) = 0 and p(1) = u1 can beconnected in any neighborhood N of K# , as claimed. In particular, 0 ! K#,u1 ! K#, whence ! = E(0) = 0 and also E(u1) = 0.

Proof of Theorem 10.2. Suppose, by contradiction, that K# contains no rela-tive minimizers nor critical points of mountain pass type. Then any u ! K#possesses a neighborhood N(u) such that N(u)1E# is non-empty and (path-)connected. Moreover, K# " E# . Now we have the following topological lemma;see Hofer [3; Lemma 1]:

10.4 Lemma. Let (M, d) be a metric space and let K and 9 be non-emptysubsets of M such that K is compact, 9 is open, and K " 9, the closure of9. Suppose {N(u) ; u ! K} is an open cover of K such that u ! N(u) andN(u) 1 9 is connected for each u ! K.Then there exists a finite, disjoint open cover {U1, . . . , UL} of K such thatUl 1 9 for each l is contained in a connected component of 9.

Proof. Choose " > 0 such that for any u ! K we have

(10.1) B$(u) " N(u) for some u ! K .

For instance, we may choose a finite subcover {N(ui) ; 1 & i & I} of{N(u) ; u ! K} and let

" = minu!K

max1&i&I

(dist(u, M \ N(ui))

)> 0 .

Define an equivalence relation '2 on K by letting

u'2 u ;

There exist a number m ! IN and points ui ! K, 0 & i &m + 1, such that u0 = u, um+1 = u and d(ui, ui+1) < " for0 & i & m.

Since K is compact there are only finitely many equivalence classes, sayK1, . . . , KL. Let

Ul =(x ! M ; dist(x, Kl) <

"

4), 1 & l & L .

Then it is immediate that Uk 1Ul = - if k '= l and9L

l=1 Ul 7 K. It remains toshow that each U'

l = Ul 1 9 is contained in a connected component of 9.Define another equivalence relation 2 on 9 by letting

v 2 w ; v and w belong to the same connectedcomponent of 9.


Fix l and let v, w ! Ul 1 9. We wish to show that v 2 w. By definition of Ul

and Kl there exists a finite chain ui ! Kl, 0 & i & m + 1, such that

d(v, u0) <"

4, d(w, um+1) <

"

4,

andd(ui, ui+1) < " , for 0 & i & m .

Set # = " )max1&i&m d(ui, ui+1) > 0. Since K " 9, for each 0 & i & m + 1there exists vi ! 9 such that d(ui, vi) < !

2 < ", whence also

d(vi, vi+1) < " for 0 & i & m .

But now by (10.1) we have

v, v0 !%B$(u0) 1 9

&"%N(u0) 1 9

&

for some u0 ! K, and our assumption about the cover {N(u) ; u ! K} impliesthat v 2 v0. Similarly, w 2 vm+1. Finally, for 0 & i & m we have

d(ui, vi+1) & d(ui, ui+1) + d(ui+1, vi+1) < " ,

whencevi, vi+1 !

%B$(ui) 1 9

&"%N(ui) 1 9

&,

and vi 2 vi+1 for all i = 0, . . . , m. In conclusion

v 2 v0 2 v1 2 . . . 2 vm+1 2 w ,

and the proof is complete

Proof of Theorem 10.2 (completed). Let {U1, . . . , UL} be a disjoint open coverof K# as in Lemma 10.4 and set N =

9Ll=1 Ul. Choose # = & > 0 and let

# > 0, % be determined according to Theorem 3.4, corresponding to !, #,and N . Let p ! 6 satisfy p([0, 1]) " E#+!. Then p) = %(·, 1) 6 p ! 6 andp)([0, 1]) " %(E#+!, 1) " E#(! 5 N = E#(! 5 U1 5 . . . 5 UL. Smoothing p) ifnecessary, we may assume p) ! C1. Choose , !]!)#, ![ such that , is a regularvalue of E 6 p).

Let 0 < t1 < t2 < . . . < t2k(1 < t2k < 1 denote the successive pre-imagesof , under E 6p), and let Ij = [t2j(1, t2j], 1 & j & k. Note that p)|Ij "

9Ll=1 Ul.

Since the latter is a union of disjoint sets, for any j there is l ! {1, . . . , L} suchthat p)(Ij) " Ul. But E 6 p)(*Ij) = , < ! and Ul 1E# is connected. Hence wemay replace p)|Ij by a path p: Ij # Ul 1 E# with endpoints p|/Ij = p|/Ij , forany j = 1, . . . k. In this way we obtain a path p ! 6 such that supu!p E(u) < !,which yields the desired contradiction.


Multiple Solutions of Coercive Elliptic Problems

We apply these results to a semilinear elliptic problem. Let 3 be a boundeddomain in IRn, g: IR # IR a continuous function. For 4 ! IR consider theproblem

)5u = 4u) g(u) in 3 ,(10.2)u = 0 on *3 ,(10.3)

Let 0 < 41 < 42 & 43 . . . denote the eigenvalues of )5 with homogeneousDirichlet boundary condition. Then we may assert

10.5 Theorem. Suppose g is locally Lipschitz with g(0) = 0 and assume thatthe map u 3# g(u)

|u| is non-decreasing with

(1%) limu#0g(u)

u = 0, and(2%) lim|u|#$

g(u)u = $.

Then for any 4 > 42 problem (10.2), (10.3) admits at least three distinct non-trivial solutions.

Note that by assumption (1%) the problem (10.2), (10.3) for any 4 ! IR admitsu * 0 as (trivial) solution. Moreover, by (2%) , the functional E related to(10.2), (10.3) is coercive. The latter stands in contrast with the examplesstudied earlier in this chapter. As a consequence, the existence of multiplesolutions for problem (10.2), (10.3) heavily depends on the behavior of thefunctional E near u = 0, governed by the parameter 4, whereas in previousexamples the behavior near $ had been responsible for the nice multiplicityresults obtained.

Theorem 10.5 is due to Ambrosetti-Mancini [1] and Struwe [5]. Later,Ambrosetti-Lupo [1] were able to simplify the argument significantly, and weshall basically follow their approach in the proof below. See also Rabinowitz[11; Theorem 2.42], Chang [1; Theorem 3], and Hofer [1] for related results.

If g is odd, then for 4k < 4 & 4k+1 problem (10.2), (10.3) possessesat least k pairs of distinct non-trivial solutions, see for example Ambrosetti[1]. (This can also be deduced from a variant of Theorem 5.7 above, appliedto the sub-level set M = E0 of the functional E related to (10.2), (10.3).Note that M is forwardly invariant under the pseudo-gradient flow for E andhence the trajectories of this flow are complete in forward time. Moreover, for4k < 4 & 4k+1 by (1%) it follows that the genus ,(M) % k.)

Without any symmetry assumption optimal multiplicity results for (10.2),(10.3) are not known. However, results of Dancer [2] suggest that in generaleven for large 4 one can expect no more than four non-trivial solutions.

Proof of Theorem 10.5. Let u± = min{u > 0 ; g(±u)±u % 4}. Then we may

replace g by the truncated function

g(u) =*

g(u), )u( & u & u+

4u, u < )u( or u+ < u.


Observe that if u satisfies

)5u = 4u) g(u) =*4u) g(u), if u ! [)u(, u+]0, else

then by the weak maximum principle in fact u satisfies (10.2), (10.3).Thus we may assume that |g(x, u)) 4u| & c and

%4u) g(u)

&u % 0 for all

u ! IR. Hence the functional

E(u) =12

#

+

%|+u|2 ) 4|u|2

&dx +

#

+G(x, u) dx

is well-defined, and E ! C1%H1,2

0 (3)&. Moreover, (P.-S.) is satisfied. In fact,

E is coercive, and, if we identify H1,20 (3) with its dual via the inner product,

DE is of the form id + compact. Therefore, (P.-S.) follows from Proposition2.2.

In a first step we now want to exhibit a positive solution u of (10.2), (10.3),and a negative solution u, respectively. This can be done in various ways byusing the methods outlined in the previous chapters. For instance, we mighttruncate the nonlinearity 4u ) g(u) below or above 0 and proceed as in theproof of Theorem 6.2. However, we can also use the trivial solution u = 0 asa sub-(super-) solution to problem (10.2), (10.3) and minimize the functionalE in the cone of non-negative (non-positive) functions, as we did earlier inSections I.2.3, I.2.4. We choose this latter approach.

Let u, u minimize E in M = {u ! H1,20 (3) ; u % 0}, respectively M =

{u ! H1,20 (3) ; u & 0}. Then, since 0 is a trivial solution to (10.2), (10.3),

by Theorem I.2.4 the functions u, u also solve (10.2), (10.3). In particular,u, u ! C2,"(3) for some & > 0; see Appendix B. Moreover, if 4 > 41 and if(1 > 0, .(1.L2 = 1, denotes a normalized eigenfunction corresponding to 41,we have

E(#(1) =12(41 ) 4)#2 ) o(#2) ,

where o(s)/s # 0 as s # 0, and this is < 0 for su!ciently small |#| '= 0;hence u, u '* 0, and in fact, by the strong maximum principle (Theorem B.4 ofAppendix B), the functions u, u cannot have an interior zero.

We claim: u and u are relative minimizers of E in H1,20 (3). It su!ces to show

that for su!ciently small ' > 0 we have

E(u) & E(u) for all u ! M% := {u ! H1,20 (3) ; .(u) u)(.H1,2

0& '} ,

where (s)± = ±max{±s, 0}. Fix ' > 0 and let u% ! M% be a minimizer of Ein M%. Then

/v, DE(u%)0 % 0

for all v % 0; that is, u% is a (weak) supersolution to (10.2), (10.3), satisfying

)5u% % 4u% ) g(u%) .


Choosing v = )(u%)( % 0 as testing function, we obtain#

+|+u%& |2 ) 4|u%& |2 dx +

#

+g(u%&)u%& dx & 0 .

But by (1%) there is a constant c % 0 such that g(u) u % )c|u|2 for all u; thus#

+|+(u%&)|2 ) (4 + c)|u%& |2 dx & 0 .

Let 3%& = {x ! 3 ; u% < 0}. Then, since u > 0, we have Ln(3%&) # 0 as'# 0. Now by Holder’s and Sobolev’s inequalities we can estimate

#

+|u|2 dx &

%Ln(3)

& 2n

,#

+|u|

2nn&2 dx

-n&2n

& C(n)%Ln(3)

& 2n

#

+|+u|2 dx

for all u ! H1,20 (3), and for any 3 " 3. Thus,

4%& := inf*:

+ |+u|2 dx:+ |u|2 dx

; u ! H1,20 (3%&) \ {0}

;#$ as '# 0 ,

and for ' > 0 su!ciently small such that 4%& > (4 + c) we obtain u%& * 0;that is, u% % 0. Hence E(u%) % E(u), and u is a relative minimizer of E inM%, whence also in H1,2

0 (3). The same conclusion is valid for u.Now let

6 =4

p ! C01[0, 1]; H1,2

0 (3)2

; p(0) = u, p(1) = u5

and denote! = inf

p!.supu!%

E(u) .

If ! '= 0 we are done, because Theorem 10.3 guarantees either the existence ofinfinitely many relative minimizers or the existence of a critical point u ! K#not of minimum type and thus distinct from u, u. Since ! '= 0, this point umust also be distinct from the trivial solution u = 0.

If ! = 0, Theorem 10.3 may yield the third critical point u = 0. However;by Theorem 10.2 there exists a critical point u with E(u) = ! of minimum ormountain pass type. But for 4 > 42, by assumption (1%) and since the mapu 3# g(u)

|u| is non-decreasing, the set E0 where E is negative is connected; thus,u = 0 is not of one of these types, and the proof is complete.

Remark. Since in the case ! = 0 we have to show only that K# '= {0}, insteadof appealing to Theorem 10.2 it would su!ce to show that, if u = 0 were theonly critical point besides u and u, then 0 can be avoided by a path joining uwith u without increasing energy. For 4 > 42 this is easily shown by a directconstruction in the spirit of Theorem 10.2.


Notes 10.6. (1%) In the context of Hamiltonian systems (8.5), Ekeland-Hofer[1] have applied Theorem 10.2 to obtain the existence of periodic solutions withprescribed minimal period for certain convex Hamiltonian functions H; see alsoGirardi-Matzeu [1], [2]. Similar applications to semilinear wave equations asconsidered in Section I.6.6 have recently been given by Salvatore [1].(2%) Generalizations of Theorem 10.2 to higher-dimensional minimax methodslike Theorem 6.3 have been obtained by Bahri-Lions [1], Lazer-Solimini [1],and Viterbo [2]. See Remark 7.3 for an application of these results. Recently,Ghoussoub [1], [2] has presented a unified approach to results in the spirit ofTheorems 10.2 and 10.3.

11. Non-di!erentiable Functionals

Sometimes a functional E: V # IR5{±$} may fail to be Frechet-di"erentiableon V but may only be Gateaux-di"erentiable on its domain

Dom(E) = {u ! V ; E(u) < $}

in direction of a dense space of “testing functions” T " V .

11.1 Nonlinear scalar field equations: The zero mass case. As an example weconsider the problem

(11.1) )5u = g(u) in IRn, n % 3 ,

with the asymptotic boundary condition

(11.2) u(x) # 0 (|x|#$ ) .

The associated energy integral is

(11.3) E(u) =12

#

IRn

|+u|2 dx)#

IRn

G(u) dx ,

where, as usual, we denote

G(u) =# u

0g(v) dv

a primitive of g. In the case of “positive mass”, that is

lim supu#0

g(u)/u & )m < 0 ,

similar to the problem studied in Section I.4.1, problem (11.1), (11.2) can bedealt with as a variational problem in H1,2(IRn), where E is di"erentiable;see Berestycki-Lions [1]. In contrast to Section I.4.1, however, now we do notexclude the “0-mass case”

11. Non-di!erentiable Functionals 151

g(u)u

# 0 (u # 0) .

Then a natural space on which to study E is the space D1,2(IRn); that is, theclosure of C$

0 (IRn) in the norm

.u.2D1,2 =#

IRn

|+u|2 dx .

By Sobolev’s inequality, an equivalent characterization is

D1,2(IRn) =4u ! L

2nn&2 (IRn) ; +u ! L2(IRn)

5.

Note that, unless G satisfies the condition

|G(u)| & c|u|2n

n&2 ,

the functional E may be infinite on a dense set of points in this space and hencecannot be Frechet-di"erentiable on D1,2(IRn). To overcome this di!culty, inStruwe [6], [7] a variant of the Palais-Smale condition was introduced and acritical point theory was developed, giving rise to existence results for saddlepoints for a broad class of functionals where standard methods fail. Below, wegive an outline of the abstract scheme of this method; then we apply it to ourmodel problem (11.1), (11.2) above to obtain a simple proof for the followingresult of Berestycki and Lions [1], [2].

11.2 Theorem. Suppose g is continuous with primitive G(u) =: u0 g(v) dv and

satisfies the conditions(1%) )$ & lim supu#0 g(u)u

E|u|

2nn&2 & 0 ,

(2%) )$ & lim sup|u|#$ g(u)uE|u|

2nn&2 & 0 ,

and suppose there exists a constant 81 such that(3%) G(81) > 0 .Moreover, assume that g is odd, that is,(4%) g()u) = )g(u) .Then there exist infinitely many radially symmetric solutions ul ! D1,2(IRn) of(11.1), (11.2) and E(ul) #$ as l #$.

Remark. Observe that by the maximum principle we may replace g by thefunction g, given by

g(u) =

67

8

g(;'), if u > ;'

g(u), if |u| & ;'

g();'), if u < );'

where ;' = inf{; % ;1 ; g(;) & 0} & +$. Indeed, if u solves (11.1), (11.2)for g, by the maximum principle u solves (11.1), (11.2) for g. Hence we mayassume that instead of (2%) g satisfies the stronger hypothesis


(5%) lim|u|#$ g(u)uE|u|

2nn&2 = 0

together with the assumption(6%) (;2 > 0 : G(;) > 0 for all ;, |;| > ;2.

For the proof of Theorem 11.2 we now follow Struwe [6].

11.3 The abstract scheme. Suppose that

E: Dom(E) " V # IR

is a densely defined functional on a Banach space V with norm . ·. . Moreover,assume there is a family (TL)L!IN of Banach spaces

T1 " . . . " TL " TL+1 " . . . " V

with norms . ·. L such that

.u. & . . . & .u.L+1 & .u.L for u ! TL ,

and whose union is dense in V :

(11.4) T :="

L!IN

TLdense" V .

(By default, all topological statements refer to the norm-topology of V .) Alsosuppose that for any u ! Dom(E) the restricted functional E|{u}+TL

! C1(TL),for any L ! IN, and the partial derivative

DLE: Dom(E) : u 3# DLE(u) ! T 'L

is continuous in the topology of V for any L ! IN.We define u ! Dom(E) " V to be critical if DLE(u) = 0 for all L ! IN,

and we denote

K# = {u ! Dom(E) ; E(u) = !, DLE(u) = 0, 4L ! IN}

the set of critical points with energy !.Suppose that E satisfies the following variant of the Palais-Smale condi-

tion:

F(P.-S.)Any sequence (um) in Dom(E) such that E(um) # !,while DLE(um) # 0 in T '

L (m # $), for any L ! IN,possesses an accumulation point in K# .

Note the following:


11.4 Lemma. Suppose V satisfies (11.4) and E: Dom(E) " V # IR satisfiesF(P.-S.). Then for any ! ! IR the set K# is compact and any neighborhood N

of K# contains a member of the family

N#,L = {u ! Dom(E) ; |E(u)) !| < 1/L, .DLE(u).'L < 1/L}, L > 0 .

Moreover, the systemU#,% = {u ! V ; (v ! K# : .u) v. < '}

is a fundamental system of neighborhoods of K#.

Proof. By F(P.-S.) any sequence (um) in K# is relatively compact and accu-mulates at some point u ! K#. To prove the second assertion, suppose bycontradiction that for some neighborhood N of K# and any L ! IN there is apoint uL ! N#,L \ N . Consider the sequence (uL). Since for any L) we have

.DL!E(uL).'L! & .DLE(uL).'L & 1/L # 0

as L #$, L % L), from F(P.-S.) we conclude that (uL) accumulates at a pointu ! K#, contrary to assumption. The proof for U#,% is similar.

DenoteRegL(E) = {u ! Dom(E) ; DLE(u) '= 0}

the set of regular points of E with respect to variations in TL. Then exactly asin Lemma 3.2, using continuity of the partial derivative DLE, we can constructa locally Lipschitz continuous pseudo-gradient vector field vL: RegL(E) # TL

for E satisfying the conditions

.vL(u). < 2 min(1, .DLE(u).'L

),

/vL(u), DLE(u)0 > min(1, .DLE(u).'L

).DLE(u).'L .

Now the deformation lemma Theorem 3.4 can be carried over easily. Given! ! IR, N 7 K# we determine L ! IN, ' > 0 such that

N 7 U#,% 7 U#,%/2 7 N#,L .

Choose a locally Lipschitz continuous function $, 0 & $ & 1, such that $ = 0if .DLE(u).'L & 1

2L and such that $(u) = 1 if .DLE(u).'L % 1L . Thus, in

particular, we have $(u) = 1 for u '! N#,L satisfying |E(u)) !| & 1L . Let %L

be the flow corresponding to the vector field eL(u) = )$(u)vL(u), defined bysolving the initial value problem

*

*t%L(u, t) = eL

%%L(u, t)

&

%L(u, 0) = u


for u ! Dom(E), t % 0. By local Lipschitz continuity and uniform boundednessof eL, the flow %L exists globally on Dom(E), [0,$[, is continuous, and fixescritical points of E. Moreover, E is non-increasing along flow-lines, and wehave

%L(E#+!, 1) " E#(! 5N ,

respectively%L(E#+! \ N, 1) " E#(! ,

where # = min(

1L , %

4L2

); see the proof of Theorem 3.4.

Note that we do not require E to be continuous on its domain. Thus,we have to be careful with truncating the vector field eL outside some energyrange. However, with this crude tool already, many of our abstract existenceresults may be carried over.

Proof of Theorem 11.2. Let us now implement the above scheme with E givenby (11.3) on

V = {u ! D1,2(IRn) ; u(x) = u(|x|)} =: D1,2rad(IR

n) ,

with norm . ·. = . ·.D1,2 . (We focus on radially symmetric solutions to removetranslation invariance.) Also let

TL =(u ! D1,2

rad(IRn) ; u(x) = 0 for |x| % L)

,

with norm . ·.L = . ·. = . ·.D1,2 , L ! IN. Note that in this way T =9

L!IN TL

simply consists of all functions u ! D1,2rad(IR

n) with compact support. Sincevariations in TL for any L only involve the evaluation of g, respectively G on acompact domain BL(0), it is clear that E(u+ ·) is Frechet-di"erentiable in TL,for any u ! Dom(E), any L ! IN. Moreover, the di"erential DLE: Dom(E) #T '

L is continuous in the topology of V for any L ! IN.Note that by radial symmetry any function u ! D1,2

rad(IRn) is representedby a function (indiscriminately denoted by u), which is continuous on IRn\{0}.Moreover, by Holders’s inequality there holds

(11.6)

|u(x)|2 &# $

|x|

d

dr|u(r)|2 dr

& 2

># $

|x|r(2(n)n(1 dr

? 1n># $

|x||u(r)|

2nn&2 rn(1 dr

?n&22n

,># $

|x|| d

dru(r)|2rn(1 dr

? 12

& C |x|(2(n).u.2 ,

for any u ! D1,2rad(IR

n).


Decompose g = g+ ) g(, G = G+ ) G( with g±(u)u = max{±g(u)u, 0}and G±(u) =

: u0 g±(v) dv. We assert that, if um / u weakly in D1,2

rad(IRn),

then

(11.7)#

IRn

G+(u) dx = limm#$

#

IRn

G+(um) dx .

Indeed, by (11.6), for any " > 0 there exists R > 0 such that |um(x)| & " for|x| % R. Moreover, by assumption (1%) , for any # > 0 there is " > 0 such that

G+(um) & #|um|2n

n&2 ,

if |u| & ". Hence for large R we can estimate

#

IRn\BR(0)G+(um) dx & #

#

IRn\BR(0)|um|

2nn&2 dx

& C # .um.2n

n&2 & C # ,

uniformly in m.On the other hand, by assumption (2%) , for any # > 0 there is a constant

C(#) such that for all u ! IR there holds

G+(u) & # |u|2n

n&2 + C(#) .

Hence for 3 " IRn with su!ciently small measure Ln(3) < !C(!) , we have

#

+G+(um) dx & #

#

+|u|

2nn&2 dx + C(#)Ln(3) & C # ,

uniformly in m, and (11.7) follows by Vitali’s convergence theorem.Since G( % 0, by Fatou’s lemma of course also

#

IRn

G((u) dx & lim infm#$

#

IRn

G((um) dx ,

and together with (11.7) we obtain that, if um / u weakly, then

(11.8)#

IRn

G(u) dx % lim supm#$

#

IRn

G(um) dx .

Moreover, in order to verify F(P.-S.) we need the following estimate reminiscentof the “Pohozaev identity”; see Lemma III.1.4.


11.5 Lemma. Suppose that (1%) and (2%) of Theorem 11.2 hold. Then anyweak solution u ! D1,2

rad(IRn) of equation (11.1) satisfies the estimate#

IRn

|+u|2 dx % 2n

n) 2

#

IRn

G(u) dx % )$ .

Proof. By (2%) and our local regularity result Lemma B.3 of Appendix B, anyweak solution to (11.2) is in Lp

loc and hence also in H2,p locally, for any p < $.Moreover, by (1%),(2%) we have G+(u) ! L1(IRn). Testing Equation (11.1) withthe function x ·+u (the generator of the family uR(x) = u(Rx) of dilations ofthe function u) we may write the product as

(11.9)n) 2

2|+u|2 ) div

,x|+u|2

2)+u (x ·+u)

-= (x ·+u) 5u

= )(x ·+u) g(u) = )x ·+G(u) = )div%xG(u)

&+ nG(u) .

Integrating over BR(0) and using the radial symmetry of u, we thus obtainthat #

BR(0)|+u|2 dx =

2n

n) 2

#

BR(0)G(u) dx

) 2R

n) 2

#

/BR(0)

%12|+u|2 + G(u)

&do

% 2n

n) 2

#

BR(0)G(u) dx

) 2R

n) 2

#

/BR(0)

%12|+u|2 + G+(u)

&do .

Since +u ! L2(IRn), G+(u) ! L1(IRn), if we let R #$ in a suitable way theboundary integral tends to zero. Moreover,

#

BR(0)|+u|2 dx #

#

IRn

|+u|2 dx ,

#

BR(0)G+(u) dx #

#

IRn

G+(u) dx ,

while from Beppo Levi’s theorem it follows that#

IRn

G((u) dx % lim supR#$

#

BR(0)G((u) dx .

The proof is complete.

Remark that under scaling u 3# uR(x) = u(Rx) the functional E behaves like

E(uR) =12R2(n

#

IRn

|+u|2 dx)R(n

#

IRn

G(u) dx ;


that is,

d

dRE(uR)|R=1 =

2) n

2

#

IRn

|+u|2 dx + n

#

IRn

G(u) dx .

Hence we may perform a preliminary normalization of admissible functions byrestricting E to the set

M = {u ! Dom(E) ; u '= 0,

#

IRn

|+u|2 dx =2n

n) 2

#

IRn

G(u) dx}

of functions which are stationary for E with respect to dilations. Note that foru ! M we have

E(u) =1n

#

IRn

|+u|2 dx ;

that is, E|M is continuous and coercive with respect to the norm in D1,2(IRn).Moreover, E|M satisfies the following compactness condition:

11.6 Lemma. Suppose that for a sequence (um) in M as m # $ we haveE(um) # ! while DLE(um) # 0 ! T '

L for any L. Then (um) accumulates ata critical point u ! M of E and E(u) = !. That is, E satisfies F(P.-S.) onM (while it seems unlikely that one can even show boundedness of a F(P.-S.)sequence in general).

Proof. Let (um) be a F(P.-S.) sequence for E in M . By coerciveness, (um) isbounded and we may assume that um / u weakly in D1,2

rad(IRn), which implies

strong convergence g(um) # g(u) in L1(3) for any 3 "" IRn. Thus for any( ! C$

0 (IRn) and su!ciently large L, as m #$ we have

/(, DLE(um)0 =#

IRn

%+um+() g(um)(

&dx

##

IRn

%+u+() g(u)(

&dx = /(, DLE(u)0 = 0 ;

that is, u ! D1,2rad(IRn) is a critical point of E and hence weakly solves (11.1).

By Lemma 11.5 and (11.8) therefore#

IRn

|+u|2 dx % 2n

n) 2

#

IRn

G(u) dx % 2n

n) 2lim supm#$

#

IRn

G(um) dx

= lim supm#$

#

IRn

|+um|2 dx %#

IRn

|+u|2 dx .

Here we have also used the normalization condition for um ! M . Hence um # ustrongly in D1,2

rad(IRn), and also:IRn G(um) dx #

:IRn G(u) dx; in particular,

u ! M and E(um) # E(u) = !, as claimed.

Now we investigate the set M more closely.


Denote k: IR+ ,Dom(E) # IR the mapping

(11.10)k(R, u) =

2Rn+1

2) n

d

d'E(u%)|%=R

= R2

#

IRn

|+u|2 dx) 2n

n) 2

#

IRn

G(u) dx .

Then for any L ! IN and any u ! Dom(E) we have k( · , u + · ) ! C1(IR+ ,TL). Moreover, by Holder’s inequality the partial derivatives *Lk(R, u) !T '

L, //Rk(R, u), and /2

/R2 k(R, u) are continuous and uniformly bounded onbounded sets {(R, u) ! IR+ ,M + TL ; R + .u. & C}. Finally, for u ! M , atR = 1 we have

*

*Rk(R, u)|R=1 = 2

#

IRn

|+u|2 dx = 2n E(u) > 0 .

Thus, by the implicit function theorem, for any ! > 0 and any L ! IN thereexists ' = '#,L > 0 and a continuous map R = R#,L on a neighborhood

V#,L = {u ! M ; !/2 & E(u) & 2!} + B2%(0; TL)

such that R(v+·) ! C1(TL) and k%R(v), v

&* 0 for v ! V#,L; that is, vR(v)(x) =

v%R(v)x

&! M . Denote

-#,L: V#,L # M

the map v 3# -#,L(v) = vR(v). Remark that -#,L is continuous. For the nextlemma let

K# = {u ! M ; E(u) = !, DLE(u) = 0 for all L} ,

N#,L = {u ! M ; |E(u)) !| < 1/L, .DLE(u).'L < 1/L} .

Note that by Lemma 11.6 the assertions of Lemma 11.4 hold for K# , N#,L asabove. We now construct a pseudo-gradient flow for E on M .

11.7 Lemma. For any ! > 0, any # > 0 and any neighborhood N of K# thereexist # !]0, #[ and a continuous family %: M , [0, 1] # M of odd continuousmaps %( · , t): M # M such that(1%) %(u, t) = u if DLE(u) = 0 for all L ! IN, or t = 0, or if |E(u)) !| % #,(2%) E

%%(u, t)

&is non-increasing in t for any u ! M ,

(3%) %(E#+! \ N, 1) " E#(!.

Proof. Choose integers L) < L such that

N 7 N#,L! 7 N#,L

and let vL: {u ! M ; DLE(u) '= 0} # TL be an odd, continuous pseudo-gradient vector field for E, satisfying


.vL(u).L < 2 min(1, .DLE(u).'L

),

/vL(u), DLE(u)0 > min(1, .DLE(u).'L

).DLE(u).'L ,

for all u ! M such that DLE(u) '= 0. Let $,( be continuous cut-o" functions0 & $,( & 1, $(u) = $()u), $ * 0 on N#,L, $ * 1 o" N#,L! , ((s) = 0 for|s)!| % 2#, ((s) = 1 for |s)!| < #, where # & min{ !2 , 1

4L} will be determinedin the sequel. We truncate vL as usual and let

eL(u) = )(%E(u)&$(u)vL(u) .

Note that eL: M # TL is odd and continuous. Let '#,L, and -#,L be definedas above. Provided that # & #

4 , for t & '#,L then we may let

%(u, t) = -#,L

%u + teL(u)

&.

% is continuous, odd, and satisfies (1%). Moreover, for fixed u ! M the termE%%(u, t)

&is di"erentiable in t. Indeed, letting

R(t) = R%u + teL(u)

&, uR(x) = u

%Rx&

for brevity, we have

d

dtE%%(u, t)

&..t=t0

=d

dtE%(u + teL(u))R(t)

&|t=t0

=,

d

dRE%%

u + t0eL(u)&R

&..R=R(t0)

-d

dtR(t)|t=t0

+C%

eL(u)&R(t0)

, DLE%%(u, t0)

&D.

Since%u + t0eL(u)

&R=R(t0)

= %(u, t0) ! M the first term vanishes. Moreover,the second up to a factor )(

%E(u)&$(u) equals

C%vL(u)

&R(t0)

, DLE%%(u, t0)

&D

= R(t0)2(n

#

IRn

+%u + t0eL(u)

&+vL(u) dx

)R(t0)(n

#

IRn

g%u + t0eL(u)

&vL(u) dx

=:CvL(u), DLE(u)

D) ,(t0) ,

where the last line defines the “error function” ,.Note that t # R(t) is di"erentiable with d

dtR(t) uniformly bounded onbounded sets and that by condition (5%) of the theorem and Vitali’s convergencetheorem also

#

BL(0)

..g%u + teL(u)

&) g(u)

.. 2nn+2 dx # 0 as t # 0 ,

uniformly on bounded sets of functions u ! M . Hence also the error


,(t) & c%|R(t)) 1| + |t|

&+....#

IRn

%g%u + teL(u)

&) g(u)

&eL(u) dx

....

# 0 as t # 0 .

In particular, we can achieve that uniformly in u ! E#+! we have ,(t) & 12L2

for 0 & t & t su!ciently small. By choice of ( and $ this implies that

d

dtE%%(u, t)

&& )

(%E(u)&$(u)

2L2for u ! M, 0 & t & t .

Hence, with # & t4L2 , (2%) and (3%) follow. Rescaling time we may assume

t = 1.

Finally, we can conclude the proof of Theorem 11.2. For l ! IN let

7l = {A " M ; A closed, A = )A, ,(A) % l} ,

where , denotes the Krasnoselskii genus introduced in Section 5.1, and define

!l = infA!-l

supu!A

E(u) .

11.8 Lemma. (1%) For any l ! IN the class 7l is nonvoid, in particular, M '= -.(2%) The numbers !l are critical values of E for any l ! IN. Moreover, !l #$as l #$.

Proof. (1%) Fix l ! IN. By condition (6%) on G we can find an l-dimensionalsubspace W " C$

0,rad(IRn) and a constant &1 > 0 such that for w ! S = {w !

W ; .w. = &1} we have #

IRn

G(w) dx > 0 .

By (11.10) we can find . > 0 such that

(11.11) k(., w) < 0

for all w ! S. Scaling x with . , if necessary, we may assume that . = 1. SinceW " C$

0 there exists another constant &2 such that .w.L( & &2 for w ! S.Consider the truncation mapping ": W # D1,2

rad(IRn) given by

"(w)(x) =

67

8

&2, w(x) > &2,w(x), |w(x)| & &2,)&2, w(x) < )&2.

Note that " is continuous and odd. Since the functions "(w) are uniformlybounded and have uniform compact support, clearly

(11.12)#

IRn

G%"(w)&

dx & c


uniformly in w ! W . On the other hand, for any w ! S there holds

(11.13)#

IRn

|+"(µw)|2 dx #$ (µ #$) ,

as is easily verified.For w ! W , with .w. % &1, let

J(w) = k%1, "(w)

&

and extend J continuously to an even map J : W # IR such that J(w) < 0for .w. < &1. (Note that by (11.11) we have J(w) = k(1, w) < 0 for w ! S;that is, for .w. = &1.) By (11.12), (11.13) the set 3 = {w ! W ; J(w) < 0}then is an open, bounded, symmetric neighborhood of 0 ! W . Hence, fromProposition 5.2 we deduce that the boundary A of 3 relative to W has genus,(A) % l. Since the mapping " is odd and continuous, by supervariance of thegenus, Proposition 5.4(4%), also ,

%"(A)&% l. Moreover, since J(A) = {0} and

"(A) ': 0, we clearly have "(A) " M , concluding the proof of (1%).(2%) By part (1%) and Lemma 11.7 the numbers !l are well-defined and crit-ical; see the proof of Theorem 4.2. To show that !l # $ (l # $) assumeby contradiction that !l & ! uniformly in l. Thus, we can find a sequence ofsets Al ! 7l such that E(u) & 2! for u ! Al, l ! IN. Letting A =

9l Al

by Proposition 5.4(2%) and Proposition 5.3 there exists an infinite sequence ofmutually orthogonal vectors um ! A. By coerciveness and uniform bounded-ness of E on A, there holds .um. & C for all m, and hence we may extracta weakly convergent subsequence (um) (relabeled). By mutual orthogonal-ity, um / 0 weakly (m # $). Decomposing g = g+ ) g( as above, withu · g±(u) = max{0,±u · g(u)}, by (11.7) therefore

(11.14)#

IRn

G+(um) dx # 0 (m #$) .

But for any u ! M

(11.15)

#

IRn

|+u|2 dx &#

IRn

|+u|2 dx +2n

n) 2

#

IRn

G((u) dx

=2n

n) 2

#

IRn

G+(u) dx .

Thus, um # 0 strongly in D1,2rad(IR

n) as m # $. On the other hand, byassumptions (1%), (2%) there exists a constant c > 0 such that for all u thereholds

g(u)u & c|u|2n

n&2 ,

and consequentlyG(u) & c|u|

2nn&2 .

Hence, for u ! M , by the Sobolev embedding theorem there holds


.u.2 =#

IRn

|+u|2 dx =2n

n) 2

#

IRn

G(u) dx & c.u.2n

n&2

L2n

n&2& c.u.

2nn&2 .

Dividing by .u.2, we conclude that

.u. % c > 0

is uniformly bounded away from 0 for u ! M , and a contradiction to (11.14),(11.15) results. This concludes the proof.

11.9 Notes. In a recent paper, Duc [1] has developed a variational approachto singular elliptic boundary value problems which is similar to the methodoutlined above. Duc only requires continuity of directional derivatives of E.However, in exchange, only a weaker form of the deformation lemma can beestablished; see Duc [1; Lemma 2.5]. Very interesting new ideas in this regardcan also be found in Duc [2].

Lack of di"erentiability is encountered in a di"erent way when dealingwith functionals involving a combination of a di"erentiable and a convex term,as in the case of free boundary problems. For such functionals E, using thenotion of sub-di"erential introduced in Section I.6, a di"erential DE may bedefined as a set-valued map. Suitable extensions of minimax techniques to suchproblems have been obtained by Chang [2] and Szulkin [1]. More generally, us-ing the concept of generalized gradients introduced by Clarke [1], [5], Chang[2] develops a complete variational theory also for Lipschitz maps satisfying a(P.-S.)-type compactness condition. In Ambrosetti-Struwe [2] these results, incombination with the technique of parameter variations described in SectionII.9 above, are used to establish the existence of steady vortex rings in an idealfluid for a prescribed, positive, non-decreasing vorticity function. Previously,this problem had been studied by Fraenkel-Berger [1] by a constrained mini-mization technique; however, their approach gave rise to a Lagrange multiplierthat could not be controlled.

12. Lusternik-Schnirelman Theory on Convex Sets

In applications we also frequently encounter functionals on closed and convexsubsets of Banach spaces. In fact, this is the natural setting for variationalinequalities where the class of admissible functions is restricted by inequalityconstraints; see Sections I.2.3, I.2.4. Functionals on closed convex sets alsoarise in certain geometric problems, as we have seen in our discussion of theclassical Plateau problem in Sections I.2.7–I.2.10. In fact it was precisely forthe latter problem, with the aim of re-deriving the mountain pass lemma forminimal surfaces due to Morse-Tompkins [1], [2] and Shi"man [2], [3], that vari-ational methods for functionals on closed convex sets were first systematicallydeveloped; see Struwe [9], [18].

12. Lusternik-Schnirelman Theory on Convex Sets 163

Suppose M is a closed, convex subset of a Banach space V , and supposethat E: M # IR possesses an extension E ! C1(V ; IR) to V . For u ! M define

g(u) = supv"M

#u&v#<1

/u) v, DE(u)0

as a measure for the slope of E in M . Clearly, if M = V we have g(u) =.DE(u).. More generally, we obtain

12.1 Lemma. If E ! C1(V ), the function g is continuous in M .

Proof. Suppose um # u (m # $), where um, u ! M . Then for any v ! Msuch that .u) v. < 1 and su!ciently large m there also holds .um ) v. < 1.Hence for any such v ! M we may estimate

/u) v, DE(u)0 = limm#$

/um ) v, DE(um)0

& lim supm#$

g(um) .

Passing to the supremum with respect to v in this inequality, we infer that

g(u) & lim supm#$

g(um) .

On the other hand, if for #m @ 0 we choose vm ! M such that .um ) vm. < 1and

/um ) vm, DE(um)0 % g(um)) #m ,

by convexity of M , also the vectors

wm = .um ) u.um +%1) .um ) u.

&vm

= um +%1) .um ) u.

&(vm ) um)

belong to M and satisfy

.wm ) u. & .um ) u.+%1) .um ) u.

&.vm ) um.

< .um ) u.+%1) .um ) u.

&= 1 ,

while.vm ) wm. = .um ) u. .vm ) um. & .um ) u. # 0 .

Thus,g(u) % lim sup

m#$/u) wm, DE(u)0

= lim supm#$

/um ) vm, DE(um)0 = lim supm#$

g(um) ,

and the proof is complete.


12.2 Definition. A point u ! M is critical if g(u) = 0, otherwise u is regular. IfE(u) = ! for some critical point u ! M of E, the value ! is critical; otherwise! is regular.

This definition coincides with the definition of regular or critical points (values)of a functional given earlier, if M = V .

Moreover, as usual we let

M# = {u ! M ; E(u) < !} ,

K# = {u ! M ; E(u) = !, g(u) = 0} ,

N#,$ = {u ! M ; |E(u)) !| < ", g(u) < "} ,

U#,% = {u ! M ; (v ! K# : .u) v. < '}

denote the sub-level sets, critical sets and families of neighborhoods of K#, forany ! ! IR.

12.3 Definition. E satisfies the Palais-Smale condition on M if the followingis true:

(P.-S.)MAny sequence (um) in M such that |E(um)| & c uni-formly, while g(um) # 0 (m # $), is relatively com-pact.

12.4 Lemma. Suppose E satisfies (P.-S.)M . Then for any ! ! IR the set K#is compact. Moreover, the families {N#,$ ; " > 0}, respectively {U#,% ; ' > 0}constitute fundamental systems of neighborhoods of K#.

The proof is identical with that of Lemma 2.3.

DenoteM = {u ! M ; g(u) '= 0}

the set of regular points of E, and let

K = {u ! M ; g(u) = 0} = M \ M

be the set of critical points .

12.5 Definition. A locally Lipschitz vector field v: M # V is a pseudo-gradientvector field for E on M if there exists c > 0 such that(1%) u + v(u) ! M,(2%) .v(u). < min{1, g(u)} ,(3%) /v(u), DE(u)0 < )c min

(1, g(u)

)g(u) ,

for all u ! M .

Arguing as in the proof of Lemma 3.2 we establish:


and since M is convex, this initial value problem may be solved by Euler’smethod.

Assertions (1%), (2%) are trivially satisfied by definition of e, (3%) is provedexactly as in Theorem 3.4.

The results from the preceding sections now may be carried over to functionalson closed, convex sets. In particular, similar to Theorem 10.3 we have:

12.8 Theorem. Suppose M is a closed, convex subset of a Banach space V, E !C1(V ) satisfies (P.-S.)M on M , and suppose E admits two distinct relativeminima u1, u2 in M . Then either E(u1) = E(u2) = ! and u1, u2 can beconnected in any neighborhood of the set of relative minima u ! M of E withE(u) = !, or there exists a critical point u of E in M which is not a relativeminimizer of E.

Applications to Semilinear Elliptic Boundary Value Problems

Here we will not enter into a detailed discussion of applications of these methodsto the Plateau problem for minimal surfaces and for surfaces of constant meancurvature for which they were developed. The reader will find this materialin the lecture notes of Struwe [18], devoted exclusively to this topic, and inChang-Eells [1], Jost-Struwe [1], Struwe [13].

Nor will we touch upon applications to variational inequalities. In thiscontext, variational methods first seem to have been applied by Miersemann[1] to eigenvalue problems in a cone; see also Kucera [1] and Quittner [1]. Usingthe methods outlined above, a unified approach to equations and inequalitiescan be achieved.

Instead, we re-derive Amann’s [2], [3] famous “three solution theorem”on the existence of “unstable” solutions of semilinear elliptic boundary valueproblems, confined in an order interval between sub- and supersolutions, in thevariational case.

12.9 Theorem. Suppose 3 is a bounded domain in IRn and g: IR # IR is ofclass C1, satisfying the growth condition

(1%) |gu(u)| & c%1 + |u|p(2

&, for some p <

2n

n) 2.

Also suppose that the problem

)5u = g(u) in 3 ,(2%)u = 0 on *3 ,(3%)


admits two pairs of sub- and supersolutions u1 & u1 & u2 & u2 ! C21H1,20 (3).

Then either u1 or u2 weakly solves (2%), (3%), or (2%), (3%) admits at least threedistinct solutions ui, u1 & ui & u2, i = 1, 2, 3.

Proof. By Theorem I.2.4 the functional E ! C1%H1,2

0 (3)&

related to (2%), (3%),given by

E(u) =12

#

+|+u|2 dx)

#

+G(u) dx ,

admits critical points ui which are relative minima of E in the order intervalsui & ui & ui, i = 1, 2. Let u = u1, u = u2, and define

M = {u ! H1,20 (3) ; u & u & u a.e.} .

Observe that – unless u1 respectively u2 solves (2%),(3%) – u1 and u2 are alsorelative minima of E in M . To see this, for i = 1, 2 consider

Mi = {u ! H1,20 (3) ; ui & u & ui almost everywhere}

and for ' > 0, as in the proof of Theorem 10.5, let

M%i = {u ! M ; (v ! Mi : .u) v.H1,2

0& '} .

Note that M%i is closed and convex, hence weakly closed, and E is coercive and

weakly lower semi-continuous on M%i with respect to H1,2

0 (3). By TheoremI.1.2, therefore, E admits relative minimizers u%i ! M%

i , i = 1, 2, for any ' > 0,and there holds

C(u%1 ) u1)+, DE(u%1)

D& 0 &

C(u2 ) u%2)+, DE(u%2)

D,

with (s)+ = max{0, s}, as usual. Subtracting the relationsC(u%1 ) u1)+, DE(u1)

D% 0 %

C(u2 ) u%2)+, DE(u2)

D,

we obtain that

.(u%1 ) u1)+.2H1,20

=#

+

..+(u%1 ) u1)+..2 dx &

#

+(%g(u%1)) g(u1)

&(u%1 ) u1)+ dx

&#

+

,# 1

0gu(u1 + s(u%1 ) u1)) ds

-(u%1 ) u1)2+ dx

& supv!M%

1

.gu(v).L

pp&2

.(u%1 ) u1)+.2Lp

& C%1 + sup

v!M%1

.v.p(2Lp

&·

Ln%{x ; u%1(x) > u1(x)}

&2( · .(u%1 ) u1)+.2L

2nn&2

where , = 1 ) n(22n p > 0, and an analogous estimate for (u2 ) u%2)+. By

Sobolev’s embedding theorem


12.6 Lemma. There exists a pseudo-gradient vector field v: M # V , satisfying(3%) of Definition 12.5 with c = 1

2 .

Proof. For u ! M choose w = w(u) ! M such that

.u) w. < min(1, g(u)

),(12.1)

/u) w, DE(u)0 >12

min(1, g(u)

)g(u) .(12.2)

Now, as in the proof of Lemma 3.2, let {U& ; ) ! I} be a locally finite opencover of M such that for any ) ! I and any u ! U& the conditions (12.1), (12.2)hold with w = w(u&), for some u& ! U&. Then let {(& ; ) ! I} be a locallyLipschitz partition of unity on M subordinate to {U&} and define

v(u) =$

&!I

(&(u)%w(u&)) u

&,

for u ! V . The resulting v is a pseudo-gradient vector field on M , as claimed.

12.7 Theorem (Deformation Lemma). Suppose M " V is closed and convex,E ! C1(V ) satisfies (P.-S.)M on M , and let ! ! IR, # > 0 be given. Then forany neighborhood N of K# there exist # !]0, #[ and a continuous deformation%: M , [0, 1] # M such that(1%) %(u, t) = u if g(u) = 0, or if t = 0, or if |E(u)) !| % #;(2%) E

%%(u, t)

&is non-increasing in t, for any u ! M ;

(3%) %(M#+!, 1) " M#(! 5N , respectively %(M#+! \ N, 1) " M#(! .

Proof. For # < min{#/2, "/4}, where

N 7 U#,% 7 U#,%/2 7 N#,$ ,

as in the proof of Theorem 3.4, let $ be locally Lipschitz with 0 & $ & 1,$(u) = 0 in N#,$/2, $(u) = 1 on N#,$ and choose a smooth cut-o" function0 & ( & 1 such that ((s) = 1 for |s) !| & #, ((s) = 0 for |s) !| % 2#. Define

e(u) =*$(u)(

%E(u)&v(u), u ! M

0, u ! K .

The vector field e is Lipschitz continuous, uniformly bounded, and induces aglobal flow %: M , [0, 1] # M such that

*

*t%(u, t) = e

%%(u, t)

&

%(u, 0) = 0 .

Note that since v (and therefore e) satisfies the condition

u + v(u) ! M , for all u ! M ,


.v.Lp & c.v.L

2nn&2

& c infw!M1

%.v ) w.

L2n

n&2+ .w.

L2n

n&2

&

& c infw!M1

.v ) w.H1,20

+ C & C < $

for v ! M%1 , ' & 1. Similarly, .(u%1 ) u1)+.

L2n

n&2& c.(u%1 ) u1)+.H1,2 , whence

with a uniform constant C for all ' > 0 there holds

(12.5) .(u%1 ) u1)+.2H1,20& CLn

%{x ; u%1(x) > u1(x)}

&2(.(u%1 ) u1)+.2H1,20

,

and an analogous estimate for (u2 ) u%2)+.As '# 0 the functions u%i accumulate at minimizers ui of E in Mi, i = 1, 2.

Arguing as in the proof of Theorem I.2.4, and using the regularity result LemmaB.3 of Appendix B, these functions ui are classical solutions of (2%),(3%). Now,if u1, u2 do not solve problem (2%),(3%), then in particular we have u1 '= u1, u2 '=u2. From the strong maximum principle we then infer that u1 < u1, u2 < u2.Hence

Ln%{x; u%1(x) > u1(x)}

&# 0 ('# 0) ,

and from (12.5) it follows that u%1 ! M1 for su!ciently small ' > 0, showingthat u1 is relatively minimal for E in M . Similarly for u2. Thus, u1 and u2

are relative minima of E in M .By Theorem 12.8 the functional E either admits infinitely many relative

minima in M or possesses at least one critical point u3 ! M which is not arelative minimizer of E, hence distinct from u1, u2.

Finally, observe that any critical point u of E in M weakly solves (2%),(3%): Indeed for any ( ! H1,2

0 (3), # > 0, let

v! = min(u, max{u, u + #(}

)= u + #() (! + (! ! M

with (! = max{0, u + #( ) u} % 0, (! = max{0, u) (u + #()} % 0, as in theproof of Theorem I.2.4. Then, if u ! M is critical, as ## 0 we have

/u) v!, DE(u)0 & .u) v!.H1,20

supv"M

#u&v#H

1,20

$1

/u) v, DE(u)0

& O(#)g(u) = 0 .

From this inequality the desired conclusion follows exactly as in the proof ofTheorem I.2.4.

12.10 Notes. A di"erent variational approach to this result was suggested byChang [5], who used the regularizing properties of the heat flow to reduce theproblem to a variational problem on an open subset of a suitable Banach spaceto which the standard methods could be applied.

A combination of topological degree and variational methods was used byHofer [1] to obtain even higher multiplicity results for certain problems.

Chapter III

Limit Cases of the Palais-Smale Condition

Condition (P.-S.) may seem rather restrictive. Actually, as Hildebrandt [4;p. 324] records, for quite a while many mathematicians felt convinced thatinspite of its success in dealing with one-dimensional variational problems likegeodesics (see Birkho!’s Theorem I.4.4, for example, or Palais’ [3] work onclosed geodesics), the Palais-Smale condition could never play a role in thesolution of “interesting” variational problems in higher dimensions.

Recent advances in the Calculus of Variations have changed this view andit has become apparent that the methods of Palais and Smale apply to manyproblems of physical and/or geometric interest and – in particular – that thePalais-Smale condition will in general hold true for such problems in a broadrange of energies. Moreover, the failure of (P.-S.) at certain energy levels reflectshighly interesting phenomena related to internal symmetries of the systems un-der study. Geometrically, we might attribute the non-compactness of certainPalais-Smale sequences to the “separation of spheres”, or to “bubbling”, per-haps associated with a “change in topology”. With some twist of imagination,and speaking again in physical terms, we might observe “phase transitions” or“particle creation” at the energy levels where (P.-S.) fails.

Such phenomena seem to have first been observed by Sacks-Uhlenbeck [1]and – independently – by Wente [5] in the context of two-dimensional harmonicmaps, respectively, in the context of surfaces of prescribed constant mean cur-vature. (See Sections 5 and 6 below.) In these cases the terms “separationof spheres” and “bubbling” have a clear geometric meaning. More recently,Sedlacek [1] has uncovered similar results also for Yang-Mills connections. Ifinterpreted appropriately, very early indications of such phenomena alreadymay be found in the work of Douglas [2], Morse-Tompkins [2] and Shi!man[2] on minimal surfaces of higher genus and/or connectivity. In this case, a“change in topology” in fact sometimes may be observed even physically asone tries to realize a multiply connected or higher genus minimal surface ina soap film experiment. See Jost-Struwe [1] for a modern approach to theseresults.

Mathematically, it seems that non-compact group actions give rise to thesee!ects. In physics and geometry, of course, such group actions naturally arise as“symmetries” from the requirements of scale or gauge invariance; in particular,in the examples of Sacks-Uhlenbeck and Wente cited above, from conformalinvariance.

170 Chapter III. Limit Cases of the Palais-Smale Condition

A symmetry may either be “manifest” or “broken” by interaction terms.As we shall see, existence results in the spirit of Theorem II.6.1 for problemswith non-compact internal symmetries often depend on the extent to whichthe symmetry is broken or perturbed, as measured by whether or not certainmountain-pass energy levels di!er from the energy levels where “bubbling”may occur. As in the case of the non-compact minimization problems studied inSection I.4, in many instances the latter question can be answered by comparingthe problem at hand with a suitable (family of) “limiting problem(s)” wherethe symmetry is acting. We start with a simple example.

1. Pohozaev’s Non-existence Result

Let ! be a domain in IRn, n > 2. Consider the limit case p = 2! = 2nn"2 in

Theorem I.2.1. Given " ! IR, we would like to solve the problem

"#u = "u + u|u|2!"2 in ! ,(1.1)

u > 0 in ! ,(1.2)u = 0 on $! .(1.3)

(Note that in order to be consistent with the literature, in this section wereverse the sign of " as compared with Section I.2.1 or Section II.5.8.) As inTheorem I.2.1 we can approach this problem by a direct method and attempt toobtain non-trivial solutions of (1.1), (1.3) as relative minima of the functional

I!(u) =12

!

"

"|#u|2 " "|u|2

#dx ,

on the unit sphere in L2!(!),

M = {u ! H1,20 (!) ; $u$2!

L2! = 1} .

Equivalently, we may seek to minimize the Sobolev quotient

S!(u;!) =$"

"|#u|2 " "|u|2

#dx

"$" |u|2! dx

#2/2! , u ! H1,20 (!) \ {0}.

Note that for " = 0, as in Section I.4.4,

S(!) = infu"H1,2

0 (!)u#=0

S0(u;!) = infu"H1,2

0 (!)u#=0

$" |#u|2 dx

"$" |u|2! dx

#2/2!

is related to the (best) Lipschitz constant for the Sobolev embedding H1,20 (!) %

L2!(!).Recall that for any u ! H1,2

0 (!) & D1,2(IRn) the ratio S0(u; IRn) is invari-ant under scaling u '% uR(x) = u(Rx); that is, we have

(1.4) S0(u; IRn) = S0(uR; IRn), for all R > 0 .

In particular, as we already observed in Remark I.4.5, we have:

1. Pohozaev’s Non-existence Result 171

1.1 Lemma. S(!) = S is independent of !.

Recalling Remark I.4.7, we note that Lemma 1.1 implies:

1.2 Theorem. S is never attained on a domain ! ( IRn, ! )= IRn.

Hence, for " = 0, the proof of Theorem I.2.1 necessarily fails in the limitcase p = 2!. More generally, we have the following uniqueness result, due toPohozaev [1]:

1.3 Theorem. Suppose ! )= IRn is a smooth (possibly unbounded) domain inIRn, n * 3, which is strictly star-shaped with respect to the origin in IRn, andlet " + 0. Then any solution u ! H1,2

0 (!) of the boundary value problem (1.1),(1.3) vanishes identically.

Theorem 1.3 applies to any solution, whereas Theorem 1.2 is limited to minimaof S0(·;!). However, Theorem 1.2 applies to any domain.

The proof of Theorem 1.3 is based on the following “Pohozaev identity”:

1.4 Lemma. Let g: IR % IR be continuous with primitive G(u) =$ u0 g(v) dv

and let u ! C2(!) , C1(!) be a solution of the equation

"#u = g(u) in !(1.5)u = 0 on $!(1.6)

in a domain ! && IRn. Then there holds

n " 22

!

"|#u|2 dx " n

!

"G(u) dx +

12

!

#"

%%%%$u

$%

%%%%2

x · % do = 0 ,

where % denotes the exterior unit normal.

Proof of Theorem 1.3. Let g(u) = "u + u|u|2!"2 with primitive

G(u) ="

2|u|2 +

12!

|u|2!

.

By Theorems B.1 and B.2 and Lemma B.3 of Appendix B, any solution of(1.1), (1.3) is smooth on !. Hence from Pohozaev’s identity we infer that

!

"|#u|2 dx " 2!

!

"G(u) dx +

1n " 2

!

#"

%%%%$u

$%

%%%%2

x · % do

=!

"

"|#u|2 " |u|2

!#dx +

n|"|n " 2

!

"|u|2 dx

+1

n " 2

!

#"

%%%%$u

$%

%%%%2

x · % do = 0 .


However, testing the Equation (1.1) with u, we infer that!

"

"|#u|2 " "|u|2 " |u|2

!#dx = 0 ,

whence

2|"|!

"|u|2 dx +

!

#"

%%%%$u

$%

%%%%2

x · % do = 0 .

Moreover, since ! is strictly star-shaped with respect to 0 ! IRn, we havex · % > 0 for all x ! $!. Thus #u

#$ = 0 on $!, and hence u - 0 by the principleof unique continuation; see Heinz [2].

Proof of Lemma 1.4. Multiply (1.5) by x ·#u and compute

0 ="#u + g(u)

#(x ·#u)

= div"#u(x ·#u)

#" |#u|2 " x ·#

" |#u|2

2#

+ x ·#G(u)

= div"#u(x ·#u) " x

|#u|2

2+ xG(u)

#+

n " 22

|#u|2 " nG(u) .

Upon integrating this identity over ! and taking account of the fact that by(1.6) we have

x ·#u = x · % $u$%

on $! ,

we obtain the asserted identity.

1.5 Interpretation. Note that the function x · #u = ddRuR used in the proof

of Lemma 1.4 is the generator of the family of scaled maps {uR ; 0 < R <.}. This observation allows a connection between Theorem 1.3 and scaleinvariance of S = S0. Indeed, we may interpret Theorem 1.3 as reflecting thenon-compactness of the multiplicative group IR+ = {R ; 0 < R < .} actingon S via scaling. Note that this group action is a “manifest” symmetry ofS!(· ;!) only in the case " = 0 and ! = IRn. In the case of a bounded domain! not all scalings u % uR will map H1,2

0 (!) into itself. For instance, if ! isan annular region ! = {x ; a < |x| < b}, in fact, H1,2

0 (!) does not admitany of these scalings as symmetries. (In Section 3 we will see that in this case(1.1)–(1.3) does have non-trivial solutions.) However, if ! is star-shaped withrespect to the origin, all scalings u % uR, R * 1 will be symmetries of H1,2

0 (!),and compactness is lost as R % .. The e!ect is shown in Theorem 1.3.

Note that it is also possible to characterize solutions u ! H1,20 (!) of equa-

tion (1.1) as critical points of a functional E! on H1,20 (!) given by

(1.7) E!(u) =12

!

"

"|#u|2 " "|u|2

#dx " 1

2!

!

"|u|2

!dx .

2. The Brezis-Nirenberg Result 173

By continuity of the embedding H1,20 (!) &% L2!

(!) &% L2(!), the functionalE! is Frechet-di!erentiable on H1,2

0 (!). Moreover, for " <" 1, the first Dirich-let eigenvalue of the operator "#, E! satisfies the conditions (1#)–(3#) of themountain pass lemma Theorem II.6.1; compare the proof of Theorem I.2.1. Inview of Theorem II.6.1, the absence of a critical point u )= 0 of E! for any " + 0proves that E! for such " cannot satisfy the Palais-Smale condition (P.-S.) ona star-shaped domain. Again the non-compact action R '% uR(x) = u(Rx) canbe held responsible.

2. The Brezis-Nirenberg Result

In contrast to Theorem 1.3, for " > 0 problem (1.1)–(1.3) may admit non-trivialsolutions. However, a subtle dependence on the dimension n is observed.

The first result in this direction is due to Brezis and Nirenberg [2], buildingon ideas of Trudinger [1] and Aubin [2].

2.1 Theorem. Suppose ! is a domain in IRn, n * 3, and let "1 > 0 denotethe first eigenvalue of the operator "# with homogeneous Dirichlet boundaryconditions.(1#) If n * 4, then for any " !]0,"1[ there exists a solution of (1.1)–(1.3).(2#) If n = 3, there exists "! ! [0,"1[ such that for any " !]"!,"1[ problem(1.1)–(1.3) admits a solution.(3#) If n = 3 and ! = B1(0) & IR3, then "! = !1

4 and for " + !14 there is no

solution to (1.1)–(1.3).

As we have seen in Section 1, there are (at least) two di!erent approaches tothis theorem. The first, which is the one chosen by Brezis and Nirenberg [2],involves the quotient

S!(u;!) =$"

"|#u|2 " "|u|2

#dx

"$" |u|2! dx

#2/2! .

A second proof can be given along the lines of Theorem II.6.1, applied to the“free” functional E!

E!(u) =12

!

"

"|#u|2 " "|u|2

#dx " 1

2!

!

"|u|2

!dx

defined earlier. Recall that E! ! C1"H1,2

0 (!)#. As we shall see, while it is not

true that E! satisfies the Palais-Smale condition “globally”, some compactnesswill hold in an energy range determined by the best Sobolev constant S; seeLemma 2.3 below. A similar compactness property holds for the functional S!.

We will first pursue the approach involving S!.


Constrained Minimization

DenoteS!(!) = inf

u$H1,20 (")\{0}

S!(u;!) .

Note that S!(!) + S for all " * 0 (in fact, for all " ! IR), and S!(!) in generalis not attained. Similar to Theorem I.4.2 now there holds:

2.2 Lemma. If ! is a bounded domain in IRn, n * 3, and if

S!(!) < S ,

then there exists u ! H1,20 (!), u > 0, such that S!(!) = S!(u;!).

Proof. Consider a minimizing sequence (um) for S! in H1,20 (!). Normalize

$um$L2! = 1. Replacing um by |um|, if necessary, we may assume that um * 0.Since by Holder’s inequality

S!(um;!) =!

"

"|#um|2 " "|um|2

#dx *

!

"|#um|2 dx " c ,

we also may assume that um ' u weakly in H1,20 (!) and strongly in L2(!) as

m % ..To proceed, observe that like (I.4.4) by Vitali’s convergence theorem we

have

(2.1)

!

"

"|um|2

!" |um " u|2

!#dx

=!

"

! 1

0

d

dt

%%um + (t " 1)u%%2!

dt dx

= 2!! 1

0

!

"

"um + (t " 1)u

#%%um + (t " 1)u%%2!"2

u dx dt

% 2!! 1

0

!

"t u|t u|2

!"2u dx dt =!

"|u|2

!dx as m % . .

Also note that

(2.2)!

"|#um|2 dx =

!

"|#(um " u)|2 dx +

!

"|#u|2 dx + o(1) ,

where o(1) % 0 as m % .. Hence we obtain:

S!(um;!) + o(1) =!

"|#(um " u)|2 dx +

!

"

"|#u|2 " "|u|2

#dx + o(1)

* S$um " u$2L2! + S!(!)$u$2

L2! + o(1)

* S$um " u$2!

L2! + S!(!)$u$2!

L2! + o(1)

*"S " S!(!)

#$um " u$2!

L2! + S!(!) + o(1) .


Since S > S!(!) = limm%& S!(um;!) by assumption, we find that um % u inL2!

(!), and $u$L2! = 1. By weak lower semi-continuity of the H1,20 (!)-norm

then we haveS!(u;!) + lim

m%&S!(um;!) = S!(!) ,

as desired.Computing the first variation of S!(u;!), as in the proof of Theorem I.2.1

we see that a positive multiple of u satisfies (1.1), (1.3). Since u * 0, u )= 0,from the strong maximum principle (Theorem B.4 of Appendix B) we inferthat u > 0 in !. The proof is complete.

The Unconstrained Case: Local Compactness

Postponing the complete proof of Theorem 2.1 for a moment, we now alsoindicate the second approach, based on a careful study of the compactnessproperties of the free functional E!. Note that in the case of Theorem 2.1both approaches are completely equivalent – and the final step in the proof ofTheorem 2.1 actually is identical in both cases. However, for more general non-linearities with critical growth it is not always possible to reduce a boundaryvalue problem like (II.6.1), (II.6.2) to a constrained minimization problem andwe will have to use the free functional instead. Moreover, this second approachwill bring out the peculiarities of the limiting case more clearly. Our presenta-tion follows Cerami-Fortunato-Struwe [1]. An indication of Lemma 2.3 belowis also given by Brezis-Nirenberg [2; p. 463].

2.3 Lemma. Let ! be a bounded domain in IRn, n * 3, and let " ! IR. Thenany sequence (um) in H1,2

0 (!) such that

E!(um) % ( <1n

Sn/2, DE!(um) % 0 ,

as m % ., is relatively compact.

Proof. To show boundedness of (um), compute

o(1)"1 + $um$H1,2

0

#+

2n

Sn/2 * 2E!(um) " /um, DE!(um)0

=&

1 " 22!

' !

"|um|2

!dx * c

&!

"|um|2 dx

'2!/2

,

where c > 0 and o(1) % 0 as m % .. Hence

$um$2H1,2

0= 2E!(um) + "

!

"|um|2 dx +

22!

!

"|um|2

!dx

+ C + o(1)$um$H1,20

,

and it follows that (um) is bounded.


Hence we may assume that um ' u weakly in H1,20 (!), and therefore

also strongly in Lp(!) for all p < 2! by the Rellich-Kondrakov theorem; seeTheorem A.5 of Appendix A.

In particular, for any ) ! C&0 (!) we obtain that

/), DE!(um)0 =!

"

"#um#)" "um)" um|um|2

!"2)#

dx

%!

"

"#u#)" "u)" u|u|2

!"2)#

dx = /), DE!(u)0 = 0 ,

as m % .. Hence, u ! H1,20 (!) weakly solves (1.1). Moreover, choosing

) = u, we have

0 = /u, DE!(u)0 =!

"

"|#u|2 " "|u|2 " |u|2

!#dx ,

and hence

E!(u) =&

12" 1

2!

' !

"|u|2

!dx =

1n

!

"|u|2

!dx * 0 .

To proceed, note that by (2.1) and (2.2) we have!

"|#um|2 dx =

!

"|#(um " u)|2 dx +

!

"|#u|2 dx + o(1) ,

!

"|um|2

!dx =

!

"|um " u|2

!dx +

!

"|u|2

!dx + o(1) ,

where o(1) % 0 (m % .). Hence

E!(um) = E!(u) + E0(um " u) + o(1).

Similarly, again using (2.1) we have!

"um|um|2

!"2(um " u) dx

=!

"

"|um|2

!" |u|2

!#dx + o(1) =

!

"|um " u|2

!dx + o(1) ,

whence

o(1) = /um " u, DE!(um)0 = /um " u, DE!(um) " DE!(u)0

=!

"

"|#(um " u)|2 " |um " u|2

!#dx + o(1) .

In particular, from the last equation it follows that

E0(um " u) =1n

!

"|#(um " u)|2 dx + o(1).

On the other hand we have the bound


E0(um " u) = E!(um) " E!(u) + o(1)

+ E!(um) + o(1) + c <1n

Sn/2 for m * m0 .

Therefore$um " u$2

H1,20

+ c < Sn/2 for m * m0 .

But then Sobolev’s inequality $um " u$2L2! + S"1$um " u$2

H1,20

yields theestimate

$um " u$2H1,2

0

(1 " S"2!/2$um " u$2!"2

H1,20

)

+!

"

"|#(um " u)|2 " |um " u|2

!#dx = o(1) ,

showing that um % u strongly in H1,20 (!), as desired.

Lemma 2.3 motivates one to introduce the following variant of (P.-S.), whichseems to appear first in Brezis-Coron-Nirenberg [1].

2.4 Definition. Let V be a Banach space, E ! C1(V ), ( ! IR. We say that Esatisfies condition (P.-S.)%, if any sequence (um) in V such that E(um) % (while DE(um) % 0 as m % . is relatively compact. (Such sequences in thesequel, for brevity, will be referred to as (P.-S.)%-sequences.)

In terms of this definition, Lemma 2.3 simply says that the functional E!satisfies (P.-S.)% for any ( < 1

nSn/2. Now recall that E! for " <" 1 satisfiesconditions (1#)–(3#) of Theorem II.6.1.

By Lemma 2.3, therefore, the proof of the first two parts of Theorem 2.1will be complete if we can show that for " > 0 (respectively " >" !) thereholds

(2.3) ( = infp$P

supu$p

E!(u) <1n

Sn/2 ,

where, for a suitable function u1 satisfying E!(u1) + 0, we let

P = {p ! C0"[0, 1] ; H1,2

0 (!)#

; p(0) = 0, p(1) = u1} ,

as in Theorem II.6.1.Of course, (2.3) and the condition S!(!) < S of Lemma 2.2 are related.

Given u ! H1,20 (!) with $u$L2! = 1, we may let p(t) = t u, u1 = t1 u for

su"ciently large t1 to obtain

( + sup0't<&

E!(t u) = sup0't<&

&t2

2S!(u;!) " t2

!

2!

'=

1n

Sn/2! (u;!) .

Hence ( + 1nSn/2

! (!). Likewise, any p ! P contains some u )= 0 such that


/u, DE!(u)0 =!

"

"|#u|2 " "|u|2 " |u|2

!#dx = 0 .

Indeed, since " <" 1, for u = p(t) )= 0 with t close to 0 we have /u, DE!(u)0 > 0,while for u1 = p(1) we have

/u1, DE!(u1)0 < 2E!(u1) + 0 ,

and by the intermediate value theorem there exists u, as claimed. But for suchu we easily compute

S!(!) + S!(u;!) =&!

"|#u|2 " "|u|2 dx

'1"2/2!

= (n E!(u))2/n +&

n supu$p

E!(u)'2/n

.

Upon passing to the infimum with respect to p ! P we find that

(2.4)1n

Sn/2! (!) + ( = inf

p$Psupu$p

E!(u) + 1n

Sn/2! (!) ;

that is, (2.3) and the condition S! < S are in fact equivalent.

Proof of Theorem 2.1(1#). It su"ces to show that S! < S. Consider the family

(2.5) u!&(x) =

[n(n " 2)*2]n$2

4

[*2 + |x|2]n$22

, * > 0 ,

of functions u!& ! D1,2(IRn). Note that u!

&(x) = *2$n

2 u!1

"x&

#, and u!

& satisfiesthe equation

(2.6) "#u!& = u!

& |u!&|2

!"2 in IRn ,

as is easily verified by a direct computation. We claim that S0

"u!&; IR

n#= S;

that is, the best Sobolev constant is achieved by the family u!& , * > 0. Indeed,

let u ! D1,2(IRn) satisfy S0(u; IRn) = S. (The existence of such a function u canbe deduced for instance from Theorem I.4.9.) Using Schwarz-symmetrizationwe may assume that u is radially symmetric; that is, u(x) = u(|x|). (In fact,any positive solution of (2.6), which decays to 0 su"ciently rapidly as |x| % .,by the result of Gidas-Ni-Nirenberg quoted in Section I.2.2 above is radiallysymmetric.) Moreover, u solves (2.6). Choose *0 > 0 such that u!

&0(0) = u(0).Then u and u!

&0 both are solutions of the ordinary di!erential equation of secondorder in r = |x|,

r1"n $

$r

&rn"1 $

$r

'u = u|u|2

!"2 for r > 0 ,

sharing the initial data u(0) = u!&0(0), $ru(0) = $ru!

&0(0) = 0. It is not hard toprove that this initial value problem admits a unique solution. Thus u = u!

&0 ,which implies that S = S0(u; IRn) = S0(u!

&0 ; IRn) = S0(u!

& ; IRn) for all * > 0.


Since (2.6) also yields that $u!&$2

H1,20

= $u!&$2!

L2! it follows that

$u!&$2

H1,20

= $u!&$2!

L2! = Sn/2 for all * > 0 .

We may suppose that 0 ! !. Let + ! C&0 (!) be a fixed cut-o! function, + - 1

in a neighborhood B'(0) of 0. Let u& = + u!& and compute

(2.7)

!

"|#u&|2 dx =

!

"|#u!

& |2+2 dx + O"*n"2

#

=!

IRn

|#u!& |2 dx + O

"*n"2

#= Sn/2 + O

"*n"2

#.

!

"|u&|2

!dx =

!

IRn

|u!& |2

!dx + O

"*n

#= Sn/2 + O

"*n

#

!

"|u&|2 dx *

!

B"(0)|u!&|2 dx

*!

B#(0)

[n(n " 2)*2]n$2

2

[2*2]n"2dx

+!

B"(0)\B#(0)

[n(n " 2)*2]n$2

2

[2|x|2]n"2dx

= c1 · *2 + c2*n"2

! '

&r3"n dr

=

*+

,

c *2 + O"*n"2

#, if n > 4

c *2| ln *| + O"*2

#, if n = 4

c *+ O"*2

#, if n = 3

with positive constants c, c1, c2 > 0. Thus, if n * 5

S!(u&;!) + (Sn/2 " c"*2 + O(*n"2))(Sn/2 + O(*n))2/2!

= S " c"*2 + O"*n"2

#< S ,

if * > 0 is su"ciently small. Similarly, if n = 4, we have

S!(u&;!) + S " c"*2| ln *| + O"*2

#< S

for * > 0 su"ciently small.

Remark on Theorem 2.1(2#),(3#). If n = 3, estimate (2.7) shows that the“gain” due to the presence of " and the “loss” due to truncation of u!

& may beof the same order in *; hence S! can only be expected to be smaller than Sfor “large” ". To see that "! < "1, choose the first eigenfunction u = )1 of("#) as comparison function. The non-existence result for ! = B1(0), " + !1

4follows from a weighted estimate similar to Lemma 1.4; see Brezis-Nirenberg[2; Lemma 1.4]. We omit the details.


Theorem 2.1 should be viewed together with the global bifurcation result ofRabinowitz [1; p. 195 f.]. Intuitively, Theorem 2.1 indicates that the branch ofpositive solutions found by Rabinowitz in dimension n * 4 on a star-shapeddomain bends back to " = 0 and becomes asymptotic to this axis.

Fig. 2.1. Solution “branches” for (1.1), (1.3) depending on !

However, Equation (1.1) may have many positive solutions. In Section 3we shall see that if ! is an annulus, positive radial solutions exist for any valueof " < "1. But note that, by Theorem 1.2, for " = 0 these cannot minimizeS!. Hence we may have many di!erent branches of (or secondary bifurcationsof branches of) positive solutions, in general.

Multiple Solutions

2.5 Bifurcation from higher eigenvalue. If we drop the requirement of positiv-ity, the local Palais-Smale condition Lemma 2.3 permits us to obtain bifurcationof non-trivial solutions of (1.1), (1.3) from higher eigenvalues, as well. Recallthat by a result of Bohme [1] and Marino [1] it is known that any eigenvalue of"# on H1,2

0 (!) is a point of bifurcation from the trivial solution of (1.1), (1.3).However, the variational method may give better estimates for the "-intervalof existence for such solutions.

The following result is due to Cerami-Fortunato-Struwe [1]:

2.6 Theorem. Let ! be a bounded domain in IRn, n * 3, and let 0 < "1 <"2 + . . . denote the eigenvalues of "# in H1,2

0 (!). Also let

% = S ·"Ln(!)

#"2/n> 0 .

Then, if


m = m(") = ,{j ; " <" j < "+ %} ,

problem (1.1), (1.3) admits at least m distinct pairs of non-trivial solutions.

Remark. From the Weyl formula "j 1 C(!)j2/n for the asymptotic behaviorof the eigenvalues "j we conclude that m(") % . as "% ..

Proof of Theorem 2.6. Consider the functional

S!(u) := S!(u;!) =$"

"|#u|2 " "|u|2

#dx

"$" |u|2! dx

#2/2!

on the unit sphere

M = {u ! H1,20 (!) ; $u$L2! = 1} .

M & H1,20 (!) is a complete Hilbert manifold, invariant under the involution

u % "u. Recall that S! is di!erentiable on M and that if u ! M is a criticalpoint of S! with S!(u) = ( > 0, then u = (

12!$2 u solves (1.1), (1.3) with

E!(u) =1n

S!(u)n/2 ;

see the proof of Theorem I.2.1.Moreover, (um) is a (P.-S.)%-sequence for S! if and only if the sequence

(um), where um = (1

2!$2 um, is a (P.-S.)% sequence for E! with ( = 1n(

n/2.In particular, by Lemma 2.3 we have that S! satisfies (P.-S.)% on M for any( !]0, S[.

Now let - denote the Krasnoselskii genus, and for j ! IN such that "j !]","+ %[ let

(j = infA%M

$(A)&j

supu$A

S!(u) .

Note that by Proposition II.5.3 for any A & M such that -(A) * j there existat least j mutually orthogonal vectors in A, whence (j > 0 for all j as above.Moreover, if we denote by )k ! H1,2

0 (!) the kth eigenfunction of "# and letAj = span{)1, . . . ,)j} , M , we obtain that

(j + supu$Aj

S!(u) + ("j " ") supu$Aj

$" |u|2 dx

"$" |u|2! dx

#2/2!

< %"Ln(!)

# 2n = S .

Hence the theorem follows from Theorem II.4.2 and Lemma II.5.6.

We do not know how far the solution branches bifurcating o! the trivial solutionu - 0 at " = "j extend. However, the following result has been obtained byCapozzi-Fortunato-Palmieri [1]:


2.7 Theorem. Let ! be a bounded domain in IRn, n * 4. Then for any " > 0problem (1.1), (1.3) admits a non-trivial solution.

The proof of Capozzi-Fortunato-Palmieri is based on a linking argument andthe local Palais-Smale condition Lemma 2.3. A simpler proof for " not be-longing to the spectrum of "#, using the duality method, was worked out byAmbrosetti and Struwe [1].

Theorem 2.7 leaves open the question of multiplicity. For n * 4, " > 0Fortunato-Jannelli [1] present results in this direction using symmetries of thedomain to restrict the space of admissible functions to certain symmetric sub-spaces, where the first Dirichlet eigenvalue is increased su"ciently for Theorem2.1 to become applicable.

For instance, if ! = B1(0; IR2) 2 !( & IRn, n * 4, " > 0, essentiallyFortunato-Jannelli solve (1.1), (1.3) by constructing a positive solution u to(1.1), (1.3) on a “slice”

!m = {x = rei( ! B1(0; IR2) ; 0 + ) + .

m}2!(

and reflecting u in the “vertical” edges of !m a total of 2m times. Since thefirst eigenvalue "(m)

1 of ("#), acting on H1,20 (!m), tends to . as m % .,

we can achieve that "(m)1 > " for m * m0. Hence the existence of a positive

solution u to (1.1), (1.3) on !m is guaranteed by Theorem 2.1.

Fig. 2.2. A slice "m of the pie "

A di!erent idea is used by Ding [1] to construct infinitely many solutions ofvarying sign of Equation (2.6) on IRn. In fact, Ding’s result involves a beautifulcombination of analysis and geometry. Conformally mapping IRn to the n-sphere and imposing invariance with respect to a subgroup SO(k) & SO(n)of rotations of the sphere as a constraint on admissible comparison functions,

3. The E!ect of Topology 183

Ding e!ectively reduces Equation (2.6) to a sub-critical variational problem towhich Theorem 4.2 may be applied.

For the existence of multiple radially symmetric solutions on balls ! =BR(0) & IRn a subtle dependence on the space dimension is observed, relatedto the fact that the best Sobolev constant S is attained in the class of radiallysymmetric functions (on IRn). The following result is due to Cerami-Solimini-Struwe [1] and Solimini [1] :

2.8 Theorem. Suppose ! = BR(0) is a ball in IRn, n * 7. Then for any " > 0problem (1.1), (1.3) admits infinitely many radially symmetric solutions.

The proof of Theorem 2.8 uses a characterization of di!erent solutions by thenodal properties they possess, as in the locally compact case; see Remark II.7.3.

By recent results of Atkinson-Brezis-Peletier [1] the restriction on the dimensionn in Theorem 2.8 appears to be sharp; see also Adimurthi-Yadava [1].

In 2002, finally, Devillanova-Solimini [1] showed that for any " > 0 problem(1.1), (1.3) admits infinitely many solutions also on an arbitrary smoothlybounded domain ! & IRn, provided that n * 7. Their method of proof relieson a precise analysis as p 3 2! of solutions u = up to the sub-critical equations

"#u = "u + u|u|p"2 in ! ,

with u = 0 on $!. Again their method indicates that the restriction n * 7may be sharp.

2.9 Notes. (1#) Egnell [1], Guedda-Veron [1], and Lions-Pacella-Tricarico[1] have studied problems of the type (1.1), (1.3) involving the (degenerate)pseudo-Laplace operator and partially free boundary conditions.(2#) The linear term "u in Equation (1.1) may be replaced by other compactperturbations; see Brezis-Nirenberg [2]. In this regard we also mention theresults by Mancini-Musina [1] concerning obstacle problems of type (1.1), (1.3).Still a di!erent kind of “perturbation” will be considered in the next section.

3. The E!ect of Topology

Instead of a star-shaped domain, as in Theorem 1.3, consider an annulus

! = {x ! IRn ; r1 < |x| < r2} ,

and for " ! IR let E! be given by (1.7). Note that if we restrict our attentionto radial functions

H1,20,rad(!) = {u ! H1,2

0 (!) ; u(x) = u(|x|)} ,

by estimate (II.11.6) the embedding


H1,20,rad(!) &% Lp(!)

is compact for any p < .. Hence DE!: H1,20,rad(!) % H"1(!) is of the form

id + compact. Since any (P.-S.)-sequence for E! by the proof of Lemma 2.3 isbounded, from Proposition II.2.2 we thus infer that E! satisfies (P.-S.) globallyon H1,2

0,rad(!) and hence from Theorem II.5.7 or Theorem II.6.5 that problem(1.1), (1.3) possesses infinitely many radially symmetric solutions on !, for anyn * 2, and any " ! IR, in particular, for " = 0. This result stands in strikingcontrast with Theorem 1.3 (or Theorem 2.8, as regards the restriction of thedimension).

In the following we shall investigate whether the solvability of (1.1), (1.3)on an annulus is a singular phenomenon, observable only in a highly symmetriccase, or is stable and survives perturbations of the domain.

A Global Compactness Result

Remark that by Theorem 1.2, for " = 0 no non-trivial solution u ! H1,20 (!)

of (1.1) can satisfy S!(u;!) + S. Hence the local compactness of Lemma 2.3will not su"ce to produce such solutions and we must study the compactnessproperties of E!, respectively S!, at higher energy levels as well. The nextresult can be viewed as an extension of P.-L. Lions’ concentration-compactnessprinciple for minimization problems (see Section I.4) to problems of minimaxtype. The idea of analyzing the behavior of a (P.-S.)-sequence near points ofconcentration by “blowing up” the singularities seems to appear first in papersby Sacks-Uhlenbeck [1] and Wente [5], where variants of the local compactnesscondition Lemma 2.3 are obtained (see Sacks-Uhlenbeck [1; Lemma 4.2]). In thenext result, due to Struwe [8], we systematically employ the blow-up techniqueto characterize all energy values ( of a variational problem where (P.-S.)% mayfail.

3.1 Theorem. Suppose ! is a smoothly bounded domain in IRn, n * 3, and for" ! IR let (um) be a (P.-S.)-sequence for E! in H1,2

0 (!) & D1,2(IRn). Thenthere exist a number k ! IN0, sequences (Rj

m), (xjm), 1 + j + k, of radii

Rjm % . (m % .) and points xj

m ! !, a solution u0 ! H1,20 (!) & D1,2(IRn)

to (1.1), (1.3) and non-trivial solutions uj ! D1,2(IRn), 1 + j + k, to the“limiting problem” associated with (1.1) and (1.3),

(3.1) "#u = u|u|2!"2 in IRn ,

such that a subsequence (um) satisfies

----um " u0 "k.

j=1

ujm

----D1,2(IRn)

% 0 .

Here ujm denotes the rescaled function


ujm(x) = (Rj

m)n$2

2 uj"Rj

m(x " xjm)

#, 1 + j + k, m ! IN .

Moreover, we have

E!(um) % E!(u0) +k.

j=1

E0(uj) .

3.2 Remarks. (1#) In particular, if ! is a ball ! = BR(0), um ! H1,20,rad(!),

from the uniqueness of the family (u!&)&>0 of radial solutions to (3.1) – see the

proof of Theorem 2.1(1#) – it follows that each uj is of the form (2.5) withE0(uj) = 1

nSn/2 =: (!. Hence in this case (P.-S.)% holds for E! for all levels (which cannot be decomposed

( = (0 + k(! ,

where k * 1 and (0 = E!(u0) is the energy of some radial solution of (1.1),(1.3). Similarly, if ! is an arbitrary bounded domain and um * 0 for all m,then also uj * 0 for all j, and by a result of Gidas-Ni-Nirenberg [1; p. 210 f.]and Obata [1] again each function uj will be radially symmetric about somepoint xj . Therefore also in this case each uj is of the form uj = u!

&(·" xj) forsome * > 0, and (P.-S.)% holds for all ( which are not of the form ( = (0 +k(!

where k * 1 and (0 = E!(u0) is the energy of some non-negative solution u0

of (1.1), (1.3).(2#) For some time it was believed that the family (2.5) gives all non-trivialsolutions of (3.1). However, Ding’s [1] result shows that (3.1) also admitsinfinitely many solutions of changing sign which are distinct modulo scaling.(3#) In general, decomposing a solution v of (3.1) into positive and negativeparts v = v+ +v", where v± = ±max{±v, 0}, upon testing (3.1) with v± fromSobolev’s inequality we infer that

0 =!

IRn

""#v " v|v|2

!"2#v± dx

=!

IRn

"|#v±|2 " |v±|2

!#dx *

(1 " S"2!/2$v±$2!"2

D1,2

)$v±$2

D1,2 .

Hence, either v - 0, or

E0(v±) =1n$v±$2

D1,2 * 1n

Sn/2 = (!.

In fact, E0(v±) > (!; otherwise S would be achieved at v±, contradictingTheorem 1.2. Therefore any solution v of (3.1) that changes sign satisfies

E0(v) = E0(v+) + E0(v") > 2(!,

and in Theorem 3.1 we can assert that E0(uj) ! {(!} 4 ]2(!,.[.


In particular, if (1.1), (1.3) does not admit any solution but the trivialsolution u - 0, the local Palais-Smale condition (P.-S.)% will hold for all ( <2(!, except for ( = (!.

When " = 0, as noted above, any non-trivial solution u of problem (1.1),(1.3) on a bounded domain satisfies E0(u) > (!. Theorem 3.1 then likewiseguarantees that condition (P.-S.)% will hold for all ( < 2(!, except for ( = (!.See Weth [1] for a recent improvement of this result.(4#) Bahri-Coron [1] observe that in Theorem 3.1 in addition we can assert

Rim

Rjm

+Rj

m

Rim

+ RimRj

m|xim " xj

m|2 % . as m % . for all i )= j.

Proof of Theorem 3.1. First recall that as in the proof of Lemma 2.3 any (P.-S.)-sequence for E! is bounded. Hence we may assume that um ' u0 weaklyin H1,2

0 (!), and u0 solves (1.1), (1.3). Moreover, if we let vm = um " u0 wehave vm % 0 strongly in L2(!), and by (2.1), (2.2) also that

!

"|vm|2

!dx =

!

"|um|2

!dx "

!

"|u0|2

!dx + o(1) ,

!

"|#vm|2 dx =

!

"|#um|2 dx "

!

"|#u0|2 dx + o(1) ,

where o(1) % 0 (m % .). Hence, in particular, we obtain that

E!(um) = E!(u0) + E0(vm) + o(1) .

Also note that

DE!(um) = DE!(u0) + DE0(vm) + o(1) = DE0(vm) + o(1) ,

where o(1) % 0 in H"1(!) (m % .). Using the following lemma, we can nowproceed by induction:

3.3 Lemma. Suppose (vm) is a (P.-S.)-sequence for E = E0 in H1,20 (!) such

that vm ' 0 weakly. Then there exists a sequence (xm) of points xm ! !, asequence (Rm) of radii Rm % . (m % .), a non-trivial solution v0 to thelimiting problem (3.1) and a (P.-S.)-sequence (wm) for E0 in H1,2

0 (!) suchthat for a subsequence (vm) there holds

wm = vm " Rn$2

2m v0

"Rm(· " xm)

#+ o(1) ,

where o(1) % 0 in D1,2(IRn) as m % .. In particular, wm ' 0 weakly.Furthermore,

E0(wm) = E0(vm) " E0(v0) + o(1) .

Moreover,Rm dist(xm, $!) % . .


Finally, if E0(vm) % ( < (!, the sequence (vm) is relatively compact and hencevm % 0 strongly in H1,2

0 (!), E0(vm) % ( = 0 as m % ..

Proof of Theorem 3.1 (completed). Apply Lemma 3.3 to the sequences v1m =

um " u0, vjm = um " u0 "

/j"1i=1 ui

m = vj"1m " uj"1

m , j > 1, where

uim(x) = (Ri

m)n$2

2 ui"Ri

m(x " xim)

#.

By induction

E0(vjm) = E!(um) " E!(u0) "

j"1.

i=1

E0(ui)

+ E!(um) " (j " 1)(! .

Since the latter will be negative for large j, by Lemma 3.3 the induction willterminate after some index k * 0. Moreover, for this index we have

vk+1m = um " u0 "

k.

j=1

ujm % 0

strongly in D1,2(IRn), and

E!(um) " E!(u0) "k.

j=1

E0(uj) % 0 ,

as desired.

Proof of Lemma 3.3. If E0(vm) % ( < (!, by Lemma 2.3 the sequence (vm)is strongly relatively compact and hence vm % 0, ( = 0. Therefore, we mayassume that E0(vm) % ( * (! = 1

nSn/2. Moreover, since DE0(vm) % 0 wealso have

1n

!

"|#vm|2 dx = E0(vm) " 1

2!/vm, DE0(vm)0 % ( * 1

nSn/2

and hence that

(3.2) lim infm%&

!

"|#vm|2 dx = n( * Sn/2 .

Extend vm - 0 outside !. Denote

Qm(r) = supx$"

!

Br(x)|#vm|2 dx

the concentration function of vm, introduced in Section I.4.3. Choose xm ! IRn

and scalevm '% vm(x) = R

2$n2

m vm

"x/Rm + xm

#


such that

Qm(1) = supx$IRn

!

B1(x)|#vm|2 dx =

!

B1(0)|#vm|2 dx =

12L

Sn/2 ,

where L is a number such that B2(0) is covered by L balls of radius 1. Clearly,by (3.2) we have Rm * R0 > 0, uniformly in m.

Letting !m = {x ! IRn ; x/Rm + xm ! !}, we may assume that !m %!& & IRn. Since

$vm$2D1,2 = $vm$2

D1,2 % n( < .,

we also may assume that vm ' v0 weakly in D1,2(IRn). By density of C&0 (!&)

in D1,2(!&), moreover, we can find functions v0m ! H1,2

0 (!m) such that v0 =limm%& v0

m in D1,2(!&).We claim that vm % v0 strongly in H1,2(!(), or, equivalently, that the

functions (vm " v0m) % 0 strongly in H1,2(!() for any !( && IRn. It su"ces

to consider balls !( = B1(x0) for any x0 ! IRn. For brevity in the following welet Br(x0) =: Br.

Choose cut-o! functions )i ! C&0 (IRn) such that 0 + )i + 1, i = 1, 2,

)1 - 1 in B1, )1 - 0 outside B3/2, while )2 - 1 in B3/2, )2 - 0 outside B2.Then let wi

m = (vm " v0m))i ! H1,2

0 (!m), i = 1, 2. Note that we have

$wim$H1,2("m) + c$vm " v0

m$H1,2("m) + C < ., i = 1, 2.

Moreover, w1m is supported in the ball B3/2, and w2

m is supported in B2.Recall that (vm " v0

m) ' 0 weakly in D1,2(IRn). The restrictions of thesefunctions to B2 then converge weakly to 0 in H1,2(B2). From compactness ofthe embedding H1,2(B2) &% Lp(B2) for any p < 2!, finally, we also find that(vm " v0

m) % 0 strongly in Lp(B2) for any such p.Using convergence arguments familiar by now and Sobolev’s inequality, we

then obtain

(3.3)

o(1) = /w1m)1, DE0(vm; !m)0

=!

IRn

"#vm#w1

m " vm|vm|2!"2w1

m

#)1 dx + o(1)

=!

IRn

"|#(vm " v0

m)|2 " |vm " v0m|2

!#)2

1 dx + o(1)

=!

IRn

"|#(vm " v0

m)|2)21 " |vm " v0

m|2!)2

1)2!"22

#dx + o(1)

=!

IRn

"|#w1

m|2 " |w1m|2|w2

m|2!"2

#dx + o(1)

* $w1m$2

D1,2(IRn)

(1 " S"2!/2$w2

m$2!"2D1,2(IRn)

)+ o(1) ,

where o(1) % 0 as m % .. But note that


!

IRn

|#w2m|2 dx =

!

IRn

|#(vm " v0m)|2)2

2 dx + o(1)

=!

IRn

"|#vm|2 " |#v0)|2

#)2

2 dx + o(1) +!

B2

|#vm|2 dx + o(1)

+ L Qm(1) + o(1) =12Sn/2 + o(1) .

Thus S"2!/2$w2m$2!"2

D1,2(IRn) + c < 1 for large m, and from (3.3) it follows thatw1

m % 0 in D1,2(IRn); that is, vm % v0 strongly locally in H1,2, as desired.In particular, !

B1(0)|#v0|2 dx =

12L

Sn/2 > 0 ,

and v0 )- 0. Since the original sequence vm ' 0 weakly, thus it also followsthat Rm % . as m % ., and !& either equals IRn or is a half-space. Nowwe distinguish two cases:(1#) Rm dist(xm, $!) + c < ., uniformly, in which case !& is a half-space, or(2#) Rm dist(xm, $!) % ., in which case !m % !& = IRn.

Since in each case for any ) ! C&0 (!&) we have that ) ! C&

0 (!m) forlarge m, there holds

/), DE0(v0; !&)0 = limm%&

/), DE0(vm; !m)0 + C limm%&

||DE0(vm)||H$1 = 0 ,

for all such ), and v0 ! H1,20 (!&) is a weak solution of (3.1) on !&. But

if !& = IRn+, by Theorem 1.3 then v0 must vanish identically. Thus (1#) is

impossible, and we are left with (2#). Thus we may now also choose a sequence(Rm) such that Rm := Rm(Rm)"1 % . while Rm dist(xm, $!) % . asm % ..

To conclude the proof, again let ) ! C&0 (IRn) be a cut-o! function satis-

fying 0 + ) + 1, ) - 1 in B1(0), ) - 0 outside B2(0), and let

wm(x) = vm(x) " Rn$2

2m v0

"Rm(x " xm)

#· )

"Rm(x " xm)

#! H1,2

0 (!).

That is,

wm(x) = R2$n

2m wm(x/Rm + xm) = vm(x) " v0(x))(x/Rm) .

Set )m(x) = )"x/Rm

#. Note that

!

IRn

%%#"v0()m " 1)

#%%2 dx

+ C

!

IRn

|#v0|2()m " 1)2 dx + C

!

IRn

|v0|2%%#()m " 1)

%%2 dx

+ C

!

IRn\BRm(0)

|#v0|2 dx + CR"2m

!

B2Rm(0)\BRm

(0)|v0|2 dx .


But #v0 ! L2(IRn). Therefore the first term tends to 0 as m % ., while byHolder’s inequality also the second term

R"2m

!

B2Rm(0)\BRm

(0)|v0|2 dx + C

0!

B2Rm(0)\BRm

(0)|v0|2

!dx

12/2!

% 0

as m % .. Thus we have wm = vm " v0 + o(1), where o(1) % 0 in D1,2(IRn).As in the proof of Lemma 2.3 we then obtain the expansion

E0(wm) = E0(wm) = E0(vm) " E0(v0) + o(1) = E0(vm) " E0(v0) + o(1).

Moreover, by a similar reasoning for any sequence of testing functions /m !H1,2(!m) with $/m$H1,2 + 1 with error o(1) % 0 as m % . we obtain

//m, DE0(wm; !m)0 =!

IRn

"#wm#/m " wm|wm|2

!"2/m

#dx

=!

IRn

"#vm#/m " vm|vm|2

!"2/m

#dx

"!

IRn

"#v0#/m " v0|v0|2

!"2/m

#dx + o(1)

= //m, DE0(vm; !m)0 " //m, DE0(v0; IRn)0 + o(1).

Thus, we find

$DE0(wm;!)$H$1

= $DE0(wm; !m)$H$1 = sup%"H1,2(!m)'%'

H1,2(1

//, DE0(wm; !m)0

+ $DE0(vm; !m)$H$1 + $DE0(v0; IRn)$H$1 + o(1)= $DE0(vm;!)$H$1 + o(1) % 0 (m % .) .

This concludes the proof.

Positive Solutions on Annular-shaped Regions

With the aid of Theorem 3.1 we can now show the existence of positive solutionsto (1.1), (1.3) on perturbed annular domains for " = 0.

The following result is due to Coron [2]:

3.4 Theorem. Suppose ! is a bounded domain in IRn satisfying the followingcondition: There exist constants 0 < R1 < R2 < . such that

! 5 {x ! IRn ; R1 < |x| < R2} ,(1#)! )5 {x ! IRn ; |x| < R1} .(2#)


Then, if R2/R1 is su!ciently large, problem (1.1), (1.3) for " = 0 admits apositive solution u ! H1,2

0 (!).

Again note that the solution u must have an energy above the compactnessthreshold given by Lemma 2.3.

The idea of the proof is to argue by contradiction and to use a minimaxmethod for S = S0( · ;!). Recall that the Sobolev constant in IRn is attainedat the functions u!

& defined in (2.5). Consider the sphere 0 = {x ! IRn; |x|2 =R1R2} in the annulus {x ! IRn; R1 < |x| < R2}. For suitably small * > 0 thenconsider the collection A = {u!

&(·" 1)) ! H1,20 (!); 1 ! 0} of shifted functions

u!& , truncated with a suitable cut-o! function ) ! C&

0 (!). If we regard thecenter of mass, as explained below in detail, then for su"ciently small * > 0the set A is homeomorphic to the sphere 0, which by assumption (2#) is notcontractible in !. After normalizing, A & M = {u ! H1,2

0 (!) : $u$L2! = 1}.On the other hand, any set A of positive functions in M is contractible in

M . Moreover, for a su"ciently large ratio R2/R1 we can contract A in such away that the corresponding minimax-value ( is smaller than the number 22/nS.If we assume that (1.1), (1.3) does not admit a positive solution, by Theorem3.1 and Remark 3.2 this is the smallest number above S where the Palais-Smalecondition fails in M . By applying the deformation lemma Theorem II.3.11 wethen conclude that the set A can be contracted below the level S + 2 for any2 > 0. For suitably small 2 > 0, finally, such a contraction will induce acontraction of 0 in !, and the desired contradiction will result.

Proof. We may assume R1 = (4R)"1 < 1 < 4R = R2. Consider the unit sphere

0 = {x ! IRn ; |x| = 1} .

For 1 ! 0, x ! IRn, 0 + t < 1 let

u)t (x) =2

1 " t

(1 " t)2 + |x " t1|2

3n$22

! D1,2(IRn) .

Note that S is attained on any such function u)t , and u)t “concentrates” at 1as t % 1. Moreover, letting t % 0 we have

u)t % u0 =2

11 + |x|2

3n$22

,

for any 1 ! 0. Choose a radially symmetric function ) ! C&0 (IRn) such that

0 + ) + 1 on !, ) - 1 on the annulus {x ; 12 < |x| < 2} and ) - 0 outside

the annulus {x ; 14 < |x| < 4}. Given any R * 1, scale

)R(x) =

*+

,

)(Rx), 0 + |x| < R"1

1, R"1 + |x| < R)(x/R), R + |x|

and let


w)t = u)t · )R, w0 = u0 · )R ! H1,20 (!) .

Note that!

IRn

|#(w)t " u)t )|2 dx + c

!

(IRn\B2R))B(2R)$1

|#u)t |2 dx

+ c · R"2

!

B4R\B2R

|u)t |2 dx + cR2

!

B(2R)$1

|u)t |2 dx % 0

as R % ., uniformly in 1 ! 0, t ! [0, 1[. Also consider the normalizedfunctions

v)t = w)t4$w)t $L2! , v0 = w0

4$w0$L2! .

Since $w)t " u)t $D1,2 % 0 as R % . and $u)t $D1,2 = $u0$D1,2 > 0, we have

S(v)t ;!) = S(w)t ;!) % S(u)t ; IRn) = S

as R % ., uniformly in 1 ! 0, 0 + t < 1. In particular, if R * 1 is su"cientlylarge, we can achieve that

sup),t

S"v)t ;!

#< S1 < 22/nS

for some constant S1 ! IR.Fix such an R and suppose that ! satisfies (1#), (2#) of the Theorem with

R2 = 4R = R"11 for this number R. Let M be the set

M = {u ! H1,20 (!) ;

!

"|u|2

!dx = 1} ,

and for u ! M letF (u) =

!

"x|#u|2 dx

denote its center of mass. Suppose (1.1), (1.3) does not admit a positive solu-tion. This is equivalent to the assertion that

E(u) =12

!

"|#u|2 dx " 1

2!

!

"|u|2

!dx

does not admit a critical point u > 0. By Remark 3.2, therefore, E satisfies(P.-S.)% on H1,2

0 (!) for 1nSn/2 < ( < 2

nSn/2. Equivalently, S0( · ;!) satisfies(P.-S.)% on M for S < ( < 22/nS. Moreover, S0( · ;!) does not admit a criticalvalue in this range.

By the deformation lemma Theorem II.3.11 and Remark II.3.12, therefore,for any ( in this range there exists * > 0 and a flow 3: M 2 [0, 1] % M suchthat

3(M%+&, 1) & M%"& ,

whereM% = {u ! M ; S(u;!) < (} .


Given 2 > 0, we may cover the interval [S + 2, S1] by finitely many such *-intervals and compose the corresponding deformations to obtain a flow 3: M 2[0, 1] % M such that

3(MS1 , 1) & MS+* .

Moreover, we may assume that 3(u, t) = u for all u with S(u;!) + S + 2/2.On the other hand, it easily follows either from Theorem I.4.8 or Theo-

rem 3.1 that, given any neighborhood U of !, there exists 2 > 0 such thatF (MS+*) & U . Indeed, for any sequence (um), where um ! MS+ 1

m, by Theo-

rem I.4.8 and Theorem 1.2 there exists a subsequence such that, as m % .,

|um|2!

dx' 2x(0) , |#um|2 dx ' S2x(0)

for some x(0) ! !. Since ! is smooth we may choose a neighborhood U of! such that any point p ! U has a unique nearest neighbor q = .(p) ! !and such that the projection . is continuous. Let 2 > 0 be determined forsuch a neighborhood U , and let 3: M 2 [0, 1] % M be the corresponding flowconstructed above. The map h:0 2 [0, 1] % !, given by

h(1, t) = .&

F"3(v)t , 1)

#',

then is well-defined, continuous, and satisfies

h(1, 0) = .&

F"3(v0, 1)

#'=: x0 ! ! , for all 1 ! 0

h(1, 1) = 1, for all 1 ! 0 .

Hence h is a contraction of 0 in !, contradicting (2#).

Actually, the e!ect of topology is much stronger than indicated by Theorem3.4. In a penetrating analysis, Bahri-Coron [1] have obtained the followingresult; see also Bahri [2]:

3.5 Theorem. Suppose ! is a domain in IRn such that

Hd(!, ZZ2) )= 0

for some d > 0. Then (1.1), (1.3) admits a positive solution for " = 0.

Note that if ! & IR3 is non-contractible then either H1(!, ZZ2) or H2(!, ZZ2))= 0 and the conclusion of Theorem 3.5 holds. It is conjectured that a similarresult will also hold for n * 4.

3.6 Notes. (1#) Benci-Cerami-Passaseo [1] – see, in particular, Theorem 3.1 –have further refined the “method of photography” used in the proof of Theorem3.4 above and have applied it to obtain multiple solutions of semilinear ellipticequations also on unbounded domains.


(2#) Schoen-Zhang [1] have obtained pointwise estimates for concentrating so-lutions um > 0 with um ' u0 weakly in H1,2(S3) to subcritical equations

"#um +n(n " 2)

4um = Kupm"1

m on S3,

comparing um to a “bubble” Rn$2

2m u!

1(Rm(x " xm)) of the “right” scale Rm,centered around a suitable point xm. Similar and improved pointwise estimatesfor bubbling families of solutions um > 0 to

"#um + hmum = u2!"1m

on manifolds have been obtained by Druet-Hebey-Robert [1] in the case whenhm % h0 in C0,+ while um ' u0 weakly in H1,2.(3#) Results similar to Theorem 3.1 have been established for higher orderequations of Yamabe type by Hebey-Robert [1].(4#) The work of Brezis-Merle [1] Li-Shafrir [1], Adimurthi-Struwe [1] and,finally, Druet [1] shows that similar geometric quantization patterns also char-acterize the blow-up behavior for concentrating solutions to semilinear ellipticequations of critical exponential growth on a planar domain. In recent work ofStruwe [28] the results of Druet [1] have been extended to critical semilinearelliptic equations of fourth order on four-dimensional domains.

4. The Yamabe Problem

Equation (1.1) arises in a geometric context in the problem whether a givenmetric g0 on a closed manifold M of dimension n * 3 with scalar curvatureRg0 = R0 can be deformed conformally to a metric g of constant scalar curva-ture. If we let

g = u4

n$2 g0 ,

where u > 0 gives the conformal factor, the scalar curvature R = Rg of themetric g is given by the equation

(4.1) "4(n " 1)n " 2

#0u + R0u = Ru2!"1,

where #0 = #g0 is the Laplace-Beltrami operator on the manifold M withrespect to the original metric g0; see Yamabe [1], T. Aubin [2], [3]. Observethat by its intrinsic geometric meaning Equation (4.1) is conformally invariant;that is, if u solves (4.1) on (M, g0) and if

g0 = v4

n$2 g0, v > 0

is conformal to a metric g0 on M , then u = uv satisfies (4.1) on (M, g0) withR0 replaced by the scalar curvature R0 of the metric g0; see also Aubin [3;Proposition, p. 126].

4. The Yamabe Problem 195

4.1 The sphere. Of particular interest is the case of the sphere M = Sn withthe standard metric induced by the embedding Sn &% IRn+1. We center Sn atthe origin of IRn 2 IR. Via the inverse of stereographic projection

IRn 6 x '%&

2x

1 + |x|2 ,1 " |x|2

1 + |x|2

'! Sn & IRn+1 ,

Sn induces a metric g = 4[1+|x|2]2 gIRn of constant scalar curvature Rg = n(n"1)

on IRn, conformal to the Euclidean metric. The classical solution

u(x) =[n(n " 2)]

n$24

[1 + |x|2]n$22

of (3.1) on IRn thus reappears in the conformal factor that changes the flatEuclidean metric to a metric of constant scalar curvature, and the invarianceof (3.1) under scaling

u '% ur,x0(x) = rn$2

2 u"r(x " x0)

#

reflects the action of the group of conformal di!eomorphism of Sn via transla-tions and dilations in the chart IRn.

The Variational Approach

We can give a variational formulation of (4.1) analogous to the one given for(1.1)–(1.3). Let H1,2(M, g0) be the set of Sobolev functions u: M % IR suchthat in local coordinates on M , with gij

0 (x) denoting the coe"cients of theinverse of the metric g0(x), and with volume element

dµ0 = dµg0 =5

det(g0) dx,

we have$u$2

H1,2 :=!

M

"|#u|20 + |u|2

#dµ0 < .

where |#u|20 = gij0 $iu$ju. By convention, we tacitly sum over repeated indices.

Let cn = 4(n"1)n"2 . In analogy with the Sobolev ratio in the Euclidean case,

for 0 )= u ! H1,2(M, g0) we define

S(u) = S(u; (M, g0)) =$

M

"cn|#u|20 + R0|u|2

#dµ0

"$M |u|2!dµ0

#2/2! .

Then S is of class C1 on the space H1,2(M, g0) \ {0}, and critical points u > 0of S on the unit sphere in L2!

(M, g0) correspond to solutions of (4.1) withconstant scalar curvature R = S(u), the total scalar curvature of the metricg = u

4n$2 g0. In particular, if the Yamabe constant

Y0 = Y (M, g0) = inf6S(u); 0 )= u ! H1,2(M, g0)

7,


on (M, g0) is attained at a function u > 0, Equation (4.1) will hold with R - Y0.If Y0 )= 0, a constant multiple of u then will solve (4.1) with R = 1 or

R = "1. Thus, depending on the sign of Y (M, g0), the latter question can alsobe approached as a minimax problem for the free functional

E(u) = E(u; (M, g0)) =12

!

M

"cn|#u|20 + R0|u|2

#dµ0 ±

12!

!

M|u|2

!dµ0,

where the “+”-sign (“"”-sign) is valid if Y (M, g0) is negative (positive). IfY (M, g0) = 0, also R = 0 and (4.1) reduces to a linear equation.

Note that the Yamabe constant Y (M, g0) is independent of the met-ric g0 representing a conformal class. In particular, we have n(n " 1) =Y (Sn, gSn) = Y (IRn, gIRn) = cnS, where S is the Sobolev constant for theembedding D1,2(IRn) &% L2!

(IRn). An argument as in Remark I.4.5 then alsoshows that there holds Y (M, g0) + Y (Sn, gSn) for any (M, g0).

If Y (M, g0) < 0, the functional E is coercive and weakly lower semi-continuous on H1,2(M, g0), as may easily be verified by the reader. Hence inthis case the existence of a non-negative critical point follows from the directmethods; see Chapter I. Alternatively, one may employ the method of sub- andsupersolutions to (4.1) used by Kazdan-Warner [1] or Loewner-Nirenberg [1];see also Sections I.2.3–I.2.6.

The di"cult case is the case when Y (M, g0) > 0. Years after Yamabe’s[1] first – unsuccessful – attempt to solve (4.1) for general manifolds (M, g0),Trudinger [1; Theorem 2, p. 269] obtained a rigorous existence result for smallpositive Y (M, g0). His approach then was refined by Aubin [2]. By usingoptimal Sobolev estimates, Aubin was able to show the following result onwhich Lemma 2.2 above was modeled.

4.2 Lemma. If Y (M, g0) < Y (Sn, gSn), then Y (M, g0) is attained at a solutionu > 0 to (4.1), inducing a conformal metric of constant scalar curvature.

As a companion result in the spirit of Lemma 2.3 we find that E satisfies(P.-S.)% on H1,2(M, g0) for ( < 1

nY (Sn, gSn)n/2.Aubin succeeded in showing that the condition Y (M, g0) < Y (Sn, gSn) is

satisfied if (M, g0) is of dimension n * 6 and not locally conformally flat. Bymeans of the “positive mass theorem” Schoen [1], finally, was able to show thatthis condition also holds in all the remaining cases when (M, g0) )= (Sn, gSn).Thus, we obtain the following result.

4.3 Theorem. On any smooth, compact Riemannian manifold (M, g0) withoutboundary of dimension n * 3 there exists a conformal metric of constant scalarcurvature, a “Yamabe metric”.


The Locally Conformally Flat Case

Particularly powerful tools are available for dealing with the Yamabe problemon a locally conformally flat manifold (M, g0). In fact, a complete Morse theoryfor the Yamabe problem may be obtained in this case.

4.4 The developing map. A manifold (M, g0) is locally conformally flat iffor every point p ! M there is a neighborhood U of p in M and a confor-mal di!eomorphism 3 from (U, g0) to Sn, endowed with the standard metric.Locally conformally flat manifolds were studied by Kuiper and, in particular,by Schoen-Yau [1], whom we follow in the sequel. If n * 3 and if M is sim-ply connected, by a standard monodromy argument the local di!eomorphisms3: U & M % Sn can be extended to a conformal immersion 3: M % Sn,the developing map, which is unique up to composition with a conformal, orMobius, transformation of Sn. Similarly, for an arbitrary locally conformallyflat n-manifold (M, g0), n * 3, there is a developing map 3 from the universalcover M of M to Sn, and the fundamental group .1(M) acting on M is mappedinto the Mobius group by a homomorphism called the holonomy representation.

One of the main accomplishments in Schoen-Yau [1] is to exhibit a largeclass of locally conformally flat manifolds where the developing maps are injec-tive.

4.5 The conformal Laplace operator. The left-hand side of (4.1) defines theconformal Laplace operator on (M, g0),

L0 = Lg0 = "cn#0 + R0 ,

scaled with the factor cn = 4(n"1)n"2 for convenience. Recall that throughout

this section we only consider closed manifolds (M, g0), that is, compact man-ifolds without boundary. Let "0 be the lowest eigenvalue of L0 and u0 > 0 acorresponding eigenfunction. By (4.1), the metric g = u4/(n"2)

0 g0 has a scalarcurvature Rg of constant sign, in fact, of the same sign as "0, which is the sameas the sign of the Yamabe invariant Y (M, g0). Thus, according to the sign ofY (M, g0), a closed locally conformally flat manifold (M, g0) may be classifiedconformally invariantly as scalar positive, scalar negative, or scalar flat.

In the following we may restrict ourselves to the case that (M, g0) is scalarpositive as the most interesting case. For such manifolds we have the followingresult of Schoen-Yau [1], Proposition 3.3 and 4.4.

4.6 Theorem. Suppose (M, g0) is closed, locally conformally flat and scalarpositive. Then the developing map 3: M % Sn is injective and 4 = $(3(M))has vanishing Newtonian capacity.

Via the covering map .: M % M we lift g0 to an equivariant metric g0 = .!g0.Using Theorem 4.6, we can push this metric forward conformally to a completemetric


h0 = 3!g0 = v4

n$20 gSn

on 3(M) & Sn, conformal to the standard metric gSn on Sn.By a second result of Schoen-Yau [1], Proposition 2.6, the conformal factor

v0 diverges uniformly near the boundary 4 of 3(M).

4.7 Proposition. There exists a constant c > 0 such that for x ! 3(M) thereholds v0(x) * c dist(x,4 )

2$n2 . The constant c = c(M, g0) depends only on the

conformal class of g0 and bounds for the curvature of g0 and its derivativeswith respect to g0.

As a final normalization, upon replacing g0 by a constant multiple of g0, ifnecessary, we may assume that (M, g0) has unit volume. Similarly, we willimpose the volume constraint

(4.2) V ol(g) =!

Mdµ =

!

Mu2!

dµ0 = 1.

on comparison metrics g = u4

n$2 g0, where dµ = dµg = u2!dµ0.

The Yamabe Flow

With these prerequisites we now turn to the construction of a Yamabe metricg& on (M, g0). Instead of using one of the standard variational methods pre-sented so far, following ideas of Hamilton [1] we will approach the problem bymeans of the parabolic evolution equation corresponding to the “L2-gradient”flow for the Yamabe energy S(u), whose convergence on a locally conformallyflat scalar positive manifold was demonstrated by Ye [1].

4.8 The evolution problem. The metric g& is determined as the limit as t % .of the “Yamabe flow” of metrics (g(t))t*0 issuing from g(0) = g0. Letting

(4.3) g(t) = u(·, t)4

n$2 g0

with Rg(t) = R(t), this flow is defined by letting u: M 2 [0,.[% IR solve theevolution problem

(4.4)$up

$t= sup " L0u = (s " R)up

for p = 2! " 1 = n+2n"2 with initial condition

u(·, 0) = 1

and with a function s = s(t) determined in such a way that the volume con-straint (4.2) is preserved. From the equation


0 =p

2!d

dtV ol(g(t)) =

!

Mu$up

$tdµ0 =

!

Mu("L0u + sup) dµ0

= "!

M

"cn|#u|20 + R0u

2#dµ0 + s

!

Mu2!

dµ0

we then deduce that

(4.5) s(t) = S(u(t)) for all t.

Multiplying (4.4) by ut and integrating over M , in view of (4.2) we alsoobtain the energy identity

2p

!

Mup"1|ut|2 dµ0 +

d

dtS(u(t)) =

2s

2!d

dtV ol(g(t)) = 0 .

Hence, the function t '% s(t) is non-increasing with a well-defined limit

s& = limt%&

s(t) * Y (M, g0) > 0.

In addition, we have the a-priori estimate

2p

! &

0

!

Mup"1|ut|2 dµ0dt = s(0) " s& + S(1) " Y (M, g0) < .

for any smooth global solution u of (4.4), (4.5). In particular, if u together withits space-time derivatives is uniformly bounded on M 2 [0,. [, a sequence u(t)as t % . will accumulate at a time-independent solution u& of (4.4), that is,a solution of (4.1) with R = s&.

When (M, g0) is locally conformally flat the necessary a-priori estimatesare a consequence of the following logarithmic gradient bound for equation(4.4), due to Ye [1], Theorem 4.

4.9 Theorem. Suppose that (M, g0) is a closed, locally conformally flat n-manifold, n * 3. There exists a constant C = C(M, g0) such that for anysmooth solution u: M 2 [0, T [% IR of (4.4), (4.5) with u(·, 0) = 1 there holds

supM+[0,T [

|#g0u|u

+ C .

By integrating along a shortest geodesic connecting points on M where u(t)achieves its maximum, respectively, its minimum, the estimate of Theorem 4.9implies the uniform Harnack inequality

infM

u(t) * c supM

u(t)

for 0 < t < T with a uniform constant c > 0. Hence, in view of the volumeconstraint (4.2), the solution u is uniformly bounded from above and awayfrom 0. But then Equation (4.4) is uniformly parabolic, and we obtain uniform


derivative estimates for u in terms of the data, as well. In particular, any localsolution u may be extended for all time and, as t % . suitably, by our remarksabove will converge smoothly to a limit u& inducing a conformal metric g&of constant scalar curvature on M . Theorem 4.9 thus implies the existence ofa Yamabe metric on any closed, locally conformally flat manifold (M, g0) ofdimension n * 3 and completes the proof of Theorem 4.3 in this case.

The Proof of Theorem 4.9 (following Ye [1])

We only consider the scalar positive case Y (M, g0) > 0. In this case by Theorem4.6 we can transplant the problem from our abstract manifold M to the spherevia the developing map 3 defined above. We then consider the evolution of themetrics

h(t) = 3!g(t) = u(t)4

n$2 h0 = v(t)4

n$2 gSn

on 3(M) & Sn, where g(t) = .!g(t), u(y, t) = u(.(3"1(y)), t), and wherev(t) = u(t)v0. Let Lh0 denote the conformal Laplace operator on (3(M), h0).

Since . 7 3"1 defines a local isometry between (3(M), h0) and (M, g0),from (4.4) we deduce the equation

(4.6)$up

$t+ Lh0 u = sup

on 3(M) 2 [0, T [. But (4.1) implies

u"pLh0 u = Rh(t) = v"pLSnv .

Hence, in terms of the round spherical metric gSn as background metric on3(M) & Sn we can also express (4.4) as the flow equation

(4.7)$vp

$t+ LSnv = svp .

Moreover, the condition u(·, 0) = u(·, 0) = 1 yields the initial condition

(4.8) v |t=0= v0 .

Fix a fundamental domain N & 3(M) for the action of the Kleinian groupG corresponding to .1(M) via the holonomy representation. For the proof ofTheorem 4.9 it su"ces to establish a bound for

(4.9) supN+[0,T [

|#gSn v|v

.

We may assume that there is an open neighborhood V of the closure of N suchthat dist(V,4 ) > 0, where 4 = $

"3(M)

#.

Given q0 ! N , we introduce conformal charts for a neighborhood of q0

and for Sn \ {q0}, as follows. After a rotation, we may assume that q0 is the


north pole q0 = (0, . . . , 0, 1) ! IRn+1. Then let F : Sn \ {q0} % IRn denotestereographic projection from q0 with inverse

F"1(x) =&

2x

1 + |x|2,|x|2 " 1|x|2 + 1

', x ! IRn .

Via stereographic projection from the south pole "q0, similarly we obtain thecoordinate representation

G(x) =&

2x

1 + |x|2 ,1 " |x|21 + |x|2

'

of Sn \ {"q0}. Note that G(0) = q0 and that we have the relation F"1(x) =G( x

|x|2 ) for x )= 0. We pull back the metric h(t) = v(t)4

n$2 gSn via F"1 toobtain a conformal metric on IRn, given by

(F"1)!h(t) = w(t)4

n$2 gIRn .

Since

(F"1)!gSn =&

21 + |x|2

'2

gIRn ,

it follows that

w(x, t) =&

21 + |x|2

'n$22

v

&G

" x

|x|2#

, t

'.

Thus, as |x| %. the function w(t) has the asymptotic expansion(4.10)

w(x, t) =2

n$22

|x|n"2

&a0 +

aixi

|x|2 +"aij "

n " 22

a02ij#xixj

|x|4

'+ O

&1

|x|n+1

'

$w

$xi(x, t) =

2n$2

2

|x|n

&ai " xi

"(n " 2)a0 + n

ajxj

|x|2#'

+ O

&1

|x|n+1

',

where

a0 = a0(t) = v"G(0), t

#> 0, ai = ai(t) =

$"v(·, t) 7 G

#

$xi(0) ,

aij = aij(t) =12$2

"v(·, t) 7 G

#

$xi$xj(0) , 1 + i, j + n .

The point y(t) with coordinates

yi(t) =ai(t)

(n " 2)a0(t)

is called the center of w(t). Note that if we shift coordinates by y(t), thenw(x, t) = w

"x + y(t), t

#has the asymptotic expansion


(4.11)w(x, t) =

2n$2

2

|x|n"2a0 + O

&1

|x|n

',

$w

$xi(x, t) = "(n " 2)2

n$22

|x|na0xi + O

&1

|x|n+1

'

as |x| %. .Moreover, a uniform bound for y gives the desired bound for (4.9). Thus,

the proof of Theorem 4.9 will be complete once we establish the followingestimate.

4.10 Lemma. There is a constant C depending only on (M, g0) and dist(V,4 ),such that |y(t)| + C, uniformly for q ! N , 0 + t < T .

Proof. Fix some number 0 < t < T . After a rotation of coordinates anda reflection in the hyperplane {xn = 0}, if necessary, we may assume thatyn(t) = maxi |yi(t)|.

For " > 0, x = (x(, xn) ! IRn denote

x! = (x(, 2"" xn)

the image of x after reflection in the hyperplane {xn = "}. By the expansion(4.10) and the arguments of Gidas-Ni-Nirenberg [1], Lemma 4.2, there exists"0 * 1 such that for any " * "0 there holds

(4.12) w(x, 0) > w(x!, 0), if xn < ";$w(x, 0)$xn

< 0, if xn = " .

Note that "0 only depends on (M, g0) and dist(V,4 ). Here and in the followingwe extend v to all of Sn 2 [0, T [ by letting v(x, t) = . for x ! 4 , as suggestedby Proposition 4.7. Similarly, we extend w to IRn2[0, T [ by letting w(x, t) = .for x ! F (4 ). In particular, by (4.12) the number "0 has to be chosen so thatF (4 ) lies “below” the hyperplane {xn = "0}; in fact, we may choose "0 suchthat F (4 ) lies strictly below {xn = "0}.

Since, by assumption, u is smooth on M 2 [0, t] and hence v is uniformlysmooth on V 2 [0, t], the expansions (4.10) are uniform in t ! [0, t]. By thesame arguments as cited above, therefore, there exists a number "1 * "0 suchthat for every " * "1 and every t ! [0, t] there holds

(4.13) w(x, t) > w(x!, t), if xn < " .

For " * "0 define w!(x, t) = w(x!, t). We restrict w! to the region wherexn + ", x /! F (4 ), 0 + t + t. Note that the functions w and w! satisfythe evolution equation (4.4) with respect to the flat Euclidean metric on thisdomain. Moreover, w = w! for xn = ".

LetI = {" ! IR; " >" 0, " > max

0't'tyn(t), w! + w} .


By (4.13) the set I is non-empty. We will show that I is open and (relatively)closed in the interval ]"0,. [; hence I =]"0,. [, which implies that yn(t) + "0.Since t < T was arbitrary, this then implies the assertion of the lemma.

I is open. Indeed, by (4.12) for " * "0 equality w! - w is impossible.(One might also use the singular set F (4 ) to rule out w! - w.) Hence, for any" ! I the parabolic maximum principle and (4.12) imply that we have

(4.14) w!(x, t) < w(x, t) for xn < ", 0 + t + t ,

and

(4.15)$w

$xn(x, t) < 0 for xn = ", 0 + t + t .

Moreover, by Proposition 4.7 and uniform boundedness of w! near F (4 )2[0, t],uniformly in " * "0, there exists c > 0 such that there holds

lim infx%F (, )

"w(x, t) " w!(x, t)

#* c > 0 ,

uniformly in t ! [0, t] and uniformly in " * "0.By (4.10), given any 50 * "0 there is a constant r0 such that the estimates

(4.14) and (4.15) hold true for |x| * r0, for all t ! [0, t] and " ! ["0,50]. Thusthere exists * > 0 such that ]"" *,"+ * [& I; that is, I is open.

I is also closed. Indeed, suppose that " >" 0 belongs to I. By continuity,we have w! + w and " * max0't't yn(t). Suppose that " = max0't't yn(t).Then " = yn(t0) for some t0 ! [0, t]. Shift coordinates by yn(t0) and denote byw(x, t) = w(x(,"+xn, t), w! = w!(x(,"+xn, t) = w(x(,""xn, t), respectively,the transformed functions. Also let 4 be the shifted singular set F (4 ). Via Fwe lift w, w! back to the sphere by letting

z4

n$2 gSn = F !w4

n$2 gIRn ,

(z!)4

n$2 gSn = F !(w!)4

n$2 gIRn .

Then z, z! are defined on (Sn" \ F"1(4 )) 2 [0, t], where Sn

" is the hemispherecorresponding to xn < 0. (z is related to v by a conformal di!eomorphism ofSn fixing the north pole q0.) Moreover, z and z! satisfy (4.7). We also knowthat z! + z and z! = z along $Sn

".Finally, note that by (4.11) the term an in the expansion (4.10) for w(t0)

vanishes. In terms of z and z! this translates into the condition

$z

$%(q0, t0) =

$z!

$%(q0, t0) = 0 ,

where % is the outward unit normal on $Sn". But then the strong parabolic

maximum principle implies that z - z!, that is, w - w!. This contradicts(4.12). Hence " > max0't't yn(t); that is, " ! I, and I is closed.

The proof is complete.


Convergence of the Yamabe Flow in the General Case

Ye’s Theorem 4.9 depends heavily on the special symmetries of the sphere.While it is not yet known if a similar result will hold for the Yamabe flow onan arbitrary closed manifold, it is possible to show convergence of the flow indimensions 3 + n + 5. The following result is due to Schwetlick-Struwe [1] andBrendle [1]. In the case n = 3 a di!erent proof was given by Gruneberg [1].

4.11 Theorem. Let (M, g0) be a smooth compact, scalar positive manifoldwithout boundary of dimension 3 + n + 5. Then there is a unique, global,smooth solution u > 0 to the Yamabe flow (4.4) with initial data u(0) - 1 suchthat the associated metrics g(t) = u(t)

4n$2 g0 as t % . converge to a metric

g& of constant scalar curvature.

Schwetlick-Struwe [1] obtain the result for initial metrics g0 with R0 =Rg0 > 0 and whose total scalar curvature s0 is small in the sense that

(4.16) 0 < Y0 + s0 +"Y (M, g0)n/2 + Y (Sn, gSn)n/2

#2/n.

Brendle [1] later was able to remove this restriction. Moreover, he also treatedthe case when the initial metric g0 has scalar curvature of varying sign.

The proof of Theorem 4.11 for the general case is very technical. In thefollowing we will therefore focus on the case when Rg0 > 0 and when g0 satisfies(4.16). Moreover, we may assume that (M, g0) is not conformal to (Sn, gSn).

To simplify the notation, in the following we rescale time with the factorp = n+2

n"2 so that our flow equation (4.4) becomes

(4.17) ut = (s " R)u , u(0) = 1 .

From (4.1) we also obtain the identity

Rt = u" n+2n$2 L0ut "

n + 2n " 2

Rut

u

for the evolution of the scalar curvature. Thus, using the invariance property

u" n+2n$2 L0(uv) = Lv = ("cn#+ R)v ,

where L = Lg with # = #g, we may rephrase the former equation as

(4.18) Rt = L(s " R) +n + 2n " 2

R(R " s) = cn#R +4

n " 2R(R " s) .

Finally, in the scaled time variable the energy inequality reads

(4.19) st =d

dt

&!

MR dµ

'= 2

!

MR(s " R) dµ = "2

!

M|R " s|2 dµ + 0 .


4.12 Global existence. We readily obtain uniform upper and lower a-prioribounds for the solution u to (4.4) on any finite time interval. Standard parabolicestimates then yield global existence of the flow.

First we show that R > 0 for all t * 0. Let R(t) = e4s0n$2 tR(t). Then, as

long as R * 0, from (4.18), (4.19) we have

Rt " cn#R =4

n " 2(R " s + s0)R * 4R

n " 2R * 0.

The maximum principle now implies that

R * minM

R(0) = minM

R0,

uniformly, and we conclude that

minM

R(t) * minM

R0 · e"4s0n$2 t

as claimed.But then from (4.17) we also obtain the uniform upper bound

supM

u(t) + es0t supM

u(0) = es0t

for any t. Moreover, we have

4.13 Lemma. Suppose that u > 0 is a smooth function inducing a metricg = u

4n$2 g0 with R * 0, normalized by (4.2). Then with constants C, q * 1

depending only on (M, g0) there holds

C infM

u * ||u||1"qL) .

Proof: In view of the equation L0u = Ru2!"1 * 0, the weak Harnack inequalityimplies the estimate

C infM

u * ||u||Lp0

with uniform constants C, p0 > 0; see Schwetlick-Struwe [1], Theorem A.2. Wemay assume p0 + 2!; otherwise, (4.2) gives a uniform lower bound for u andwe are done. But then by (4.2) we have

1 = ||u||2!

L2! + ||u||p0Lp0 ||u||

2!"p0L) .

Hence we obtainC inf

Mu * ||u||1"2!/p0

L) ,

as desired.

In view of Lemma 4.13, our uniform upper bound for u(t) then also implies auniform lower bound for any finite time. Global existence of the flow (4.4) nowfollows.


4.14 Curvature decay. For q * 1 also consider the functionals

Sq(g) =!

MRq dµ , Fq(g) =

!

M|R " s|q dµ ,

so that s(g) = S1(g). Note that (4.17) and (4.18) give

$tSq = q

!

MRq"1Rt dµ + 2!

!

MRq ut

udµ

= qcn

!

MRq"1#R dµ +

4q " 2n

n " 2

!

MRq(R " s) dµ

= "4(q " 1)cn

q

!

M|#Rq/2|2 dµ +

4q " 2n

n " 2

!

M(Rq " sq)(R " s) dµ .

By convexity, moreover, we have

(Rq " sq)(R " s) * |R " s|q+1 .

Thus, for 1 + q < n2 we obtain that

(4.20)! &

0Fq+1(g(t)) dt =

! &

0

!

M|R " s|q+1 dµ dt + n " 2

2n " 4qSq(g0) .

In particular, for q = 1 we recover the estimate (4.19). Moreover, for q < n2 we

find

(4.21) lim inft%&

Fq+1(g(t)) = 0 .

In fact, the convergence is uniform.

4.15 Lemma. For any 1 + p < n+22 there holds Fp(g(t)) % 0 as t % ..

The assertion of Lemma 4.15 even holds true for any p < .; see Schwetlick-Struwe [1], Lemma 3.3.

Proof of Lemma 4.15. By Holder’s inequality, for 1 + q + p we have Fq + F q/pp .

It therefore su"ces to prove the claim for p * n+12 * 2. For such p we have

$tFp(g) = p

!

M(R " s)p"1(Rt " st) dµ + 2!

!

M|R " s|p ut

udµ

= pcn

!

M(R " s)p"1#R dµ +

4p

n " 2

!

M|R " s|pR dµ

" pst

!

M(R " s)p"1 dµ " 2!

!

M(R " s)p+1 dµ

= "4(p " 1)cn

p

!

M|#(R " s)p/2|2 dµ +

4p " 2n

n " 2

!

M(R " s)p+1 dµ

+4ps

n " 2

!

M|R " s|p dµ " pst

!

M(R " s)p"1 dµ ,


where z+ := z|z|+"1 for all z ! IR, any 6 > 0. That is, recalling that p * 2and taking account of (4.19), with constants C = C(p, g0) we have

(4.22)

$tFp(g) + "!

M

"cn|#(R " s)p/2|2 + R|R " s|p

#dµ

+ 2pF2(g)Fp"1(g) +" 4ps0

n " 2+ s

#Fp(g) +

" |4p " 2n|n " 2

+ 1#Fp+1(g)

+ "Y0Fp!(g)n$2

n + CFp(g) + CFp+1(g) ,

where p! = npn"2 . Here we also used the fact that our assumption Y0 > 0 implies

the uniform Sobolev type inequality

Y0

&!

M|v|2

!dµ

'n$2n

+!

M

"cn|#v|2 + Rv2

#dµ

for all g = g(t) and all functions v ! H1(M, g); moreover, we observe that byHolder’s inequality we can bound F2(g)Fp"1(g) + Fp+1(g).

Furthermore, by Holder’s inequality again, there holds

Fp+1(g) = ||R " s||p+1Lp+1(M,g) + ||R " s||p+

Lp!(M,g)||R " s||1+(1"+)p

Lp(M,g) ,

where 6 = n2p < 1. Applying Young’s inequality

AB + 6A1/+ + (1 " 6)B1/(1"+) + A1/+ + B1/(1"+),

for any 2 > 0 we then obtain the bound

Fp+1(g) + 2||R " s||pLp!(M,g)

+ C||R " s||p(1+%)Lp(M,g) = 2Fp!(g)

n$2n + CFp(g)1+% ,

where C = C(2) = 2"&

1$& , ( = 1p(1"+) . For su"ciently small 2 > 0 from (4.22)

we then obtain that

$tFp(g) + CFp(g) + CFp(g)1+%

and the claim follows from (4.20), (4.21).

By means of Lemma 4.15, we can deduce convergence of the flow (4.4) as aconsequence of the following proposition.

4.16 Proposition. For any sequence tk % . (k % .) there exist constants0 < - < 1, C such that for a subsequence there holds

s(tk) " s& + CF 2nn+2

(g(tk))n+22n (1+-) .

Indeed, Proposition 4.16 implies the following result.


4.17 Lemma. There exist constants 0 < - < 1, t0 such that for all t * t0 thereholds

s(t) " s& + F2(g(t))1+$2 .

Proof. We argue indirectly. In view of Lemma 4.15 we thus suppose that for asequence tk % . (k % .) we have

s(tk) " s& * F2(g(tk))1+1/k

2 .

But by Proposition 4.16 and Holder’s inequality for a subsequence with suitableconstants - > 0, C there holds

s(tk) " s& + CF 2nn+2

(g(tk))n+22n (1+-) + CF2(g(tk))

1+$2 .

We then conclude that1 + CF2(g(tk))

$$1/k2 ,

which for large k contradicts Lemma 4.15. The proof is complete.

4.18 Lemma. We have! &

0F2(g(t))

12 dt < ..

Proof. In view of (4.19), the assertion of Lemma 4.17 translates into the dif-ferential inequality

d

dt(s(t) " s&) = "2F2(g) + "C(s(t) " s&)1+*

for some 0 < 2 < 1. It follows that

s(t) " s& + Ct"1/*

for some constant C > 0. Hence from (4.19) we deduce0! 2T

TF2(g(t))

12 dt

12

+ T

! 2T

TF2(g(t)) dt + T

2(s(T ) " s(2T )) + CT 1"1/* .

By diadic decomposition we conclude! &

1F2(g(t))

12 dt =

&.

k=0

! 2k+1

2k

F2(g(t))12 dt + C

&.

k=0

2"1$'2' k < . ,

as claimed.

From Lemma 4.18 weak H1-convergence of the flow u(t) can be obtained in astraightforward manner. Standard parabolic regularity theory, moreover, yieldsuniform a-priori bounds for u(t) and its derivatives, which completes the proofof Theorem 4.11; see Brendle [1], Section 3, for details.


It thus remains to give the proof of Proposition 4.16. Fix any sequence tk %. (k % .) and let uk = u(tk). From Lemma 4.15 we conclude that thesequence (uk) is a Palais-Smale sequence for the Yamabe energy, satisfying theconditions

sk =!

M+{tk}R dµ = S(uk) % s& * Y0

and"cn#0uk + R0uk = s&u

n+2n$2k + o(1)

as k % ., with error

o(1) = (Rk " sk)un+2n$2k + (sk " s&)u

n+2n$2k % 0 in L

2nn+2 &% H"1 .

From the general concentration-compactness result Theorem 3.1 for such se-quences we then obtain the following result. Let r0 > 0 denote a lower boundfor the injectivity radius on (M, g0). Fix a function ) ! C&

0 (Br0(0)) satisfying)(x) = 1 on Br0/2(0) & IRn and for any x, y ! M let )y(x) = )(exp"1

y (x)),where exp is the exponential map in the metric g0.

4.19 Lemma. For any sequence tk % . (k % .), letting uk = u(tk), thereexist an integer L and sequences (xk,l)k$IN, (rk,l)k$IN, l = 1, . . . , L, such thatfor a subsequence as k % . there holds

uk(x) "L.

l=1

)xk,l(x) r2$n

2k,l u(r"1

k,l exp"1xk,l

(x)) % u& in H1(M, g0) ,

where u& * 0 solves

(4.23) "cn#0u& + R0u& = s&un+2n$2& on M ,

and where u is a solution to the equation

(4.24) "cn#IRn u = s&un+2n$2 on IRn.

Moreover, letting

V ol(u&) =!

Mu2!

& dµ0, V ol(u) =!

IRn

u2!dx

for brevity, as k % . we have

(4.25) 1 = V ol(uk) % V ol(u&) + L V ol(u).

By the result of Gidas-Ni-Nirenberg and Obata cited in Remark 3.2, we mayassume that

u(x) =(4n(n " 1)/s&)

n$24

(1 + |x|2)n$22

.


In particular, u achieves the Yamabe constant on IRn, which is conformal toSn. Letting

S(v) =cn

$IRn |#v|2gIRn dx

"$IRn |v|2! dx

#n$2n

,

hence we find the relation

S(u) = Y (Sn, gSn) .

Upon multiplying Equation (4.24) by u and integrating by parts we then obtainthe identity

Y (Sn, gSn) = S(u) = s&"V ol(u)

#1"2/2!

= s&"V ol(u)

#2/n.

Similarly, if u& )= 0 we may multiply Equation (4.23) by u& and integrateby parts to obtain the identity

S(u&) = s& (V ol(u&))2/n .

Solving for V ol(u&) and V ol(u), from (4.25) we then obtain

(4.26) 1 = V ol(u&) + L V ol(u) =&

S(u&)s&

'n/2

+ L

&Y (Sn, gSn)

s&

'n/2

,

that is,s& =

"S(u&)n/2 + L Y (Sn, gSn)n/2

#2/n.

But by definition of the Yamabe invariant we have S(u&) * Y (M, g0); more-over, assumption (4.16) and (4.19) imply the estimate

s& + s0 +"Y (M, g0)n/2 + Y (Sn, gSn)n/2

#2/n.

Thus, we either have s& = s0 and g0 is a Yamabe metric, or L = 0 ; in eachcase the sequence (uk) is H1-compact with limit u& inducing a Yamabe metric.Note that u& > 0 by the maximum principle.

On the other hand, if u& - 0, from (4.16) and (4.19) we obtain

L2/nY (Sn, gSn) = s& < s0 +"Y (M, g0)n/2 + Y (Sn, gSn)n/2

#2/n.

Since Y (M, g0) + Y (Sn, gSn) we then conclude that L = 1; that is, at most asingle simple bubble develops from (uk) as k % .. Moreover, from (4.26) inthis case we have

1 =&

Y (Sn, gSn)s&

'n/2

and it also follows that s& = Y (Sn, gSn) = n(n " 1).


The Compact Case u& > 0

Throughout this part of the argument we follow Brendle [1], Section 6. Theoperator

)% A&) = u" 4

n$2& L0)

is symmetric with respect to the inner product

(),/)L2(M,h)) =!

Mu

4n$2& )/ dµ0 .

The spectral theorem therefore yields the existence of a complete L2(M, h&)-orthonormal sequence of eigenfunctions )i, i ! IN, of the operator A&, witheigenvalues 0 < "i + "i+1 % . as i % .. Let i0 be the first index such that"i0 > s&

n+2n"2 and set Z = span{)i; i < i0}.

Recalling the equation

"cn#0u& + R0u& " s&un+2n$2& = u

4n$2&

"A&u& " s&u&

#= 0 ,

by the implicit function theorem we can find a radius 7 > 0 such that for allz = zi)i ! Z with |z| := ||z||L2(M,h)) < 7 there exists a unique functionuz = u& + z + o(|z|) with

(uz " u&,)i)L2(M,h)) =!

Mu

4n$2& (uz " u&))i dµ0 = zi, 1 + i < i0,

such that

(4.27)(A&uz " s&

" uz

u&

# 4n$2 uz,)i)L2(M,h))

=!

M

&L0uz " s&u

n+2n$2z

')i dµ0 = 0

for all i * i0.Moreover, the map z % uz is real analytic. According to results of Lo-

jasiewicz [1], [2], there then exists a number 0 < - < 1 such that

(4.28) S(uz) " S(u&) + C supi<i0

%%%%$

$ziS(uz)

%%%%1+-

;

see also Simon [2], Equation (2.4) on p. 538. Observe that (4.27) implies theidentity

$

$ziS(uz) =

2V ol(uz)

n$2n

!

M

&L0uz " s&u

n+2n$2z

')i dµ0 " I

for any i < i0, with error

I = 2

0S(uz)

V ol(uz)" s&

V ol(uz)n$2

n

1!

Mu

n+2n$2z

$

$ziuz dµ0 .


Expanding&

S(uz)V ol(uz)

" s&

V ol(uz)n$2

n

'

= V ol(uz)2$2n

n

!

M

&L0uz " s&u

n+2n$2z

'uz dµ0 ,

and recalling (4.27), from (4.28) we can thus estimate

(4.29) S(uz) " S(u&) + C supi<i0

%%%%!

M

&L0uz " s&u

n+2n$2z

')i dµ0

%%%%1+-

.

4.20 Normalization. For any k ! IN we choose zk ! Z such that!

M(cn|#(uzk " uk)|20 + R0|uzk " uk|2) dµ0

= min|z|<'

!

M(cn|#(uz " uk)|20 + R0|uz " uk|2) dµ0 .

Note that H1-convergence uk % u& implies that zk % 0 as k % .. Decom-posing

uk = uzk + wk ,

now !

ML0wk

$

$ziuz dµ0 =

!

M("cn#0wk + R0wk)

$

$ziuz dµ0 = 0

for any i < i0 by choice of zk. Thus with error o(1) % 0 as k % . we have

(4.30)"i

!

Mu

4n$2& )iwk dµ0 =

!

ML0)iwk dµ0

=!

ML0wk()i "

$

$ziuz) dµ0 = o(1)||wk||H1

for any such i.

4.21 Lemma. There exist constants c > 0, k0 such that for k * k0 there holds

n + 2n " 2

s&

!

Mu

4n$2& w2

k dµ0 + (1 " c)!

M(cn|#wk|20 + R0|wk|2) dµ0 .

Proof. Otherwise for a subsequence k % . we may rescale wk = akwk so that

1 =!

M(cn|#wk|20 + R0|wk|2) dµ0 + lim inf

k%&

&n + 2n " 2

s&

!

Mu

4n$2& w2

k dµ0

'.

A subsequence wk % w )= 0 weakly in H1, where!

M(cn|#w|20 + R0|w|2) dµ0 + n + 2

n " 2s&

!

Mu

4n$2& w2 dµ0 .


But (4.30) implies that!

M(cn|#w|20 + R0|w|2) dµ0 * "i0

!

Mu

4n$2& w2 dµ0 ,

and a contradiction results to the choice of i0.

4.22 Lemma. There exist constants C > 0, k0 such that for k * k0 there holds

||wk||H1 + C

&!

M+{tk}|R " s&|

2nn+2 dµ

'n+22n

.

Proof. By (4.30) there exist functions vk with (vk,)i)L2(M,h)) = 0 for all i < i0such that ||vk " wk||H1 = o(1)||wk||H1 , where o(1) % 0 as k % .. Recallingthe definition of uz, we then have

!

M(L0uzk " s&u

n+2n$2zk )vk dµ0 = 0 .

Subtracting the term

L0uk " s&un+2n$2k = (Rg(tk) " s&)u

n+2n$2k

in the integrand, from Lemma 4.21 with a uniform constant c > 0 we obtain

c||wk||2H1 +!

M(L0wk " n + 2

n " 2s&u

4n$2& wk)wk dµ0

=!

M(L0wk " n + 2

n " 2s&u

4n$2& wk)vk dµ0 + o(1)||wk||2H1

=!

M

"(Rg(tk) " s&)u

n+2n$2k + dk

#vk dµ0 + o(1)||wk||2H1 =: Ik ,

where

dk = s&(un+2n$2k " u

n+2n$2zk " n + 2

n " 2u

4n$2& wk)

= s&n + 2n " 2

(u4

n$2zk " u

4n$2& )wk + s&(u

n+2n$2k " u

n+2n$2zk " n + 2

n " 2u

4n$2zk wk) .

Since we assume that n + 5 this term can be bounded

|dk| + C(uk + |wk| + u&)6$nn$2 (|zk| + |wk|)|wk| .

By Sobolev’s embedding H1 &% L2!, thus we can estimate

!

Mdkvk dµ0 + o(1)||wk||2H1 .

It follows that


Ik + C

&!

M+{tk}|R " s&|

2nn+2 u2!

k dµ0

'n+22n

||vk||L2! + o(1)||wk||2H1

+ C

&!

M+{tk}|R " s&|

2nn+2 dµ

'n+22n

||wk||H1 + o(1)||wk||2H1 ,

and hence the claim.

Proof of Proposition 4.16 in the case u& > 0. Expand

L0uzk " s&un+2n$2zk = L0uk " Rg(tk)u

n+2n$2k

+ (Rg(tk) " s&)un+2n$2k " L0wk " s&(u

n+2n$2zk " u

n+2n$2k ) .

Recalling (4.1) and observing the estimates!

ML0wk)i dµ0 =

!

MwkL0)i dµ0

= "i

!

Mu

4n$2& wk)i dµ0 + C||wk||L2! + C||wk||H1 , 1 + i < i0 ,

together with!

Ms&(u

n+2n$2zk " u

n+2n$2k ))i dµ0 + C

!

M(uk + |wk|)

4n$2 |wk||)i| dµ0

+ C||wk||L2! + C||wk||H1 ,

for any i < i0 we have the bound%%!

M(L0uzk " s&u

n+2n$2zk ))i dµ0

%% +%%!

M(Rg(tk) " s&)u

n+2n$2k )i dµ0

%% + C||wk||H1 .

Lemma 4.22 and (4.29) then yield the estimate

(4.31) S(uzk) " S(u&) + C

&!

M+{tk}|R " s&|

2nn+2 dµ

'n+22n (1+-)

.

Likewise we expand

L0ukuk = L0(uzk + wk)(uzk + wk)= L0uzkuzk + 2L0ukwk " L0wkwk

and use (4.1) to conclude

S(uk) =!

ML0(uzk + wk)(uzk + wk) dµ0

=!

ML0uzkuzk dµ0 + 2

!

MRg(tk)u

n+2n$2k wk dµ0 "

!

ML0wkwk dµ0 .

We may write the latter in the form


S(uk) = s& + 2!

M(Rg(tk) " s&)u

n+2n$2k wk dµ0

"!

M(L0wkwk " n + 2

n " 2s&u

4n$2zk w2

k) dµ0 + Ik ,

whereIk = (S(uzk) " s&)V ol(uzk)

n$2n + s&(V ol(uzk)

n$2n " 1)

+ s&

!

M(2u

n+2n$2k wk " n + 2

n " 2u

4n$2zk w2

k) dµ0 .

By concavity of the function s % sn$2

n we have

V ol(uzk)n$2

n " 1 + n " 2n

(V ol(uzk) " 1) =n " 2

n

!

M(u2!

zk" u2!

k ) dµ0 .

Therefore we can bound the error term

Ik + (S(uzk) " s&)V ol(uzk)n$2

n

+ s&

!

M(n " 2

nu2!

zk" n + 2

n " 2u

4n$2zk w2

k + 2un+2n$2k wk " n " 2

nu2!

k ) dµ0

= (S(uzk) " S(u&))V ol(uzk)n$2

n

" 2s&2!

!

M(u2!

k " 2!un+2n$2k wk + 2!

n + 2n " 2

u4

n$2zk w2

k/2 " u2!

zk) dµ0 .

Recalling that uk = uzk + wk, we can bound

|u2!

k " 2!un+2n$2k wk + 2!

n + 2n " 2

u4

n$2zk w2

k/2 " u2!

zk| + C(uk + |wk|)

6$nn$2 |wk|3 .

In view of (4.31) and Lemma 4.22 we thus obtain the error estimate

Ik + C

&!

M+{tk}|R " s&|

2nn+2 dµ

'n+22n (1+-)

+ C||wk||3H1

+ C

&!

M+{tk}|R " s&|

2nn+2 dµ

' n+22n (1+-)

.

By Holder’s inequality and Lemma 4.22 we also may bound!

M(Rg(tk) " s&)u

n+2n$2k wk dµ0 +

&!

M+{tk}|R " s&|

2nn+2 dµ

'n+22n

||wk||L2!

+ C

&!

M+{tk}|R " s&|

2nn+2 dµ

' n+2n

,

while by Lemma 4.21 with a uniform constant c > 0 for k * k0 we have!

M(L0wkwk " n + 2

n " 2s&u

4n$2zk w2

k) dµ0 * c||wk||2H1 .


Finally, we can estimate&!

M+{tk}|R " s&|

2nn+2 dµ

'n+22n

+ F 2nn+2

(g(tk))n+22n + s(tk) " s& .

Combining the previous estimates and observing that - < 1, we obtain

s(tk) " s& = S(uk) " s& + CF 2nn+2

(g(tk))n+22n (1+-) + C(s(tk) " s&)1+- .

Since s(tk) % s& as k % ., our claim in Proposition 4.16 follows.

The idea of using the Lojasiewicz inequality for proving convergence of a geo-metric flow was pioneered by Leon Simon [1].

Bubbling: The Case u& - 0 .

In order to deal with “bubbling” the key idea is to make an optimally balancedchoice for the position xk, scale rk, and size 6k of the “bubble” separating fromthe sequence (uk) by Lemma 4.19, and to use the positive mass theorem.

4.23 The standard bubble. Following Lee-Parker [1], Theorem 5.1, or Gunther[2] we introduce local conformal normal coordinates on (M, g0). Fix somenumber N ! IN. Given a point x ! M , we can find a conformal metrichx = )

4n$2x g0 with )x(x) = 1 such that

det(D expx(8)) = 1 + O(|8|N)

near 8 = 0, where D expx(8) denotes the di!erential of the exponential map inthe metric hx at a point 8 ! TxM . Moreover, we have

|Rhx(y)| + C7x(y)2 ,

where 7x(y) denotes the Riemannian distance from x to y as measured byhx. Also let Gx be the Green’s function of the operator Lhx with pole at x,satisfying

LhxGx = "cn#hxGx + RhxGx = 0 for y )= x .

Moreover, Gx satisfies the estimates

|Gx(y) " 7x(y)2"n " Ax| + C7x(y) , |d(Gx(y) " 7x(y)2"n)| + C ,

with a constant Ax > 0 by the positive mass theorem. As shown byHabermann-Jost [1], Proposition I.1.3, the term Ax smoothly depends onx ! M , and we have infM Ax = A > 0.

Fix a smooth cut-o! function 9 such that 0 + 9 + 1, 9(s) = 1 for s + 1,9(s) = 0 for s * 2. For any small number 2 > 0 scale 9*(s) = 9(s/2). Forx ! M , * > 0 then let u!

x,&(y) = )x(y)U!x,&(y), where similar to (2.5) the profile

U!x,& is given by

U!x,&(y) =

&4n(n " 1)*2

s&

'n$24

&9*(7x(y))

(*2 + 7x(y)2)n$2

2+

"1 " 9*(7x(y))

#Gx(y)

'.

Then we have the following result of Schoen [1].


4.24 Theorem. Suppose that 3 + n + 5 and that (M, g0) is not conformallyequivalent to the standard sphere Sn. Then, with constants C, c > 0 indepen-dent of x ! M , 2 > 0, and * > 0 there holds

S(u!x,&) + Y (Sn, gSn) " c*n"2 + C2*n"2 + C2"n*n .

Proof. A detailed proof, adapting the argument of Schoen [1], thereby usingthe uniform positivity of the mass, is given by Brendle [1], Proposition B.3.

By Theorem 4.24 we can fix 2 > 0 such that for su"ciently small * > 0 wehave S(u!

x,&) < Y (Sn, gSn). To be consistent with the notation of Lemma 4.19henceforth we denote the scale as r > 0 and we let u!

x,r denote a standard bubbleof scale r > 0 centered at x. Then Lemma 4.19 provides a decomposition

uk = 6ku!xk,rk

+ wk =: vk + wk

with suitable xk, rk > 0 and 6k % 1. In fact, similar to the approach in thecase when u& > 0 we now normalize this choice by the requirement that

!

M(cn|#(uk " 6ku!

xk,rk)|20 + R0|uk " 6ku!

xk,rk|2) dµ0

= min+>0, x$M, r>0

!

M(cn|#(uk " 6u!

x,r)|20 + R0|uk " 6u!x,r|2) dµ0 .

Then we have the following result similar to (4.30).

4.25 Lemma. With error o(1) % 0 as k % . there holds!

Mv

n+2n$2k /kwk dµ0 = o(1)||wk||H1

for the functions

/k = 1 , /k =r2k

r2k + d(x, xk)2

, /k =rkexp"1

xk(x)

r2k + d(x, xk)2

related to the generators of the family (6u!x,r)+>0, x$M, r>0.

Proof. Observe that

L0vk = L0uk " L0wk = Rg(tk)un+2n$2k " L0wk = s&v

n+2n$2k + dk

with

dk = (Rg(tk) " s&)un+2n$2k + s&(u

n+2n$2k " v

n+2n$2k ) " L0wk % 0 in H"1 .

By minimality of the above decomposition with respect to 6 then we have

0 =!

ML0vkwk dµ0 = s&

!

Mv

n+2n$2k wk dµ0 + o(1)||wk||H1 .

as claimed. The other assertions follow similarly.


The following result provides the analogue of Lemma 4.21. It reflects the factthat the functions u!

& defined in (2.5) up to scaling and translation strictlyminimize the Sobolev ratio S(u) on IRn.

4.26 Lemma. There exist constants c > 0, k0 such that for k * k0 there holds

n + 2n " 2

s&

!

Mv

4n$2k w2

k dµ0 + (1 " c)!

M(cn|#wk|20 + R0|wk|2) dµ0 .

Proof. Otherwise we rescale wk = akwk so that

(4.32) 1 =!

M(cn|#wk|20 + R0|wk|2) dµ0 + n + 2

n " 2s&

!

Mv

4n$2k w2

k dµ0 + o(1)

with error o(1) % 0 for a subsequence k % .. Letting

wk(8) = rn$2

2k wk(expxk(rk8)): BR/rk

(0) & TxkM % IR ,

for some R < :0, the injectivity radius of (M, g0), by Sobolev’s inequality wehave

Y0

&!

BR/rk(0)

|wk|2!d8

'n$2n

+ 1 + o(1) ,

!

BR/rk(0)

cn|#wk|2 d8 + 1 + o(1) ,

and a subsequence wk % w weakly in H1loc(IR

n), where w ! D1,2(IRn) satisfies!

IRn

w2

(1 + |8|2)2 d8 > 0 .

Moreover, passing to the limit in (4.32) and simplifying the factors, we find!

IRn

|#w|2 d8 + n(n + 2)!

IRn

w2

(1 + |8|2)2 d8 .

But by Lemma 4.25 there holds!

IRn

1(1 + |8|2)n+2

2w d8 = 0 ,

!

IRn

1(1 + |8|2)n+2

2

1 " |8|21 + |8|2 w d8 = 0 ,

!

IRn

1(1 + |8|2)n+2

2

8

1 + |8|2w d8 = 0 .

A result of Rey [1], Appendix D, pp. 49–51, now implies that w - 0, whichgives the desired contradiction.

The proof of Proposition 4.16 now can be completed just as in the case whenu& > 0.


Proof of Proposition 4.16 in the case u& - 0. Expand

S(uk) =!

ML0(vk + wk)(vk + wk) dµ0

=!

ML0vkvk dµ0 + 2

!

MRg(tk)u

n+2n$2k wk dµ0 "

!

ML0wkwk dµ0

= s& + 2!

M(Rg(tk) " s&)u

n+2n$2k wk dµ0

"!

M(L0wkwk " n + 2

n " 2s&v

4n$2k w2

k) dµ0 + Ik ,

where the error term

Ik = (S(vk) " s&)V ol(vk)n$2

n + s&(V ol(vk)n$2

n " 1)

+ s&

!

M(2u

n+2n$2k wk " n + 2

n " 2v

4n$2k w2

k) dµ0

in view of the estimate S(vk) = S(u!xk,rk

) + Y (Sn, gSn) = s& may be bounded

Ik + "2s&2!

!

M(u2!

k " 2!un+2n$2k wk + 2!

n + 2n " 2

v4

n$2k w2

k/2 " v2!

k ) dµ0 + C||wk||3H1

similar to the case when u& > 0. Since Lemmas 4.26 and 4.19 yield that

||wk||2H1 + C

!

M(L0wkwk " n + 2

n " 2s&v

4n$2k w2

k) dµ0

for k * k0, the claim follows from Holder’s inequality as before.

4.27 Remarks. (1#) Theorems 4.9 and 4.11 show that it may not always be thebest strategy to follow the gradient flow for a variational problem; sometimesan apparently singular (possibly degenerate) evolution equation may be muchmore tractable, due in this case to the intrinsic geometric meaning of (4.4). Weencounter a similar phenomenon in Section 6.(2#) An argument analogous to Ye’s proof above works in n = 2 dimensions;see Bartz-Struwe-Ye [1]. See also Struwe [26] for a new approach in this case.(3#) For further material and references on the Yamabe problem we refer thereader to the survey by Lee-Parker [1].(4#) See Kazdan-Warner [1], Aubin [3], Bahri-Coron [1], Schoen [2], Chang-Yang [1], and - more recently - Struwe [27] for the related problem of findingconformal metrics of prescribed scalar curvature (“Nirenberg’s problem”).(5#) Another variant of the Yamabe problem is the singular Yamabe problem offinding complete conformal metrics of constant scalar curvature, for instance,on Sn \ 4 , where 4 is a smooth k-dimensional submanifold of Sn. If k > n"2

2 ,this leads to a coercive problem, reminiscent of the case S(M) < 0 for theYamabe problem. This was solved by Loewner-Nirenberg [1]. The case k < n"2

2is more di"cult and was solved only recently by Mazzeo-Smale [1], Mazzeo-Pacard [1].


5. The Dirichlet Problem for the Equation of ConstantMean Curvature

Another borderline case of a variational problem is the following: Let !be a bounded domain in IR2 with generic point z = (x, y) and let u0 !C0(!; IR3), H ! IR be given. Find a solution u ! C2(!; IR3) , C0(!; IR3)to the problem

#u = 2Hux 8 uy in ! ,(5.1)u = u0 on $! .(5.2)

Here, for a = (a1, a2, a3), b = (b1, b2, b3) ! IR3, a8b denotes the wedge producta 8 b = (a2b3 " b2a3, a3b1 " b3a1, a1b2 " b1a2) and, for instance, ux = #

#xu.(5.1) is the equation satisfied by surfaces of mean curvature H in conformalrepresentation.

Surprisingly, (5.1) is of variational type. In fact, solutions of (5.1) mayarise as “soap bubbles”, that is, surfaces of least area enclosing a given volume.Also for prescribed Dirichlet data, where a geometric interpretation of (5.1) isimpossible, we may recognize (5.1) as the Euler-Lagrange equations associatedwith the variational integral

EH(u) =12

!

"|#u|2 dz +

2H

3

!

"u · ux 8 uy dz .

For smooth “surfaces” u, the term

V (u) :=13

!

"u · ux 8 uy dz

may be interpreted as the algebraic volume enclosed between the “surface”parametrized by u and a fixed reference surface spanning the “curve” definedby the Dirichlet data u0; see Figure 5.1.

Indeed, computing the variation of the volume V at a point u ! C2(!; IR3)in direction of a vector ) ! C&

0 (!; IR3), we obtain

3d

d*V (u + *))|&=0

=!

"

") · ux 8 uy + u · )x 8 uy + u · ux 8 )y

#dz

= 3!

") · ux 8 uy dz +

!

") · (u 8 uyx + uxy 8 u) dz ,

and the second integral vanishes by anti-symmetry of the wedge product. Hencecritical points u ! C2(!; IR3) of E solve (5.1).

5. The Dirichlet Problem for the Equation of Constant Mean Curvature 221

Fig. 5.1. On the volume functional

Small Solutions

Since V is cubic, in EH the Dirichlet integral dominates if u is “small” and wecan expect that for “small data” and “small” H a solution of (5.1), (5.2) canbe obtained by minimizing EH in a suitable convex set. Generalizing earlierresults by Heinz [1] and Werner [1], Hildebrandt [2] has obtained the followingresult, which is conjectured to give the best possible bounds for the type ofconstraint considered:

5.1 Theorem. Suppose u0 ! H1,2 , L&(!; IR3). Then for any H ! IR suchthat

$u0$L) · |H| < 1 ,

there exists a solution u ! u0 + H1,20 (!; IR3) of (5.1), (5.2) such that

$u$L) + $u0$L) .

The solution u is characterized by the condition

EH(u) = min{EH(v) ; v ! u0 + H1,20 , L&(!, IR3), $v$L) |H| + 1} .

In particular, u is a relative minimizer of EH in u0 + H1,20 , L&(!; IR3).

5.2 Remark. Working with a di!erent geometric constraint Wente [1; Theorem6.1] and Ste!en [1; Theorem 2.2] prove the existence of a relative minimizerprovided

E0(u0)H2 <23. ,

whereE0(u) =

12

!

"|#u|2 dz


is the Dirichlet integral of u. The bound is not optimal, see Struwe [18; RemarkIV.4.14]; it is conjectured that E0(u0)H2 < . su"ces.

Proof of Theorem 5.1. Let

M = {v ! u0 + H1,20 (!; IR3) ; v ! L&(!; IR3), $v$L) |H| + 1} .

M is closed and convex, hence weakly closed in H1,2(!). Note that for u ! Mwe can estimate

(5.3) |H u · ux 8 uy| +12|H| $u$L) |#u|2 + 1

2|#u|2 ,

almost everywhere in !. Hence

EH(u) * 13E0(u) ;

that is, EH is coercive on M with respect to the H1,2-norm. Moreover, by (5.3)on M the functional EH may be represented by an integral

EH(u) =!

"F (u,#u) dx

whereF (u, p) =

12|p|2 +

23Hu · p1 8 p2

is non-negative, continuous in u ! IR3, and convex in p = (p1, p2) ! IR3 2 IR3

for all u ! IR3 with |H||u| + 1. Hence from Theorem 1.6 we infer that EH isweakly lower semi-continuous on M .

By Theorem I.1.2, therefore, EH attains its infimum on M at a pointu ! M . Moreover, EH is analytic in H1,2 , L&(!; IR3). Therefore we maycompute the directional derivative of EH in the direction of any vector pointingfrom u into M . Let ) ! C&

0 (!), 0 + ) + 1 and choose v = u(1 " )) ! M ascomparison function. Then by (5.3) we have

0 * /u " v, DEH(u)0 = /u), DEH(u)0

=!

"

"#u#(u)) + 2H u · ux 8 uy)

#dz

=!

"

"|#u|2 + 2Hu · ux 8 uy

#) dz +

12

!

"#

"|u|2

##) dz

* 12

!

"#

"|u|2

##) dz .

Hence |u|2 is weakly sub-harmonic on !. By the weak maximum principle,Theorem B.6 of Appendix B, |u|2 attains its supremum on $!. That is, thereholds

$u$L) + $u0$L) ,


as desired. But then u lies interior to M relative to H1,2,L& and DEH(u) = 0;which means that u weakly solves (5.1), (5.2). By a result of Wente [1; Theorem5.5], finally, any weak solution of (5.1) is also regular, in fact, analytic in !.

In view of the cubic character of V , having established the existence of a rela-tively minimal solution to (5.1), (5.2), we are now led to expect the existenceof a second solution for H )= 0. This is also supported by geometrical evidence;see Figure 5.2.

Fig. 5.2. A small and a large spherical cap of radius 1/|H| for 0 < |H| < 1 give rise todistinct solutions of (5.1) with boundary data u0(z) = z on #B1(0; IR2).

We start with an analysis of V .

The Volume Functional

In the preceding theorem we have used the obvious fact that V is smoothon H1,2 , L&(!; IR3) – but much more is true. Without proof we state thefollowing result due to Wente [1; Section III]:

5.3 Lemma. For any u0 ! H1,2 ,L&(!; IR3) the volume functional V extendsto an analytic functional on the a!ne space u0+H1,2

0 (!; IR3) and the followingexpansion holds:

(5.4) V (u0 + )) = V (u0) + /), DV (u0)0 +12D2V (u0)(),)) + V ()) .

Moreover, the derivatives

DV : H1,2(!; IR3) % H"1(!; IR3) = H1,20 (!; IR3)

!,

D2V : H1,2(!; IR3) %&8

H1,20 (!; IR3)

92'!


are continuous and bounded in terms of the Dirichlet integral

/), DV (u)0 =!

") · ux 8 uy dz + cE0(u)E0())1/2 ,

D2V (u)(),/) =!

"u · ()x 8 /y + /x 8 )y) dz + c

"E0(u)E0())E0(/)

#1/2.

Furthermore, DV and D2V are weakly continuous in the sense that, if um ' uweakly in H1,2

0 (!; IR3), then

/), DV (um)0 % /), DV (u)0, for all ) ! H1,20 (!; IR3) ,

D2V (um)(),/) % D2V (u)(),/), for all ),/ ! H1,20 (!; IR3) .

Finally, for any u ! H1,2(!; IR3) the bilinear form D2V (u) is compact; thatis, if )m ' ), /m ' / weakly in H1,2

0 (!; IR3), then

D2V (u)()m,/m) % D2V (u)(),/) .

Wente’s proof is based on the isoperimetric inequality, Theorem 5.4 below.However, with more modern tools, we can also give an entirely analytic proofof Lemma 5.3. In fact, the special properties of V are due to the anti-symmetryof the volume form which gives rise to certain cancellation properties as inour discussion of the compensated compactness scheme; compare Section I.3.In particular, the above bounds for V and its derivatives are related to theobservation of Coifman-Lions-Meyer-Semmes [1] that the cross product )x 8/y for ) ,/ ! H1,2

0 (! ; IR3) belongs to the Hardy space H1, with H1-normbounded in terms of the Dirichlet integrals of ) and /. Moreover, we haveH1,2(!; IR3) &% BMO(!; IR3), and, by a result of Fe!erman-Stein [1], BMOis the dual space of H1. Hence, for instance, we derive the estimate

!

"u · )x 8 /y dz + C$u$BMO$)x 8 /y$H1 + C

"E0(u)E0())E0(/)

#1/2.

The remaining properties of V can be derived similarly, using these techniques.

The remaining term in our functional EH is simply Dirichlet’s integral E0(u) =12

$" |#u|2 dz, well familiar from I.2.7–I.2.10. Both E0 and V are conformally

invariant, in particular invariant under scaling u % u(Rx).The fundamental estimate for dealing with the functional E is the isoperi-

metric inequality for closed surfaces in IR3; see for instance Rado [2]. This in-equality for (5.1), (5.2) plays the same role as the Sobolev inequality S$u$2

L2! +$u$2

H1,2 played for problem (1.1), (1.3).

5.4 Theorem. For any “closed surface” ) ! H1,20 (!; IR3) there holds

36.%%V ())

%%2 + E0())3 .

The constant 36. is best possible.


Remark. The best constant 36. is attained for instance on the function ) !D1,2(IR2),

)(x, y) =2

1 + x2 + y2(x, y, 1) ,

corresponding to stereographic projection of a sphere of radius 1 above (0, 0) !IR2 onto IR2, and its rescalings

(5.5) )&(x, y) = )(x/*, y/*) =2*

*2 + x2 + y2(x, y, *) .

) and )& solve Equation (5.1) on IR2 with H = 1: the mean curvature of theunit sphere in IR3.

Wente’s Uniqueness Result

Using the unique continuation property for the analytic Equation (5.1), anal-ogous to Theorem 1.2 we can show that the best constant in the isoperimetricinequality is never achieved on a domain ! ( IR2, ! )= IR2. Moreover, similarto Theorem 1.3, a sharper result holds, due to Wente [4]:

5.5 Theorem. If ! & IR2 is smoothly bounded and simply connected then anysolution u ! H1,2

0 (!; IR3) to (5.1) vanishes identically.

Proof. By conformal invariance of (5.1) we may assume that ! is a ball

B1(0; IR2). Reflecting u(z) = "u

&z

|z|2

'we extend u as a (weak) solution

of (5.1) on IR2. From Wente’s regularity result (Wente [1; Theorem 5.5]) weinfer that u is smooth and solves (5.1) classically. Now by direct computationwe see that the function

3(x + iy) ="|ux|2 " |uy|2

#" 2iux · uy

is holomorphic on C. Since!

IR2|#u|2 dz = 2

!

B1(0)|#u|2 dz < . ,

it follows that 3 ! L1(IR2), and hence that 3 - 0 by the mean-value propertyof holomorphic functions. That is, u is conformal. But then, since u - 0 on$B1(0; IR2), it follows that also #u - 0 on $B1(0; IR2) and hence, by uniquecontinuation, that u - 0; see Hartmann-Wintner [1; Corollary 1].

Theorem 5.5 – like Theorem 1.3 in the context of problem (1.1), (1.3) – provesthat EH cannot satisfy (P.-S.) globally on H1,2

0 (!), for any H )= 0. Indeed,note that EH(0) = 0. Moreover, by Theorem 5.4 for u ! H1,2

0 (!; IR3) withE0(u) = 4.

H2 there holds


EH(u) = E0(u) + 2HV (u) * E0(u)

01 "

:4H2

36.E0(u)

1

* 6 :=4.

3H2> 0 ,

while for any comparison surface u with V (u) )= 0, if HV (u) < 0, we have

EH(7u) = 72E0(u) + 2H73V (u) % ". as 7% . .

Hence, if EH satisfied (P.-S.) globally on H1,20 (!; IR3), from Theorem II.6.1 we

would obtain a contradiction to Wente’s uniqueness result Theorem 5.5.

Local Compactness

However, the following analogue of Lemma 2.3 holds:

5.6 Lemma. Suppose u0 ! H1,2 ,L&(!; IR3) is a relative minimizer of EH inthe space u0 + H1,2 , L&(!; IR3). Then for any ( < EH(u0) + 4.

3H2 condition(P.-S.)% holds on the a!ne space {u0} + H1,2

0 (!; IR3).

For the proof of Lemma 5.6 we need D2EH(u0) to be positive definite onH1,2

0 (!; IR3). By (5.3) this is clear for the relative minimizers constructed inTheorem 5.1, if $u0$L) |H| < 1

2 . In the general case some care is needed. Alsonote the subtle di!erence in the topology of H1,2 ,L& considered in Theorem5.1 and H1,2 considered here.

Postponing the proof of Lemma 5.6 for a moment we establish the followingresult by Brezis and Coron [2; Lemma 3]:

5.7 Lemma. Suppose u0 ! H1,2 , L&(!; IR3) is a relative minimizer of EH

in u0 + H1,20 , L&(!; IR3). Then u0 is a relative minimizer of EH in u0 +

H1,20 (!; IR3), and there exists a constant 2 > 0 such that

D2EH(u0)(),)) * 2E0()), for all ) ! H1,20 (!; IR3) .

Proof. We may assume that H )= 0. By density of C&0 (!; IR3) in H1,2

0 (!; IR3),since u0 is a relative minimizer in u0 + H1,2

0 , L&(!; IR3) clearly we have

2 = inf6D2EH(u0)(),)) ; ) ! H1,2

0 (!; IR3), E0()) = 17* 0 .

Note thatD2EH(u0)(),)) = 2E0()) + 2HD2V (u0)(),)) ,

and D2V (u0) is compact by Lemma 5.3. Hence, if

% := 2H inf($H1,2

0 (";IR3); E0(()=1D2V (u0)(),)) < 0 ,

then % is attained. Indeed, a minimizing sequence ()m) for % accumulatesweakly in H1,2

0 (!; IR3) at a limit ) with


2HD2V (u0)(),)) = % < 0 , E0()) + 1 .

In particular, ) )= 0, and we can consider the normalized function ) = (9E0(()

as comparison function with

2HD2V (u0)(), )) =%

E0()).

It follows that E0()) = 1; that is, % is attained. In terms of 2, we thus concludethat, if 2 = 2 + % < 2, then 2 is attained. In particular, if 2 = 0 there exists )such that E0()) = 1 and

D2EH(u0)(),)) = 0 = inf6D2EH(u0)(/,/) ; / ! H1,2

0 (!; IR3), E0(/) = 17

.

Necessarily, ) satisfies the Euler equation for D2EH(u0):

#) = 2H(u0x 8 )y + )x 8 u0y) .

It follows from a result of Wente [5; Lemma 3.1 ] that ) ! L&(!; IR3); seealso Brezis-Coron [2; Lemma A.1] or Struwe [18; Theorem III.5.1]. Hence byminimality of u0 in u0 + H1,2

0 , L&(!; IR3), for small |t| from Lemma 5.3 wehave

EH(u0) + EH(u0 + t)) = EH(u0) + t/), DEH(u0)0

+t2

2D2EH(u0)(),)) + 2Ht3V ())

= EH(u0) + 2Ht3V ()) ,

and it follows that V ()) = 0; that is, EH(u0 + t)) = EH(u0) for all t ! IR.But then for small |t| also u0 + t) is a relative minimizer of EH and satisfies(5.1), (5.2):

#(u0 + t)) = 2H(u0 + t))x 8 (u0 + t))y .

Since H )= 0, di!erentiating twice with respect to t gives

0 = )x 8 )y .

But then D2V (u0)(),)) = 2$" u0 · )x 8 )y dz = 0, and we obtain

2 = d2EH(u0)(),)) = 2E0()) = 2 ,

contrary to the assumption about 2.

Proof of Lemma 5.6. Let (um) be a (P.-S.)%-sequence in u0 + H1,20 (!; IR3).

Consider )m = um " u0 and note that by Lemma 5.3

EH(um) = EH(u0 + )m) = EH(u0) + /)m, DEH(u0)0

+12D2EH(u0)()m,)m) + 2HV ()m)

= EH(u0) +12D2EH(u0)()m,)m) + 2HV ()m) ,

while


o(1)"E0()m)

# 12 = /)m, DEH(um)0

= /)m, DEH(u0)0 + D2EH(u0)()m,)m) + 6HV ()m)= D2EH(u0)()m,)m) + 6HV ()m) ,

where o(1) % 0 (m % .). Subtracting, we obtain

D2EH(u0)()m,)m) = 6"EH(um) " EH(u0)

#+ o(1)

"E0()m)

# 12 ,

and by Lemma 5.7 it follows that ()m) and hence (um) is bounded. We mayassume that um ' u weakly. Then for any ) ! H1,2

0 (!; IR3) by Lemma 5.3also

o(1) = /), DEH(um)0 % /), DEH(u)0 ,

and it follows that u solves (5.1), (5.2). Thus, t = 1 is a critical point for thecubic function

t '% EH(u0 + t(u " u0))which also attains a relative minimum at t = 0, and we conclude that

EH(u) * EH(u0) .

Now let /m = um " u and note that by Lemma 5.3 we have

EH(um) = EH(u) + E0(/m) + HD2V (u)(/m,/m) + 2HV (/m)

= EH(u) + EH(/m) + HD2V (u)(/m,/m)= EH(u) + EH(/m) + o(1) ,

o(1) = //m, DEH(um)0 = //m, DEH(/m)0 + 2HD2V (u)(/m,/m)= //m, DEH(/m)0 + o(1) ,

with error o(1) % 0 as m % .. That is, for m su"ciently large

EH(/m) = E0(/m) + 2HV (/m) + EH(um) " EH(u) + o(1)

+ EH(um) " EH(u0) + o(1) + c <4.

3H2,

while 12//m, DEH(/m)0 = E0(/m) + 3HV (/m) = o(1) .

It follows that for m * m0 there holds

E0(/m) + c <4.H2

,

whence by the isoperimetric inequality Theorem 5.4 we have

o(1) = E0(/m) + 3HV (/m)

* E0(/m)

0

1 ":

H2E0(/m)4.

1

* cE0(/m)

for some c > 0, and /m % 0 strongly, as desired.


Large Solutions

We can now state the main result of this section, the existence of “large”solutions to the Dirichlet problem (5.1), (5.2), due to Brezis-Coron [2] andStruwe [12], [11], with a contribution by Ste!en [2].

5.8 Theorem. Suppose u0 )- const., H )= 0 and assume that EH admits arelative minimum u on u0 + H1,2

0 (!; IR3). Then there exists a further solutionu, distinct from u, of (5.1), (5.2).

Proof. We may assume that u = u0. Then by Lemma 5.6 the theorem followsfrom Theorem II.6.1 once we establish the following two conditions:(1#) There exists u1 such that EH(u1) < EH(u0).(2#) Letting

P =6p ! C0

"[0, 1]; u0 + H1,2

0 (!; IR3)#; p(0) = u0, p(1) = u1

7,

we have( = inf

p$Psupu$p

EH(u) < EH(u0) +4.

3H2.

Conditions (1#) and (2#) will be established by using the sphere-attachingmechanism of Wente [2; p. 285 f.], [3], refined by Brezis and Coron [2; Lemma5]: Since u = u0 )- const., there exists some point z0 = (x0, y0) ! ! suchthat #u(z0) )= 0. By conformal invariance of (5.1) we may assume thatz0 = 0, u(z0) = 0, and that with

ux(0) = a = (a1, a2, a3), uy = b = (b1, b2, b3)

there holdsH(a1 + b2) < 0 .

For * > 0 now let )&(x, y) = 2&&2+x2+y2 (x, y, *) be the stereographic projection

of IR2 onto a sphere of radius 1 centered at (0, 0, 1) considered earlier, and let8 ! C&

0 (!) be a symmetric cut-o! function such that 8(z) = 8("z) and 8 - 1in a neighborhood of z0 = 0. Define

ut = u&t := u + t8)& ! u + H1,20 (!; IR3) .

For small * > 0 the surface ut “looks” like a sphere of radius t attached to uabove u(0).

Now compute, using (5.4),

EH(ut) = EH(u) +t2

2D2EH(u)(8)&, 8)&) + 2Ht3V (8)&)

= EH(u) + t2E0(8)&) + 2Ht3V (8)&)

+ 2Ht2!

"u · (8)&)x 8 (8)&)y dx dy .


ClearlyE0(8)&) + E0()&; IR2) + O(*2) = 4. + O(*2) ,

whileV (8)&) = V ()&; IR2) + O(*3) =

4.3

+ O(*3) .

Expand

u(x, y) = u(0) + ux(0)x + uy(0)y + O(r2) = ax + by + O(r2) ,

where r2 = x2 +y2. Upon integrating by parts and using anti-symmetry of thewedge product, we obtain the following expression:

2H

!

"(ax + by) · (8)&)x 8 (8)&)y dx dy

= H

!

"

"a 8 (8)&)y + (8)&)x 8 b

#· (8)&) dx dy

= H

!

"

"a 8 (0, 1, 0) + (1, 0, 0)8 b

#· 82 4*2

(*2 + r2)2(x, y, *) dx dy

= 4H(a1 + b2)!

"

*3

(*2 + r2)282 dx dy

" 4H

!

"(a3x + b3y)

4*2

(*2 + r2)282 dx dy .

Since 8 is symmetric the last term vanishes, while for su"ciently small * > 0!

"

*3

(*2 + r2)282 dx dy * 1

4*

!

B#(0)dx dy =

*.

4> 0 .

Finally, since |#)&(x, y)| + c&&2+x2+y2 , we may estimate

%%%%!

"O(r2) · (8)&)x 8 (8)&)y dx dy

%%%%

+!

"O(r2)|#)&|2 dx dy + O(*2)

+ c

!

"

*2r2

(*2 + r2)2dx dy + O(*2)

+ c

!

B#(0)dx dy + c

!

"\B#(0)

*2

r2dx dy + O(*2)

+ c*2 + c*2|ln *| .

Hence, for t * 0 we may estimate

EH(ut) + EH(u)+t2"4.+H(a1+b2).*+c*2|ln *|+c*2

#+2Ht3

"4.3

+O(*3)#

.

6. Harmonic Maps of Riemannian Surfaces 231

In particular, if H < 0 and if * > 0 is chosen su"ciently small, there existsT > 0 such that uT satisfies EH(uT ) < EH(u) and, moreover,

sup0't'T

EH(ut) < EH(u) +4.

3H2,

as desired. The case H > 0 follows in a similar way by considering (ut)t'0.

5.9 Remarks. A “global” compactness condition analogous to Theorem 3.1also holds in the case of Equation (5.1); see Brezis-Coron [3], Struwe [11]. Asa consequence, it is not hard to see the analogue of Theorem 3.4 (Rentzmann[1], Patnaik [1]; if ! is an annulus the existence of non-trivial solutions u !H1,2

0 (!; IR3) to (5.1) was already shown by Wente [4]). It would be interestingto further investigate the e!ect of topology, for instance, in the spirit of Bahri-Coron [1].

For further material and references on the Dirichlet and Plateau problemsfor surfaces of prescribed mean curvature we refer the interested reader to thelecture notes by Struwe [18]. A result analogous to Theorem 5.8 for variablemean curvature H = H(u) has recently been obtained by Struwe [19]; see alsoWang, G.F. [1].

6. Harmonic Maps of Riemannian Surfaces

As our final example we now present a borderline variational problem in a non-smooth setting, where critical points under suitable conditions can be obtainedvia a flow approach similar to Section 4.

Given a smooth, closed Riemannian surface 0 with metric - and anysmooth, closed k-manifold N with metric g, a natural generalization of Dirich-let’s integral for C1-functions on a domain in IRn is the energy functional

E(u) =!

/e(u) dµ .

Here, in local coordinates on 0 and N , the energy density e(u) is given by

e(u) =.

1'+,%'2

.

1'i,j'k

12-+%(x)gij(u)

$

$xaui $

$x%uj ,

with -+% = (-+%)"1 denoting the coe"cients of the inverse of the matrix (-+%)representing the metric -, (gij) representing g, and with area element

dµ =5

|-|dx, |-| = det(-+%) .

Since we assume that both 0 and N are compact, this expression may be sim-plified considerably: First, by the Nash embedding theorem, see for instance


Nash [1] or Schwartz [2; pp. 43–53], any compact manifold N may be isometri-cally embedded into some Euclidean IRn. A new proof – avoiding the “hard”implicit function theorem – has recently been obtained by Gunther [1].

Moreover, E is invariant under conformal mappings of 0. Thus, by theuniformization theorem, we may assume that either 0 = S2 or 0 = T 2 =IR2/ZZ2, or is a quotient of the upper half-space IH, endowed with the hyperbolicmetric 1

y dx dy. In particular, if 0 = T 2, the energy density is simply givenby e(u) = 1

2 |#u|2 and E becomes the standard Dirichlet integral for mappingsu: T 2 = IR2/ZZ2 % N & IRn.

Consider the space C1(0; N) of C1-functions u:0 % N &% IRn. Note thatC1(0; N) is a manifold with tangent space given by

TuC1(0; N) =6) ! C1(0; IRn) ; )(x) ! Tu(x)N for x ! 0

7,

and E is di!erentiable on this space.In fact, if we consider variations ) ! TuC1(0; N) such that supp()) is

contained in a chart U on 0 whose image u(U) is contained in a coordinatepatch of N , then in order to compute the variation of E in direction ) it su"cesto work in one coordinate frame – both on 0 and N – and all computations canbe done as in the “flat” case. From this, the di!erentiability of E is immediateand the following definition is meaningful.

6.1 Definition. A stationary point u ! C1(0; N) of E is called a harmonicmap.

The concept of harmonic map generalizes the notion of (closed) geodesic tohigher dimensions; compare Section II.4. Moreover, if we choose N = IRn, wesee that harmonic functions simply appear as special cases of harmonic maps.However, when the target space is curved, the Euler-Lagrange equations forharmonic maps no longer will be linear, as we shall presently see.

The Euler-Lagrange Equations for Harmonic Maps

First consider the model case 0 = T 2 = IR2/ZZ2, where E reduces to thestandard Dirichlet integral for doubly periodic mappings u: IR2 % N & IRn,restricted to a fundamental domain.

In this case, if u: T 2 = IR2/ZZ2 % N & IRn is harmonic of class C2, thefirst variation of E gives

0 = /), DE(u)0 =!

T 2#u#) dx = "

!

T 2#u ) dx ,

for all doubly periodic ) ! C1(IR2; IRn) satisfying the condition

)(x) ! Tu(x)N for all x ! IR2 .


That is, "#u(x) is orthogonal to the tangent space of N at the point u(x), forany x ! T 2; in symbols:

"#u(x) =n.

i=k+1

"i(x)%i(u(x)) : Tu(x)N for all x ! T 2 ,

where %k+1, . . . , %n is a smooth local orthonormal frame for the normal bundleTN, near u(x), and where "k+1, . . . ,"n are scalar functions. In general, theLaplace operator will be replaced by the Laplace-Beltrami operator #/ on 0.

The particular structure of this equation can best be seen when N =Sn"1 & IRn. In this case, if u: T 2 % Sn"1 & IRn is of class C2 and harmonic,it follows that

"#u = "u

for some continuous function ": T 2 % IR. Testing this relation with u andnoting that |u| - 1, u ·#u - 0, we see that

" = "#u · u = "div(u ·#u) + |#u|2 = |#u|2 ;

that is, harmonic maps into spheres satisfy the relation

"#u = u|#u|2 .

For general target manifolds N we may proceed similarly. Fix an indexi ! {k + 1, . . . , n}. Then, since $+u = #

#x&u ! TuN for any 6, we have

"i = "/#u, %i 7 u0= "div/#u, %i 7 u0 + /#u, (d%i 7 u) ·#u0 = /#u, (d%i 7 u) ·#u0 ,

where /·, ·0 denotes the scalar product. That is, we have

"#u = A(u)(#u,#u) ,

where A(u): TuN 2 TuN % T,u N denotes the second fundamental form of N ,

given by

A(p)(8, +) =n.

i=k+1

%i(p)Ai(p)(8, +) , Ai(p)(8, +) = /8, d%i(p)+0 ,

for p ! N , 8, + ! TpN .For general domains, similarly the harmonic map equation reads

"#/u = A(u)(#u,#u)/ : TuN ,

withA(u)(#u,#u)/ =

.

+,%

-+%A(u)($+u, $%u) .

Note that by compactness of N the coe"cients of the form A are uniformlybounded.


Bochner Identity

Upon di!erentiating the harmonic map equation and taking the scalar productwith the components of #u, we obtain the equation

"#e(u) + |#2u|2 = "12#

"|#u|2

#+ |#2u|2

= "#(#u) ·#u = #"A(u)(#u,#u)

#·#u

= div"A(u)(#u,#u) ·#u

#" A(u)(#u,#u) ·#u

= |A(u)(#u,#u)|2 + Ce(u)2

for the energy density e(u). In the model case of harmonic maps u: T 2 =IR2/ZZ2 % Sn"1 &% IRn, this identity simply takes the form

"#e(u) + |#2u|2 = |#u|4 = 4e(u)2 .

For a general domain manifold0, additional terms related to the curvatureof 0 appear. If we do not care about the precise form of these terms we thenobtain the inequality

"#/e(u) + |#du|2/ + C(1 + e(u))e(u)

with a constant C depending on N and 0.The most interesting variant of the Bochner identity results if we work

intrinsically on the manifold N and use covariant di!erentiation instead oftaking the ordinary gradient. Then we obtain the di!erential inequality

"#/e(u) + ;Ne(u)2 + Ce(u)

for the energy density of u, where ;N * 0 denotes an upper bound for thesectional curvature of N , and where C denotes a constant depending only on0 and N . See for instance Jost [3], formula (3.2.10).

With these remarks we hope that the reader has become somewhat familiarwith the concept of harmonic maps. Moreover, to fix ideas, in the followingone may always think of mappings u: T 2 = IR2/ZZ2 % Sn"1 & IRn. In thisspecial case already, all essential di"culties appear and nothing of the flavourof the results will be lost.

The Homotopy Problem and its Functional Analytic Setting

A natural generalization of Dirichlet’s problem for harmonic functions is thefollowing “Homotopy Problem”: Given a map u0:0 % N , is there a harmonicmap u homotopic to u0?

As in the case of a scalar function u:! % IR we may attempt to approach thisproblem by direct methods. Denote


H1,2(0; N) =6u ! H1,2(0; IRn) ; u(x) ! N for a.e. x ! 0

7

the space of H1,2-mappings into N . (If 0 = T 2, then H1,2(0; N) is the spaceof mappings u ! H1,2

loc (IR2; N) of period 1 in both variables, restricted to a fun-damental domain.) Note that E is weakly lower semi-continuous and coerciveon H1,2(0; N) with respect to H1,2(0; IRn).

Moreover, by a result of Schoen-Uhlenbeck [2; Section IV] we have:

6.2 Theorem. (1#) The space C&(0; N) of smooth maps u:0 % N & IRn isdense in H1,2(0; N).(2#) The homotopy class of any map u0 ! H1,2(0; N) is well-defined.

Proof. (1#) The following argument basically is due to Courant [1; p. 214 f.].Let ) ! C&

0

"B1(0)

#satisfy 0 + ) + 1,

$B1(0)

) dx = 1, and for * > 0 let)& = *"2)

"x&

#. Given u ! H1,2(0; IRn), denote

u ; )&(x0) =!

B#(x0)u(x))&(x0 " x) dx

its mollification with )& (in local coordinates on 0). Note that u ; )& !C&(0; IRn) and u ; )& % u in H1,2(0; IRn) as * % 0. Let –

$denote average

and let dist(P, N) = inf{|P " Q| ; Q ! N} denote the distance of a point Pfrom N .

For x0 ! 0, * > 0 and y ! B&(x0) now estimate

dist2"u ; )&(x0), N

#+

%%%%

!

B#(x0)

"u(x) " u(y)

#)&(x0 " x) dx

%%%%2

+ C —!

B#(x0)

%%u(x) " u(y)%%2 dx .

Taking the average with respect to y ! B&(x0), therefore we can bound

dist2"u ; )&(x0), N

#+ C —

!

B#(x0)—!

B#(x0)|u(x) " u(y)|2 dx dy

+ C*2 —!

B#(x0)—!

B#(x0)

! 1

0

%%#u"y + t(x " y)

#%%2 dt dx dy

+ C

!

B#(x0)|#u|2 dx .

By absolute continuity of the Lebesgue integral, the latter is small for small*, uniformly in x0. In particular, for * < *0 = *0(u, N) the distance of u& =u ; )& from N is smaller than d/2, where d = d(N) > 0 is the focal distancefrom N . We then can smoothly project u& down to N to obtain a functionu& ! C&(0; N), satisfying

(u& " u&) : Tu#N , $u& " u&$L) % 0 as *% 0 .


Since u& % u in H1,2 as *% 0,

$u& " u$L2 + $u& " u&$L2 + $u& " u$L2 % 0 as *% 0 .

Moreover, representing u& locally as u& = u& +/

i "i&%i 7 u&, by orthogonalitywe have

|#u&|2 * |#u& +.

i

"i&#(%i 7 u&)|2 *"1 " C

.

i

$"i&$L)#|#u&|2 .

Hencelim sup&%0

$#u&$L2 + lim&%0

$#u&$L2 = $#u$L2 ,

and it follows that u& % u in H1,2(0; N), as desired.(2#) Define the homotopy class of u ! H1,2(0; N) as the homotopy class ofany map v ! C&(0; N) with $v " u$H1,2 + 2 for some 2 = 2(u, N) > 0 tobe determined. To see that this is well-defined, let v0, v1 ! C&(0; N) satisfy$vi"u$H1,2 + 2, i = 0, 1, and let vt = (1"t)v0 +tv1 be a homotopy connectingv0 and v1 in C&(0; IRn). Note that

$vt " u$H1,2 + (1 " t)$v0 " u$H1,2 + t$v1 " u$H1,2 + 2

for 0 + t + 1. Moreover, for any w ! H1,2, * > 0 we can estimate

$w ; )&$L) + C$w ; )&$H2,2 + C(1 + *"1)$w$H1,2 .

Thus for 0 < *0 = *0(u, N) + 1 as determined in (1#) and 0 + t + 1 we have

dist(vt ; )&0 , N) + dist(u ; )&0 , N) + $(vt " u) ; )&0$L)

+ dist(u ; )&0 , N) + C1*"10 $vt " u$H1,2 + d/2 + C12/*0 + 3d/4 ,

provided that we choose 4C12 + d*0. We may then project the maps vt ; )&0to maps ;vt ; )&0 ! C&(0; N), 0 + t + 1, connecting ;v0 ; )&0 and ;v1 ; )&0 .

For i = 0, 1 and 0 < *<* 0(u, N), finally, as in (1#) we can estimate

dist2(vi ; )&(x0), N) + C

!

B#(x0)|#vi|2 dx + C

!

B#(x0)|#u|2 dx + C2 + 3d/4 ,

provided 2 = 2(u, N) > 0 is chosen su"ciently small. Thus v0 and ;v0 ; )&0 arehomotopic through ;v0 ; )&, 0 + * + *0, and similarly for v1, showing that v0

and v1 belong to the same homotopy class.

Bethuel [1] showed that on an m-ball B for any m > 2 the space C&(B; N) isdense in H1,2(B; N) if and only if .2(N) = 0. For general domain manifolds,Bethuel’s work was completed by Hang-Lin [1], after they uncovered a mistakein Bethuel’s original work dealing with this case.

Motivated by Theorem 6.2 one could attempt to solve the homotopy prob-lem by minimizing E in the homotopy class of u0. However, while H1,2(0; N) is


weakly closed in the topology of H1,2(0; IRn), in general this will not be the casefor homotopy classes of non-constant maps. Consider for example the family()&)&>0 of stereographic projections to the standard sphere, introduced in (5.5);projecting back with )1, we obtain a family of maps u& = )& 7 ()1)"1: S2 % S2

of degree 1, converging weakly to a constant map. Therefore, the direct methodfails to be applicable for solving the homotopy problem.

Also note that the space H1,2(0; N) is not a manifold; therefore the stan-dard deformation lemma Theorem II.3.4 or II.3.11 cannot be applied. (Aninteresting variant was recently proposed by Duc [2].)

Existence and Non-existence Results

In fact, the infinum of E in a given homotopy class in general need not beattained. (See White [1] for further results in this regard.) Even worse, thehomotopy problem may not always have a solution. The following result, anal-ogous to the results of Pohozaev (Theorem 1.3) and Wente (Theorem 5.1) inour previous examples, is due to Eells-Wood [1].

6.3 Theorem. Any harmonic map u ! C1(T 2; S2) necessarily has topologicaldegree )= 1.

In particular, there is no harmonic map homotopic to a map u0: T 2 % S2 ofdegree +1.

This result shows that we may encounter some lack of compactness in at-tempting to find critical points of E. However, compactness can be restoredunder certain conditions. In a pioneering paper Eells-Sampson [1] have ob-tained the following result.

6.4 Theorem. Suppose the sectional curvature ;N of N is non-positive. Thenfor any map u0:0 % N there exists a harmonic map homotopic to u0.

Surprisingly, also a topological condition on the target may su"ce to solvethe homotopy problem. The following result was obtained independently byLemaire [1] and Sacks-Uhlenbeck [1].

6.5 Theorem. If .2(N) = 0, then for any u0 ! H1,2(0; N) there is a smoothharmonic map homotopic to u0.

For the proof of Theorem 6.4 Eells and Sampson consider an evolution problemsimilar to the flow approach that we have encountered in Section 4.

In fact, also Theorem 6.5 can be obtained from an in-depth analysis ofthe Eells-Sampson flow. Moreover, as we shall see, this analysis reveals a deepanalogy between the harmonic map problem and the problems (1.1), (1.3), theYamabe problem, and the Dirichlet problem for the equation of constant meancurvature (5.1).


The Heat Flow for Harmonic Maps

Given a map u0 ! C&(0; N), Eells and Sampson propose to consider a solutionu: M 2 [0,.[% N &% IRn of the initial value problem

ut "#/u = A(u)(#u,#u)/ : TuN in 0 2 IR+ ,(6.1)u|t=0 = u0 .(6.2)

We write ut = $tu for brevity. Equation (6.1) may be interpreted as the “L2-gradient flow” for E; in particular, we have the energy inequality

! T

0

!

/|ut|2 dµ dt + E

"u(T )

#+ E(u0) ,

for all T > 0; see Lemma 6.8 below. Moreover, like the standard linear heatequation, the flow defined by (6.1) has certain smoothing properties. For exam-ple, analogous to the stationary (time-independent) case, for (6.1) there holdsthe Bochner-type inequality

($t "#/)e(u) + |#du|2/ + C(1 + e(u))e(u) .

Working intrinsically, as in Jost [3], formula (3.2.11), the bound for the leadingterm on the right can be improved and we obtain the di!erential inequality

(6.3) ($t "#/)e(u) + ;Ne(u)2 + Ce(u) ,

where ;N * 0 again denotes an upper bound for the sectional curvature of N .

Proof of Theorem 6.4. If ;N + 0, estimate (6.3) implies a linear di!erential in-equality for the energy density, and we obtain the existence of a global solutionu ! C2(0 2 IR+, N) to the evolution problem (6.1), (6.2). Moreover, by theweak Harnack inequality for sub-solutions of parabolic equations (see Moser [2;Theorem 3]) and the energy inequality, the maximum of |#u| may be a-prioribounded in terms of the initial data. Again by the energy inequality, we canfind a sequence of times tm % . such that ut(tm) % 0 in L2 as m % ., anda subsequence (u(tm)) converges to a harmonic map homotopic to u0.

By Theorem 6.3, in general we cannot expect that (6.1), (6.2) admits a globalsmooth solution, converging asymptotically to a harmonic map. However, wepropose to establish that (6.1), (6.2) always admits a global weak solutionwhich is unique in its class, for arbitrary initial data u ! H1,2(0; N) and with-out imposing any topological or geometric restrictions on the target manifold.Moreover, space-time singularities of the flow or failure of u(t) to convergeas t % . may be uniquely associated with the “bubbling o!” of harmonicspheres, as we will now make precise.


The Global Existence Result

Let expx: Tx0 % 0 denote the exponential map at a point x ! 0. (If 0 = T 2,then expx(y) = x + y.) The following result is due to Struwe [10].

6.6 Theorem. For any u0 ! H1,2(0; N) there exists a distribution solutionu:0 2 IR+ % N of (6.1) which is smooth on 0 2 IR+ away from at mostfinitely many points (xk, tk), 0 < tk < ., 1 + k + K, which satisfies theenergy inequality E(u(t)) + E(u(s)) for all 0 + s + t, and which assumes itsinitial data continuously in H1,2(0; N). The solution u is unique in this class.

At a singularity (x, t) a smooth harmonic map u: S2 % N separates in thesense that for sequences xm % x, tm < t, Rm = 0 as m % . the family

um(x) - u"expxm

(Rmx), tm#% u in H2,2

loc (IR2; N) ,

where u: IR2 % N is smooth and harmonic with finite energy. By compositionwith stereographic projection, u induces a non-constant, smooth harmonic mapu: S2 1= IR2 % N . See Fig. 6.1 for illustration.

As tm % . suitably, the sequence of maps u(tm) converges weakly inH1,2(0; N) to a smooth harmonic map u&:0 % N , and smoothly away fromfinitely many points xk, 1 + k + K, where again harmonic spheres separate inthe sense described above. Moreover, we have

E(u&) + E(u0) " K *0 ,

where K = K + K is the total number of singularities and where

*0 = inf6E(u) ; u ! C1(S2; N) is non-constant and harmonic

7> 0

is a constant depending only on the geometry of N . In particular, the numberof singularities of u is a-priori bounded, K + *"1

0 E(u0).

Theorem 6.6 implies Theorem 6.5:

Proof of Theorem 6.5. We follow Struwe [15; p. 299 f.]. Let [u0] be a homotopyclass of maps from 0 into N . We may suppose that [u0] is represented by asmooth map u0:0 % N such that

E(u0) + infu$[u0]

E(u) +*02

.

Let u:0 2 IR+ % N be the solution to the evolution problem (6.1), (6.2)constructed in Theorem 6.6. Suppose u first becomes singular or concentratesat a point (x, t), t + .. Then for sequences tm < t, xm % x, rm = 0 we have

um(x) := u"expxm

(rmx), tm#% u in H2,2

loc (IR2; N)

where u may be extended to a smooth harmonic map u: S2 % N .


Fig. 6.1. “Separation of spheres”

In the image covered by um we now replace a large part of the “harmonicsphere” u by its “small” complement, “saving” at least *0/2 in energy. If.2(N) = 0, this change of um will not a!ect the homotopy class [um] = [u0],and a contradiction will result.

More precisely, since um % u in H2,2loc and since E(u) < ., we can find a

sequence of radii Rm % . such that as m % . we have rmRm % 0 while

sup#BRm

|um " u| + Rm

!

#BRm

|#(um " u)|2 do % 0 ,

!

IR2\BRm

|#u|2 dx % 0 ,

!

BRm

|#(um " u)|2 dx % 0 ,

where BRm = BRm(0). Now let / ! H1,2(BRm ; IRn) solve the Dirichlet prob-lem

"#/m = 0 in BRm

with boundary data/m = um " u on $BRm .

By the maximum principle

supBRm

|/m| = sup#BRm

|um " u| % 0 .


Moreover, by classical potential estimates and interpolation (see Lions-Magenes[1; Theorem 8.2]), the Dirichlet integral of /m may be estimated in terms of thesemi-norm of the trace /m|#BRm

! H1/2,2($BRm) and as m % . we obtain!

BRm

|#/m|2 dx + c|/m|2H1/2,2(#BRm )

+ c

!

#BRm

!

#BRm

|/m(x) " /m(y)|2

|x " y|2dx dy

+ c Rm

!

#BRm

|#(/m|2 do + c Rm

!

#BRm

|#(um " u)|2 do % 0 ,

where #( denotes the tangential gradient. (See for instance Adams [1; Theorem7.48] for the equivalent integral representation of the H1/2,2-semi-norm.)

Hence if we replace u(tm) by the map vm, where

vm

"expxm

(rmx)#

=

<um(x), |x| * Rm

u"R2

mx

|x|2#

+ /m(x), |x| < Rm ,

is defined via the exponential map in a small coordinate chart U around xm,and where vm - u(tm) on 0 \ U , we obtain that, as m % .,

dist"vm(x), N

#+ sup

BRm

|/m| % 0 ,

uniformly in x ! 0. But then we may project vm onto N to obtain a mapwm ! H1,2(0; N) satisfying, with error o(1) % 0 as m % .,

E(wm) + E(vm) + o(1)

+ E(um) " 12

!

BRm

|#um|2 dx +!

IR2\BRm

|#u|2 dx

+!

BRm

|#/m|2 dx + o(1)

+ E(u0) "12

!

BRm

|#u|2 dx + o(1)

+ infu$[u0]

E(u) + *0/2 " *0 + o(1)

and the latter is strictly smaller than infu$[u0] E(u) for large m.On the other hand, since .2(N) = 0, the maps x '% u(x) and x '%

u(R2mx/|x|2) for any m are homotopic as maps from BRm into N with fixed

boundary; hence wm is homotopic to u(tm), and the latter is homotopic to u0

via the flow (u(t))0't'tm , which is regular for all m by assumption. Thus, ifthe flow (u(t))t>0 were to develop a singularity at some finite time t or a con-centration as t % ., there results a contradiction to our choice of u0. Hence,the flow (u(t))t>0 exists and is regular for all time and, as tm % . suitably,


u(tm) % u& strongly in H1,2(0; N) where u& is harmonic. But by Theorem6.2, for su"ciently large m the maps u& and u(tm) are homotopic, and thelatter is homotopic to u0 via the flow (u(t))0't'tm .

The Proof of Theorem 6.6

The following Sobolev interpolation estimate independently due to Gagliardi-Nirenberg and Ladyzhenskaya [1] plays a key role in the proof of Theorem6.6.

6.7 Lemma. For any function u ! H1,2loc (IR2), any R > 0, and any function

) ! C&0 (BR) with 0 + ) + 1 and such that |#)| + 4/R, there holds

!

IR2|u|4)2 dx + c0

&!

BR

|u|2dx

'·&!

BR

|#u|2)2 dx + R"2

!

BR

|u|2dx

',

with c0 independent of u and R.

Proof. The function |u|2) has compact support. Thus, for any x = (8, +) ! IR2

we have%%"|u|2)

#(8, +)

%% =

%%%%%

! 0

"&

$

$8

"|u|2)

#(8(, +) d8(

%%%%%

+ 2"! &

"&

"|#u| |u|)+ |u|2|#)|

#(8(, +) d8( ,

and an analogous estimate with integration in +-direction.Hence by Fubini’s theorem and Holder’s inequality we obtain!

IR2|u|4)2(x) dx =

! &

"&

! &

"&|u|4)2(8, +) d8 d+

+ 4 ·! &

"&

! &

"&

=&! &

"&

%%#u| |u|)+ |u|2 |#)|#(8(, +) d8(

'·

·&! &

"&

"|#u| |u|)+ |u|2 |#)|

#(8, +() d+(

'>d8 d+

= 4! &

"&

! &

"&

"|#u| |u|)+ |u|2|#)|

#(8(, +) d8( d+·

·! &

"&

! &

"&

"|#u| |u|)+ |u|2|#)|

#(8, +()d8 d+(

= 4&!

IR2

"|#u| |u|)+ |u|2|#)|

#dx

'2

+ 8&!

supp(|u|2 dx

'·!

IR2

"|#u|2)2 + |u|2|#)|2

#dx .

Since supp) & BR and |#)| + 4/R, the claim follows.


To see how we may apply Lemma 6.7, observe that by compactness of N witha uniform constant C = C(N) for any solution u of (6.1) there holds

(6.4) |ut "#/u| + C|#u|2 .

Hence, as shown in Lemmas 6.10 and 6.11 below, if the energy of u(t) is uni-formly small on small balls, from Lemma 6.7 applied to the function |#u| weobtain an L2-estimate for |#2u|, and then also higher regularity.

It is then of crucial importance to obtain energy bounds for u(t), bothglobally and locally. First we note the following energy inequality.

6.8 Lemma. Let u ! C2"0 2 [0, T [; N

#solve (6.1), (6.2). Then there holds

! t

0

!

/|ut|2 dµ dt( + E

"u(t)

#+ E(u0) for all t < T .

Proof. Multiply (6.1) by ut ! TuN and use orthogonality A(u)(·, ·) : TuN toobtain the identity

(6.5) /ut "#/u, ut0 = |ut|2 +d

dte(u) " div

"/#u, ut0

#= 0 .

The claim follows upon integrating over 0 2 [0, t].

Lemma 6.7 also calls for control of the energy density, locally. For this letı/ be the injectivity radius of the exponential map on 0. Then on any ballBR(x0) & 0 of radius R < ı/ we may introduce Euclidean coordinates by aconformal change of variables. Recall that E is conformally invariant. Withreference to such a conformal chart then introduce the local energy

E"u; BR(x0)

#=

12

!

BR(x0)|#u|2 dx , R < ı/ .

In the following we shift x0 = 0 and again let BR = BR(0) for brevity.

6.9 Lemma. There is an absolute constant c1 such that for any R < 12 ı/ and

any solution u ! C2(B2R2 [0, T ]; N) to (6.1) with sup0't'T E"u(t); B2R

#+ E0

there holdsE

"u(T ); BR

#+ E(u0; B2R) + c1

T

R2E0 .

Proof. Consider 0 = T 2, for simplicity. Choose ) ! C&0 (B2R) such that

0 + ) + 1, ) - 1 in BR and |#)| + 2/R. Multiplying (6.5) by )2 andintegrating by parts, we obtain

! T

0

!

B2R

&|ut|2)2 +

12

d

dt

"|#u|2)2

#'dx dt = "2

! T

0

!

B2R

#uut#)) dx dt

+! T

0

!

B2R

|ut|2)2 dx dt +! T

0

!

B2R

|#u|2|#)|2 dx dt .


Therefore we can estimate

E"u(T ); BR

#" E(u0, B2R) + 1

2

!

B2R

"|#u|2)2

#dx

%%Tt=0

+ 4R"2

! T

0

!

B2R

|#u|2 dx dt + 8T

R2E0 ,

as claimed.

If the energy of a solution u to (6.1) is uniformly small on small balls, we canbound its second derivatives in terms of the energy.

6.10 Lemma. There exists *1 = *1(0, N) > 0 with the following property: Ifu ! C2

"B2R 2 [0, T [; N

#solves (6.1) on B2R 2 [0, T [ for some R !]0, 1

2 ı/[, andif

sup0't'T

E"u(t), B2R

#< *1 + E0 ,

then we have ! T

0

!

BR

|#2u|2 dx dt + c E0

&1 +

T

R2

',

where c depends only on 0 and N .

Proof. Let Q = B2R2[0, T [. Choose ) ! C&0 (B2R) such that 0 + ) + 1, ) - 1

on BR(x), |#)| + 2/R and multiply (6.1) by #u)2. Then from (6.4) we obtainthat

!

Q

&$t

" |#u|2)2

2#

+ |#u|2)2

'dx dt

+ C

!

Q|#u|2|#u|)2 dx dt " 2

!

Qut#u#)) dx dt

+ C

!

Q

"|#u|4)2 + |#u|2|#)|2

#dx dt +

12

!

Q|#u|2)2 dx dt

+ C1*1

!

Q|#2u|2)2 dx dt +

12

!

Q|#u|2)2 dx dt

+ CR"2

!

Q|#u|2 dx dt .

Here we have used the fact that ut#u = #u#u by (6.1). Young’s inequality2|ab| + 2a2 + 2"1b2 for any a, b ! IR, 2 > 0 was used to pass from the secondto the third line, and we invoked Lemma 6.7 in the final estimate.

Integrating by parts twice and again using Young’s inequality, we find theestimate

!

Q|#u|2)2 dx * 1

2

!

Q|#2u|2)2 dx " C

!

Q|#u|2|#)|2 dx .


Hence, if we choose *1 > 0 such that C1*1 + 18 , we obtain

!

Q|#2u|2)2 dx dt + 4

!

/|#u0|2)2 dx + CR"2

! T

0

!

B2R

|#u|2 dx dt

+ C E0

"1 +

T

R2

#.

By choice of ) this implies the assertion of the lemma.

As final preparation we show that if u ! H1,2 weakly solves (6.1) on someinterval ]0, T [ with |#2u| ! L2, then u is smooth and smoothly extends up totime T .

6.11 Lemma. Let Q = BR2]0, T [. If u ! H1,2(Q; N) solves (6.1) on Q with|#2u| ! L2(Q) and with sup0<t<T E

"u(t); BR

#+ E0, then u ! C&(Q; N) and

u extends smoothly to BR2]0, T ].

Proof. In order to avoid using cut-o! functions, we consider the case 0 = T 2,Q = 02]0, T [, for simplicity.

In a first step we then show that #u ! Lq"0 2 [t!, T ]

#for any q < .

and any t! > 0. We proceed by Moser iteration. For any L * 1 let |#u|L =min{|#u|, L}. Multiply (6.1) by "div

"#u|#u|sL

#and integrate by parts at any

(fixed) time t < T . Dropping the term

s

4

!

/#(|#u|2) ·#(|#u|2L)|#u|s"2

L dx =s

4

!

/|#(|#u|2L)|2|#u|s"2

L dx * 0 ,

on account of (6.4) for any s * 0 we obtain the estimate!

/

&$t

" |#u|22

#|#u|sL + |#2u|2|#u|sL

'dx + C

!

/|#2u||#u|sL|#u|2 dx

+ 12

!

/|#2u|2|#u|sL dx + C

!

/|#u|4|#u|sL dx .

The first term on the right is easily absorbed in the left hand side. DenotingwL,s = |#u|s/2

L |#u|, upon integrating in t, for any 0 < t0 < t1 < T we have! t1

t0

!

/$t

"|#u|2

#|#u|sL dx dt =

!

/|wL,s(t1)|2 dx "

!

/|wL,s(t0)|2 dx

" s

s + 2

! t1

t0

!

/$t

"|#u|s+2

L

#dx dt

* 2s + 2

!

/|wL,s(t1)|2 dx "

!

/|wL,s(t0)|2 dx .

Thus, with constants depending only on 0, N , and s we find


supt0<t<t1

!

/|wL,s|2 dx +

! t1

t0

!

/|#wL,s|2 dx dt

+ C

! t1

t0

!

/|#u|2|wL,s|2 dx dt + C

!

/|wL,s(t0)|2 dx

+ C

&! t1

t0

!

/|#u|4 dx dt ·

! t1

t0

!

/|wL,s|4 dx dt

' 12

+ C

!

/|wL,s(t0)|2 dx .

But by Lemma 6.7 for any L * 1, s * 0 on Q0 = 0 2 [t0, t1] we can bound

$wL,s$4L4(Q0) + c0 sup

t0't't1

$wL,s(t)$2L2(/) ·

($#wL,s$2

L2(Q0) + $wL,s$2L2(Q0)

).

Provided that t1 + t0 + 1, this yields the estimate!

Q0

|wL,s|4dx dt + c0

&sup

t0<t<t1

!

/|wL,s|2 dx +

!

Q0

|#wL,s|2dx dt

'2

.

Together with our previous bounds we then obtain! t1

t0

!

/|wL,s|4dx dt + c1

! t1

t0

!

/|#u|4 dx dt ·

! t1

t0

!

/|wL,s|4 dx dt

+ C

&!

/|wL,s(t0)|2 dx

'2

.

Since u has finite energy and |#2u| ! L2(Q), Lemma 6.7 in particular yieldsthat |#u| ! L4(Q). By absolute continuity of the Lebesgue integral we maythen choose - = -(s,0, N) > 0 such that for 0 + t0 < t1 + T with t1 " t0 < -there holds

2c1

! t1

t0

!

/|#u|4 dx dt + 1 .

For any L * 1, any s * 0, and any 0 < t0 < t1 < t0 + -, with a constant Cdepending only on the geometry and s there then results the estimate

! t1

t0

!

/|wL,s|4 dx dt + C

&!

/|wL,s(t0)|2 dx

'2

.

Let s0 = 0, si = 2(1 + si"1), i ! IN. Given t! > 0, suppose that |#u(ti"1)| !L2+si$1(0) for some i ! IN, where 0 = t0 + ti"1 < t!. We may assume thatt! < min{ti"1 + -, T}. Upon letting L % . then we can bound

! t!

ti$1

!

/|#u|2+si dx dt + C

&!

/|#u(ti"1)|2+si$1 dx

'2

.

Covering the interval ]ti"1, T [ with finitely many intervals of length -, we findthat #u ! L2+si

"02 [t!, T ]

#. Moreover, by Fubini’s theorem, there exists ti !

]ti"1, t![ such that |#u(ti)| ! L2+si(0). Induction then yields that |#u(ti)| !


L2+si(0) for suitable 0 + ti < t! for any i ! IN, and we conclude that #u !Lq

"0 2 [t!, T ]

#for any q < . and any t! > 0, as claimed.

The linear theory for (6.4) then yields that |ut|, |#2u| ! Lq"02 [t!, T ]

#for

any q < ., and #u is uniformly Holder continuous on 02 [t!, T ], for any t! >0; see Ladyzhenskaya-Solonnikov-Ural’ceva [1; Theorem IV.9.1 and LemmaII.3.3]. Higher regularity up to t = T then follows from the Schauder estimatesfor the heat equation, applied to (6.1), and the usual bootstrap argument.

Proof of Theorem 6.6. For simplicity, we always consider the case 0 = T 2.(1#) We first consider smooth initial data u0 ! C&(0; N). By the local solv-ability of ordinary di!erential equations in Banach spaces, problem (6.1), (6.2)has a local solution u ! C&(0 2 [0, T [; N); see Hamilton [1; p. 122 !.]. ByLemma 6.8 we have ut ! L2(0 2 [0, T ]) and E

"u(t)

#+ E

"u(s)

#+ E(u0) for

all 0 + s + t + T .Moreover, if R > 0 can be chosen such that

supx$/, 0't'T

E"u(t); BR(x)

#< *1 ,

it follows from Lemma 6.10 that! T

0

!

/|#2u|2 dx dt + c E(u0)

&1 +

T

R2

',

and hence from Lemma 6.11 that u extends to a C&-solution of (6.1) on theclosure 0 2 [0, T ]. But then u can be extended beyond T as a smooth solutionof (6.1). Thus, if T > 0 is maximal, there exist points x1, x2, . . . such that

lim supt-T

E"u(t), BR(xk)

#* *1

for any R > 0 and any index k. Choose any finite collection xk, 1 + k + K, ofsuch points and for any R > 0, k = 1, . . . , K, let tk < T be chosen such that

E"u(tk); BR(xk)

#* *1/2 .

We may assume that tk * T " &1R2

4c1E0=: t0 * 0 and that B2R(xk),B2R(xj) = >

for j )= k. With the constant c1 from Lemma 6.9 then we have

(6.6)

E"u(t0)

#*

K.

k=1

E"u(t0); B2R(xk)

#

*K.

k=1

&E

"u(tk); BR(xk)

#" c1

tk " t0R2

E0

'

* K

&*1/2 " c1

T " t0R2

E0

'= K*1/4 .

Since by Lemma 6.8 we have E"u(t0)

#+ E(u0), this gives an upper bound


K + 4E(u0)/*1

for the number K of singular points x1, . . . , xK at time T . Moreover for anyQ && 0 2 [0, T ] \ {(x1, T ), . . . , (xK, T )} there exists R = RQ > 0 such that

sup(x,t)$Q

E"u(t); BR(x)

#< *1

and by Lemma 6.10, 6.11 our solution u extends to a C&-solution of (6.1) on0 2 [0, T ] \ {(x1, T ), . . . (xK , T )}.

(2#) For initial data u0 ! H1,2(0; N) choose a sequence u0m ! C&(0; N)approximating u0 in H1,2(0; N). This is possible by Theorem 6.2. For eachm let um be the associated solution of (6.1) with um(0) = u0m and Tm > 0 itsmaximal time of existence. Let E0 = supmE(u0m). Choose R0 > 0 such that

supx$/

E"u0; B2R0(x)

#+ *1/4 .

This inequality will also hold with *1/2 instead of *1/4 for u0m if m * m0. ByLemma 6.9 then for T = &1R2

04c1E0

we have

supx"(

0(t(min{Tm,T}

E"um(t); BR0(x)

#+ sup

x$/E

"u0m, B2R0(x)

#+ c1

T

R20

E0

+ *1/2 + *1/4 < *1 .

By Lemma 6.10 it follows that #2um is uniformly bounded in L2"02 [0, t]

#for

t + min{T, Tm} in terms of E(u0) and R0 only. By Lemma 6.11, therefore, theinterval of existence of um is both open and closed in [0, T ]; that is, Tm * T > 0.Moreover ! T

0

!

/|#2um|2 dx dt + c E0

&1 +

T

R20

'

uniformly. Also using Lemma 6.8, we may assume that um converges weaklyto a solution u of (6.1), (6.2) with |ut|, |#2u|2 ! L2

"0 2 [0, T ]

#, and such that

E"u(t)

#+ lim inf

m%&E

"um(t)

#+ lim

m%&E(u0m) = E(u0) ,

uniformly in t ! [0, T ]. Since |ut| ! L2"0 2 [0, T ]

#, the solution u attains

its initial data u0 continuously in L2(0; N); by the uniform energy boundE

"u(t)

#+ E(u0), moreover, this is also true in the H1,2(0; N)-topology.

By Lemma 6.11 we have u ! C&(02]0, T1[, N) for some maximal T1 > T .By Lemma 6.8, we then also have the energy estimate E

"u(t)

#+ E

"u(s)

#+

E(u0) for 0 + s + t < T1. By part (1#) of this proof, u extends smoothly to02]0, T1] \ {(x1, T1), . . . , (xK1 , T1)} for some finite collection of singular pointsxk, 1 + k + K1. Moreover, as t < T1 we have u(t) % u(1)

0 ! H1,2(0; N)weakly and strongly in H1,2

loc

"0 \ {x1, . . . , xK1}; N

#. Thus, by (6.6),


E(u(1)0 ) = lim

R%0E

"u(1)

0 ;0 \K1?

k=1

B2R(xk)#

= limR%0

limt-T1

0E

"u(t);0

#"

K1.

k=1

E"u(t); B2R(xk)

#1

+ E(u0) " K1*1/4 .

Letting T0 = 0, u(0) = u, u(0)0 = u0, by iteration we now obtain a sequence

u(m) of smooth solutions to (6.1) on 02]Tm, Tm+1[ with initial data u(m)0 at

t = Tm and such that u(m)(t) % u(m+1)0 weakly in H1,2(0; N) as t < Tm+1.

Moreover, u(m) has finitely many singularities x(m)1 , . . . , x(m)

Km+1at t = Tm+1,

where

(6.7) E"u(m+1)

0

#+ E

"u(m)

0

#" Km+1*1/4 + . . . + E(u0) "

m+1.

l=1

Kl*1/4 ,

and u(m)(t) % u(m+1)0 smoothly away from x(m)

1 , . . . , x(m)Km+1

as t < Tm+1, forany m ! IN. In particular, the total number of singularities of the flows u(m) isfinite, and Tm = . for some m ! IN. Piecing the u(m) together, for any initialu0 ! H1,2(0; N) we thus obtain a weak solution u to (6.1), (6.2) on 02]0,.[satisfying the energy inequality E

"u(t)

#+ E

"u(s)

#+ E(u0) for 0 + s + t < .,

attaining the initial data continuously in H1,2"0; N

#, and smooth on 02]0,.[

up to finitely many points.

(3#) Asymptotics: If for some T > 0, R > 0 we have

supx$/, t>T

E"u(t); BR(x)

#< *1 ,

then, by Lemma 6.10, for any t > T there holds! t+1

t

!

/|#2u|2 dx dt + c E(u0)

"1 + R"2

#

with a uniform constant c = c(0, N), while by Lemma 6.8 we have! t+1

t

!

/|ut|2 dx dt % 0 .

Hence we may choose a sequence tm % . such that u(tm) % u& weaklyin H2,2(0; N), while ut(tm) % 0 in L2. Moreover, by the Rellich-Kondrakovtheorem u(tm) % u& also strongly in W 1,p(0; N) for any p < .. Testing (6.1)with #(u " u&) and integrating by parts on 0 2 {tm}, we obtain that!

/+{tm}|#2(u " u&)|2 dx

+%%%%!

/+{tm}ut#(u " u&)dx

%%%% +%%%%!

/+{tm}A(u)(#u,#u)#(u" u&)dx

%%%% % 0 ;


that is, u(tm) % u& also strongly in H2,2(0; N). Passing to the limit m % .in (6.1), evaluated at tm, we then find that u& is harmonic.

It remains to study the case that for a sequence sm % . there exist pointsx0m ! 0 such that for any R > 0 we have

lim infm%&

"E

"u(sm); BR(x0m)

##> *1/2 .

Passing to a subsequence, we may assume that x0m % x0, and there holds

(6.8) lim infm%&

"E

"u(sm); B2R(x0)

##* *1/2

for any R > 0. Suppose that (6.8) holds for points x1, . . . , xK . Choose R > 0such that B2R(xj),B2R(xk) = > (j )= k). Then for su"ciently large m we have

(6.9) E"u(sm)

#*

K.

k=1

E"u(sm); B2R(xk)

#* K*1/4 .

It follows that K + 4E(u0)/*1. We thus may assume that the collectionx1, . . . , xK is maximal with property (6.8). By repeated selection of subse-quences of (sm) and in view of the uniform boundedness of the number K ofconcentration points (independent of the sequence (sm)), we can even achievethat x1, . . . , xK is maximal with the property that

lim supm%&

"E

"u(sm); B2R(xk)

##* *1/2

for any R > 0, 1 + k + K. By Lemma 6.9 then for any ! && 0 \{x1, . . . , xK}there exists R > 0 such that for any x ! ! with < = &1R2

4c1E(u0) there holds

supsm't'sm+1

E"u(t); BR(x)

#+ lim sup

m%&E

"u(tm); B2R(x)

#+ *1/2 + *1

for large m. By compactness, ! is covered by finitely many such balls BR(x).Hence by Lemma 6.10 there holds

(6.10)! sm+1

sm

!

"|#2u|2 dx dt + c E(u0)

(1 +

<

R2

)= c E(u0) + c

for large m, while Lemma 6.8 implies that

(6.11)! sm+1

sm

!

"|ut|2 dx dt % 0 .

Exhausting 0 \ {x1, . . . , xK} by such domains !, we may choose a sequencetm ! [sm, sm + < ] such that u(tm) % u& weakly in H2,2

loc (0 \ {x1, . . . , xK}; N),where u&:0 \ {x1, . . . , xK} % N is harmonic. Moreover, by Lemma 6.8 wealso may assume that u(tm) % u& weakly in H1,2(0; N), and u& has finiteenergy. By the regularity result of Sacks-Uhlenbeck [1; Theorem 3.6], then u&extends to a smooth harmonic map u& ! C&(0; N).


(4#) Singularities: Suppose (x, t) is singular in the sense that for any R > 0 wehave

lim supt-t

E"u(t); BR(x)

#* *1 ,

or that x is a concentration point at t = . in the sense that for a sequencesm % . as in (3#) for any R > 0 we have

lim infm%&

"E

"u(sm); B2R(x)

##* *1/2 .

By finiteness of the singular set, x is isolated among concentration points.Hence there is R0 > 0 such that when we let Rm % 0 and set <m = &1R2

m16c1E(u0)

,there are tm < t and xm % x such that

E"u(tm); BRm(xm)

#= sup

x"B2R0(x)

tm$)m(t(tm

E"u(t); BRm(x)

#= *1/4 .

We may assume BRm(xm) & BR0(x); moreover, if t = . we may suppose thatsm + tm " <m + sm + <m. Rescale

um(x, t) := u(xm + Rmx, tm + R2mt)

and note that um: B1+R0/Rm2 [t0, 0] % N , with t0 = " &1

16c1E(u0), solves (6.1)

classically with

supRm|x|(R0

t0(t(0

E"um(t); B1(x)

#+ E

"um(0); B1

#= *1/4

and ! 0

t0

!

BR0/Rm

|um,t|2 dx dt +! tm

tm"1m

!

/|ut|2 dx dt % 0 ,

as m % .. From Lemma 6.10 it follows that also! 0

t0

!

BR0/Rm

|#2um|2 dx dt + c

uniformly. Hence for a suitable sequence tm ! [t0, 0] we have convergenceum,t(tm) % 0 in L2, while um(tm) % u weakly in H2,2

loc (IR2; N) and stronglyin H1,2

loc (IR2; N). (In fact, we can even show that um(tm) % u strongly inH2,2

loc (IR2; N).) Upon passing to the limit m % . in Equation (6.1) for um att = tm we see that u is harmonic. Moreover, by Lemma 6.9 with error o(1) % 0as m % . we have

E(u; B2) = E"um(tm); B2

#" o(1)

* E"um(0); B1

#+ c1t0E(u0) " o(1) = *1/4 " *1/16 " o(1) > 0

for su"ciently large m. Thus u )- const. Finally,


E(u) + lim infm%&

E(um(tm); BR0/Rm) + lim inf

m%&E(u(tm + R2

mtm)) + E(u0) ,

and by the result of Sacks-Uhlenbeck [1; Theorem 3.6] quoted earlier, u extendsto a harmonic map u: S2 % N .

In particular, we conclude E(u) = E(u) * *0. Therefore, and usingLemma 6.9, for any r, R > 0 we obtain that

E"u(sm); B2R(xm)

#* E

"u(tm + R2

mtm); BR(xm)#" o(1)

= E"um(tm); BR/Rm

#" o(1) * E

"um(tm); Br

#" o(1)

= E"um; Br

#" o(1) * *0 " o(1) ,

where o(1) % 0 as we first let m % . and then r % .. It follows thatestimates (6.6), (6.7), and (6.9) may be improved, yielding the upper boundK + E(u0)/*0 for the total number of singularities and concentration points.

(5#) Uniqueness: It su"ces to show that two solutions u, v of (6.1) satisfying

$tu ,#2u , $tv ,#2v ! L2"0 2 [0, T ]

#

and with u |t=0= u0 = v |t=0 coincide. Let w = u" v. By (6.1), w satisfies thedi!erential inequality

|$tw "#/w| = |A(u)(#u,#u)" A(v)(#v,#v)|+ C|w|

"|#u|2 + |#v|2

#+ C|#w|

"|#u| + |#v|

#.

Multiplying by w and integrating by parts over 0 2 [0, t0], for any t0 > 0 weobtain

(6.12)

12

!

/|w(t0)|2 dx +

! t0

0

!

/|#w|2 dx dt

+ C

! t0

0

!

/|w|2

"|#u|2 + |#v|2

#dx dt +

12

! t0

0

!

/|#w|2 dx dt .

Here we used Young’s inequality to estimate

C|w||#w|"|#u| + |#v|

#+ 1

2|#w|2 + C|w|2

"|#u|2 + |#v|2

#,

and we also used the fact that w |t=0= 0.By Lemma 6.7, the functions |#u|, |#v|, and w belong to L4

"0 2 [0, t0]

#,

and for any 2 > 0 we can estimate! t0

0

!

/

"|#u|4 + |#v|4

#dx dt

+ CE(u0)! t0

0

!

/

"|#2u|2 + |#2v|2 + |#u|2 + |#v|2

#dx dt + 22 ,

if 0 < t0 + T0 = T0(2) is su"ciently small. We may assume that T0 + 1. Thensimilarly for such t0 we can bound


! t0

0

!

/|w|4 dx dt

+ C

&sup

0't't0

!

/|w(t)|2 dx

'·&! t0

0

!

/

"|#w|2 + |w|2

#dx dt

'

+ C

&sup

0't't0

!

/|w(t)|2 dx +

! t0

0

!

/|#w|2 dx dt

'2

.

Given 2 > 0, choose 0 < t0 + T0 = T0(2) such that!

/|w(t0)|2 dx = sup

0't'T0

!

/|w(t)|2 dx = sup

0't't0

!

/|w(t)|2 dx.

Then, from (6.12) we obtain the estimate

sup0't't0

!

/|w(t)|2 dx +

! t0

0

!

/|#w|2 dx dt

+ C

&! t0

0

!

/|w|4 dx dt

'1/2 &! t0

0

!

/

"|#u|4 + |#v|4

#dx dt

'1/2

+ C12

&sup

0't't0

!

/|w(t)|2 dx +

! t0

0

!

/|#w|2 dx dt

'.

with a constant C1 = C1(0, N). Thus, if we choose 2 > 0 such that C12 < 1, itfollows that w - 0 on 02 [0, T0(2)]. More generally, the above argument showsthat the maximal interval I & [0, T ] containing t = 0 and such that u(t) = v(t)for t ! I is relatively open. Since I is trivially closed, uniqueness follows.

6.12 Remark. Chang [6] has obtained the analogue of Theorem 6.6 for theevolution problem (6.1), (6.2) on manifolds 0 with boundary $0 )= > and withDirichlet boundary data. His result – in the same way as we used Theorem 6.6to prove Theorem 6.5 – can be used to prove, for instance, the existence andmultiplicity results of Brezis-Coron [1] and Jost [1] for harmonic maps withboundary.

Finite-time Blow-up.

The question whether singularities actually may occur in finite time remainedopen for quite a while. It was finally settled by Chang-Ding-Ye [1] who gavethe following example.

6.13 The co-rotational setting. Let x = rei( be the representation of pointsx ! IR2 in polar coordinates (r,)). Similarly, we denote as = the longitudinalangle on S2 and let 7 denote the distance traveled along any fixed geodesicfrom the north pole N . A map u: B = B1(0) % S2 is co-rotational (of degree1), if u is represented by maps


=(rei() = ) , 7(rei() = h(r)

so thatu(rei() = expN (h(r)ei() .

The energy of u then may be expressed as

(6.13) E(u) =12

!

B|#u|2 dx = .

! 1

0

"|hr|2 +

sin2h

r2

#dr ,

and u is harmonic if and only if h satisfies the equation

"1r(rhr)r +

sin 2h

2r2= 0 on [0, 1[ .

Likewise, a smooth family u(t), 0 + t < T , of such maps, represented by a maph: [0, 1]2 [0, T [% IR, evolves by the heat flow for harmonic maps from B to S2,that is, solves the equation

(6.14) ut "#u = u|#u|2 on B 2 [0, T [ ,

if and only if h solves the scalar evolution equation

(6.15) ht "1r(rhr)r +

sin 2h

2r2= 0 on [0, 1[2[0, T [ .

The representationh(r) = 2 arctan(r)

of the harmonic map u: IR2 % S2 obtained as the inverse of stereographicprojection then is a stationary (time-independent) solution of (6.15), as are allfunctions

h!(r) = h(r/") , " > 0 ,

corresponding to the scaled maps u!(x) = u(x/"), " > 0.Chang-Ding-Ye [1] show the following result.

6.14 Theorem. For any smooth co-rotational data u0: B % S2, represented bya map h0: [0, 1] % IR with h0(0) = 0 and h0(1) = b > ., the solution u to theEquation (6.14) with inital and boundary data u0, or, equivalently, the solutionh to Equation (6.15) with data

h(0, r) = h0(r) , h(t, 0) = h0(0) = 0 , h(t, 1) = h0(1) = b ,

will develop a singularity at a point (0, T ) for some T < . in the sense that|#u| and |hr| will be unbounded in any neighborhood of the point (0, T ) !B 2 [0, T ].

Observe that all maps h0 appearing in the Chang-Ding-Ye result cover thesphere slightly more than once. It is an interesting open question what theprecise rate of blow-up is.


Proof. Let b > .. We show that (6.15) admits a smooth sub-solution f with

f(0, t) = 0 + f(r, t) + f(1, t) + b

on [0, 1]2 [0, Tf [ for some Tf < . such that

fr(0, t) % . as t % Tf .

For any data h0 as in the theorem such that h0 * f(·, 0) we then consider thecorresponding solution h of (6.15) with initial data h(0) = h0. Provided thata smooth solution h exists for 0 + t < Tf , by the maximum principle h willsatisfy h * f on [0, 1] 2 [0, Tf [. Since h(0, t) = 0 = f(0, t), it then also followsthat

hr(0, t) % . as t % Tf .

Hence h must blow up at some time T + Tf .For constants * > 0, µ > 0 consider the function l = l(r), given by

l(r) = hµ(r1+&) .

Note that l satisfies the equation

(6.16) "1r(rlr)r +

(1 + *)2 sin 2l

2r2= 0 .

Fix some number 0 < * < 1 once and for all. Given b > . as above, wecan fix µ > 0 su"ciently large so that we have l(1) + . + b and so that, inaddition, there holds

(6.17) cos l(r) * 11 + *

for 0 + r + 1 .

With a function " = "(t) > 0 to be specified later we then make the ansatzf = h! + l. We have 0 + f(r, t) + b for 0 + r + 1 and all t * 0 where "(t) isdefined. Moreover, in view of (6.15), (6.16) we compute

(6.18) <(f) :=1r(rfr)r "

sin 2f

2r2=

sin 2h!(t) " sin 2f + (1 + *)2 sin 2l

2r2.

Now observe that for any 6, ( ! IR there holds

sin 2(6+ () " sin 26 = sin((26+ () + () " sin((26+ () " ()= 2 cos(26+ () sin( .

Thus we may simplify (6.18) and use (6.17) to obtain

<(f) = r"2"(1 + *)2 cos l " cos(2h!(t) + l)

#sin l

* r"2"1 + *" cos(2h!(t) + l)

#sin l * r"2* sin l .

But for |6| < ./2 we have the identity


sin 26 =2 tan6

1 + tan2 6.

Since we may assume that µ > 1, it follows that

<(f) * *

r2sin l =

*

r2sin(2 arctan(r1+&/µ)) =

*

r2

2µr1+&

µ2 + r2(1+&)* *1r&"1 ,

where *1 = 2µ&µ2+1 > 0. Given "0 > 0, 2 > 0 define

"(t) =""1"&

0 " (1 " *)2t#1/(1"&)

so thatd

dt" = "2"& .

Then we compute

d

dtf =

d

dth!(t) = " 2r

r2 + "2

d

dt" =

22r"&

r2 + "2.

Hence we obtain

d

dtf " <(f) + 22"&r

r2 + "2" *1r&"1 =

&22"&r2"&

r2 + "2" *1

'r&"1 .

But by Young’s inequality we have

"&r2"& + C(*)(r2 + "2) .

Hence, for any * > 0 there is 20 > 0 such that for any 0 < 2 < 20 the functionf is a sub-solution to (6.15), as desired.

Observe that the constants µ and "0 in the above construction may be cho-sen arbitrarily large. Thus, for any h0 with h0(1) > . and such that dh0

dr (0) > 0while h0(r) > 0 for r > 0 we can arrange that f(·, 0) + h0 and conclude thatthe solution h to (6.15) must blow up in finite time.

By the maximum principle the above conditions will hold true at any timet > 0 for the solution h(t) issuing from any function h0 * 0 which satisfies thehypotheses of the theorem.

Finally, observe that for any k ! IN the function f " k. again is a sub-solution to (6.15). By iteratively employing the sub-solutions f " k. for k =k0, . . . , 1, where k0 is determined so that k0.+h0 * 0, for a general function h0

as in the theorem we then see that after some waiting-time t0 > 0 the conditionh(t) * 0 will hold true for all t * t0, provided the solution h does not blow upbefore t0. The proof is complete.


Reverse Bubbling and Nonuniqueness

The fact that the energy E(u(t)) of the solutions u constructed in Theorem 6.6to the heat flow (6.1), (6.2) is non-increasing in t is essential for the uniquenessof these solutions. In fact, and quite surprisingly, prompted by the work ofChang-Ding-Ye [1] presented above, Topping [1] and Bertsch et al. [1] inde-pendently constructed examples of weak solutions to (6.1), (6.2) which violatethe energy inequality and therefore di!er from the solutions constructed inTheorem 6.6. These solutions, moreover, are smooth with isolated singularitiesand have bounded energy.

The construction intuitively is achieved by attaching more and more con-centrated “bubbles” to some smooth given data u0 at time t = 1, and lettingthe new configuration evolve by the flow, say, for time 1 < t < 2. Under suit-able hypotheses on u0, the resulting sequence of solutions to (6.1) then canbe shown to converge to a limit u which blows up as we go backwards to theinitial time t = 1 where a “bubble” develops. This process can therefore bestbe termed “reverse bubbling”.

The following result is taken from Topping [1], Theorem 1.1. We againwork in the co-rotational setting described in Section 6.12. Recall the repre-sentation

h(r) = 2 arctan(r)

of the harmonic map obtained as the inverse of stereographic projection andthe notation

h!(r) = h(r/")," > 0

for the rescaled map h.

6.15 Theorem. Let u0: B % S2 be the co-rotational harmonic map representedby the function h0(r) = ." h(r), thus precisely covering the lower hemisphere.There exists a co-rotational weak solution u ! H1

loc(B2[0,.[; S2) to (6.1) suchthat u(t) = u0 for t + 1 and u(t) )= u0 for t > 1. Moreover, u is smooth exceptat the point (0, 1) ! B 2 [0,.[, and there holds the uniform energy bound

(6.19) 2. = E(u0) + E(u(t)) + E(u0) + 4. = limt.1

E(u(t)) for all t .

Proof. Similar to the proof of Theorem 6.14 we first construct a super-solutiong to the flow (6.15) which exhibits reverse bubbling at the time t = 0. Manydetails of the construction may be borrowed from the preceding section.

For constants * > 0, µ > 0 again consider the function l = l(r), given by

l(r) = hµ(r1+&) .

Recall that l satisfies Equation (6.16), that is,

"1r(rlr)r +

(1 + *)2 sin 2l

2r2= 0 .


We fix 0 < *< 1 and then choose µ > 0 su"ciently large so that the estimatesl(1) < ./4 and (6.17) both hold, that is,

cos l(r) * 11 + *

for 0 + r + 1 .

For suitable 2 > 0, to be determined in the sequel, we then define

"(t) ="(1 " *)2t

#1/(1"&),

satisfyingd

dt" = 2"&, "(0) = 0.

We then make the ansatz

g = h! " l .

In the identity

sin 2(6+ () " sin 26 = 2 cos(26+ () sin( ,

we now let 6 = h!(t), ( = "l. By means of (6.16), (6.17) we then compute

<(g) =1r(rgr)r "

sin 2g

2r2=

sin 2h!(t) " sin 2g " (1 + *)2 sin 2l

2r2

= r"2"cos(2h!(t) " l) " (1 + *)2 cos l

#sin l

+ r"2"cos(2h!(t) " l) " (1 + *)

#sin l

+ " *

r2sin l = " 2*µr&"1

µ2 + r2(1+&)+ "*1r&"1 ,

where *1 = 2µ&µ2+1 > 0. Moreover, we have

d

dtg =

d

dth!(t) = " 2r

r2 + "2

d

dt" = " 22r"&

r2 + "2.

Hence we obtain

d

dtg " <(g) * *1r&"1 " 22"&r

r2 + "2=

&*1 "

22"&r2"&

r2 + "2

'r&"1 * 0 ,

provided we choose 0 < 2 < 20 su"ciently small.Next, for n ! IN consider the co-rotational solutions un to the flow (6.14)

with initial data u0n represented by the functions

h0n(r) = min{h0(r), h(nr)} , 0 + r + 1 .

Note that there is a unique point 0 < rn < 1, tending monotonically to 0 asn % ., such that h0n(r) = h(nr) for r < rn and h0n(r) = h0(r) for r > rn; inparticular, we have that h0n(0) = h(0) = 0 and h0n(1) = h0(1) = ./2 for all


n. Moreover, there holds h0m(r) + h0n(r) for all r ! [0, 1], whenever m + n.Finally, from the representation (6.13) we also conclude the bound

E(u0n) = E(u0, B \ Brn) + E(u, Bnrn) < E(u0) + E(u) = 6. = limn%&

E(u0n) .

Here, in addition, we use the fact that u0 is conformal; its Dirichlet energytherefore equals the area 2. covered by its image.

By the energy inequality, Lemma 6.8, we then have

(6.20) E(un(t)) < E(u0) + E(u) = 6.

for all t > 0. In particular, the flow solutions un will be smooth for t > 0, asany singularity, say at a time tn > 0, by Theorem 6.6 would absorb at least theenergy 4. and therefore not leave enough energy for the weak limit un(tn) tospan the boundary data given by u0, which would require at least energy 2..

Letting hn be the solution to (6.15) representing un, we thus may invokethe maximum principle to conclude that

(6.21) 0 + hm(r, t) + hn(r, t) + min{h0(r), g(r, t)}

for all r ! [0, 1], t * 0, whenever m + n. Indeed, since h0 is a time-independentsolution of (6.15) and since we have h0m + h0n + h0 for m + n, the firstinequalities are immediate. Moreover, from our assumption that l(1) < ./4 wededuce that h0(r) < ." l(r) = limt.0 g(r, t) for any r > 0. Thus for any n ! INthere exists a number < > 0 such that h0n + g(<) and hence hn(t) + g(< + t)for all t * 0. But by construction the function g(t) is non-increasing in t, and(6.21) follows.

In view of the unifom energy bound (6.20) and the pointwise bounds (6.21)a subsequence hn thus converges smoothly away from r = 0 to a solution h of(6.15). We can strengthen this assertion to include the axis r = 0 by means ofLemma 6.10 once we show the following result.

6.16 Lemma. There is a time T > 0 and an index N ! IN such that for anyt0 !]0, T [ there exists a radius r0 > 0 so that

supt0't'T

E(un(t), Br0(0)) < *1 for all n * N ,

where *1 > 0 is the constant determined in Lemma 6.10 for N = S2.

Proof. We may of course assume that5&14. < .

2 .Since sup0'r'1 h0n(r) % . as n % ., we can find an index N and a

radius r1 := rN > 0 such that

h0N (r1) = sup0'r'1

h0N (r) * . " 12

:*14.

.

We may then fix T > 0 su"ciently small to insure the inequality


hN (r1, t) * . ":*14.

for any t ! [0, T ]. By (6.21) we then also have

(6.22) hn(r1, t) * . ":*14.

for any n * N , uniformly in t ! [0, T ]. Reducing T , if necessary, we also mayassume that there holds g(1, t) > ./2 for 0 + t + T .

For any t0 !]0, T [ now we choose 0 < r0 < 1 so that

g(r, t0) <

:*14.

for r ! [0, r0]. Recalling that g(r, t) is monotonically decreasing in t for any r,by (6.21) then for any r ! [0, r0], any t ! [t0, T ], and any n * N we have

(6.23) hn(r, t) + g(r, t) + g(r, t0) <

:*14.

.

In particular, in view of (6.22) and our assumption about *1 then it followsthat 0 < r0 < r1 < 1. Since for each t ! [t0, T ] the map hn(t), restrictedto [r0, r1] thus covers the interval [

5 &14. , . "

5 &14. ], we conclude that the map

un(t), restricted to the annulus Br1 \Br0 covers the sphere with caps of radius5 &14. removed from each pole. In particular, since the area of a geodesic disk

on S2 of radius r > 0 is less than .r2, the area covered by un(t) over this regionis larger than 4. " 2.

"5&14.

#2 = 4. " &12 , and hence

E(un(t), Br1 \ Br0) * 4. " *12

.

By a similar reasoning, and since we have hn(1, t) = ./2 for all t and n, weconclude that the map hn(t), restricted to [r1, 1] covers the interval [./2, . "5&14. ], and thus the map un(t), restricted to the annulus B \ Br1 covers the

lower hemisphere with a cap of radius5&14. removed from the south pole. It

follows that we have

E(un(t), B \ Br1) * 2. " .&:

*14.

'2

* 2. " *14

,

leaving the energy

E(un(t), Br0) = E(un(t)) ""E(un(t), B \ Br1) + E(un(t), Br1 \ Br0)

#

+ 6. " (6. " 3*14

) =3*14

< *1 ,

for the ball Br0 for any t ! [t0, T ], any n * N , as claimed.


Proof of Theorem 6.15, completed. In view of Lemma 6.16, from Lemma 6.10we obtain uniform bounds for #2un in L2(B2 [t0, T ]) for any t0 > 0. Togetherwith the smooth estimates on B2]0, T ] away from x = 0 these estimates su"ceto pass to the limit in Equation (6.14) for un and we obtain a solution u to thisequation which splits o! a bubble of area 4. when t decreases to 0. Moreover,u(t) % u0 weakly in H1(B) and hence strongly in L2(B) as t ? 0. We maytherefore shift time by 1 and join this solution of (6.14) with the constantsolution u(t) - u0 for 0 + t + 1 at time t = 1 to obtain a weak solutionof (6.14) on the interval [0, 1 + T ] satisfying (6.19). In view of the inequalityE(u(t)) + 6., which by Lemma 6.8 must be strict for t > 1, again the flowu cannot become singular at any time t > 1 and therefore can be smoothlyextended for all t > 1. This finishes the proof.

6.17 Uniqueness issues. Theorem 6.15 also shows that if we drop the require-ment that E(u(t)) be non-increasing, the initial value problem for (6.14) withdata u0 has more than just the constant solution u(t) - u0; in fact, since wemay shift time by an arbitrary amount, Theorem 6.15 gives infinitely manydistinct solutions that exhibit reverse bubbling at an arbitrary time t0 > 0.

On the other hand, results of Riviere [1] and Freire [1], [2] establish unique-ness for the Cauchy problem (6.1), (6.2) also in the “energy class” of weaksolutions u to (6.1) such that E(u(t)) is bounded and $tu ! L2

"0 2 [0, T ]

#,

provided that the energy E(u(t)) is a non-increasing function of t. In partic-ular, in view of the smoothing property of the flow (6.1) for short time, theirresults give a new proof of Helein’s [1] regularity result for weakly harmonicmaps of surfaces. Topping [1] conjectured that uniqueness holds in the energyclass if we impose suitable hypotheses to rule out reverse bubbling, for instance,if we suppose that

(6.24) lim supt.t0

E(u(t)) < E(u(t0)) + *0 ,

where *0 is defined in Theorem 6.6. As shown by Rupflin [1], this conjectureholds true if, in addition to (6.24), we assume that the function E(u(t)) isof locally bounded variation, or if the constant *0 in (6.24) is replaced by apossibly smaller constant *1 > 0, depending only on the target manifold.

The preceding remarks refer to arbitrary smooth and closed target mani-folds N . In the special case when N = S2, from a physical interpretation of theDirichlet energy E(u,!) as the Frank-Oseen energy of a nematic liquid crystalcontained in a cylinder of unit length with cross-section !, Bertsch et al. [1]conjecture that there is a unique “physical time” t1 * t0 when the energy con-centrated in a spherical bubble separating at time t0 from a flow solution u of(6.14) may be “released” as the bubble is re-attached. It may be conjecturedthat suitable approximations, like the Ginzburg-Landau approximation stud-ied by Harpes [1], can give rise to such uniqueness results for the flow (6.14)in a restricted energy class of solutions that allows both forward and backwardbubbling.


6.18 Energy Quantization. At a singular time t < . the flow constructed inTheorem 6.6 splits o! a finite number of bubbles uj : S2 % N , j = 1, . . . , k, andwe have the estimate

(6.25) limt/tE(u(t)) * E(u(t)) +k.

j=1

E(uj) .

Similar to the quantization result for the Yamabe equation that we obtainedin Theorem 3.1, we might expect (6.25) to hold with equality. When N = Sn,this was actually shown by Jie Qing [2]. His result was generalized to arbitrarytarget manifolds by C. Wang [1] and Ding-Tian [1], independently, with laterimprovements by Lin-Wang [1].

The analogous quantization result

(6.26) limm%&

E(um) = E(u0) +k.

j=1

E(uj)

for sequences of harmonic maps um ' u0 weakly in H1(B, N) with bubblesuj : S2 % N , j = 1, . . . , k, was shown by Parker [1]. Parker’s result was ex-tended to Palais-Smale sequences (um) with

<(um) = #um " A(um)(#um,#um) % 0 in L2

and bounded energy by Qing-Tian [1]. By the energy inequality for the heatflow, Lemma 6.8, from (6.26) we then also obtain the corresponding energyquantization in the case when the heat flow concentrates as t % ., for suitablesequences tm % ..

6.19 Remarks. (1#) Under suitable hypotheses Topping [3] shows exponentialL2-convergence as t % . of the heat flow (6.1) for harmonic maps S2 % S2 inthe presence of “bubbling at infinity”, and exponential H1-convergence locallyaway from any bubble points of the flow.(2#) Results analogous to Theorem 6.6 hold for a number of planar geometricflows and may be applied to prove the existence of minimal surfaces or, moregenerally, of surfaces of constant mean curvature with free boundaries; seeStruwe [16].(3#) See Schoen-Uhlenbeck [1], [2], Struwe [14], Chen-Struwe [1], Coron [3],Coron-Ghidaglia [1], Chen-Ding [1] for results on harmonic maps and the evo-lution problem (6.1), (6.2) in the case dim(0) > 2. A survey of results relatedto harmonic maps can be found in Eells-Lemaire [1], [2]. An overview of devel-opments for the evolution problem is given in Struwe [20], [24].

Appendix A

Here, we collect without proof a few basic results about Sobolev spaces. Ageneral reference to this topic is Gilbarg-Trudinger [1], or Adams [1].

Sobolev Spaces

Let ! be a domain in IRn. For u ! L1loc(!) and any multi-index " =

("1, . . . ,"n) ! INn0 , with |"| =

!nj=1 "j, define the distibutional derivative

D!u = "!1

"x!11

· · · "!n

"x!nn

u by letting

(A.1) < #, D!u >="

#("1)|!|uD!# dx,

for all # ! C!0 (!). We say D!u ! Lp(!), if there is a function g! ! Lp(!)

satisfying< #, D!u >=< #, g! >=

"

##g! dx

for all # ! C!0 (!). In this case we identify D!u with g! ! Lp(!).

For k ! IN0, 1 # p # $, define the space

W k,p(!) =#u ! Lp(!); D!u ! Lp(!) for all ": |"| # k

$,

with norm%u%p

W k,p =%

|!|"k

%D!u%pLp , if 1 # p < $,

respectively, with norm

%u%W k,! = max|!|"k

%D!u%L! .

Note that the distributional derivative (A.1) is continuous with respect to weakconvergence in L1

loc(!).Many properties of Lp(!) carry over to W k,p(!).

A.1 Theorem. For any k ! IN0, 1 # p # $, W k,p(!) is a Banach space.W k,p(!) is reflexive if and only if 1 < p < $. Moreover, W k,2(!) is a Hilbertspace with scalar product

(u, v)W k,2 =%

|!|"k

"

#D!u D!v dx ,

inducing the norm above.

For 1 # p < $, W k,p(!) also is separable. In fact, we have the following resultdue to Meyers and Serrin; see Adams [1; Theorem 3.16].

264 Appendix A

A.2 Theorem. For any k ! IN0, 1 # p < $, the subspace W k,p & C!(!) isdense in W k,p(!).

The completion of W k,p & C!(!) in W k,p(!) is denoted by Hk,p(!). ByTheorem A.2, W k,p(!) = Hk,p(!). In particular, if p = 2 it is customary touse the latter notation.

Finally, W k,p0 (!) is the closure of C!

0 (!) in W k,p(!); in particular,Hk,2

0 (!) is the closure of C!0 (!) in Hk,2(!), with dual H#k(!). Dk,p(!)

is the closure of C!0 (!) in the norm

%u%pDk,p =

%

|!|=k

%D!u%pLp .

Holder Spaces

A function u:! ' IRn ( IR is Holder continuous with exponent $ > 0 if

[u]($) = supx$=y%#

|u(x) " u(y)||x " y|$ < $.

For m ! IN0, 0 < $ # 1, denote

Cm,$(!) =#u ! Cm(!); D!u is Holder continuous

with exponent $ for all ": |"| = m$

.

If ! is relatively compact, Cm,$(!) becomes a Banach space in the norm

%u%Cm," =%

|!|"m

%D!u%L! +%

|!|=m

[D!u]($).

The space Cm,$(!) on an open domain ! ' IRn carries a Frechet space topol-ogy, induced by the Cm,$-norms on compact sets exhausting !. Finally, wemay set Cm,0(!) := Cm(!). Observe that for 0 < $ # 1 smooth functions arenot dense in Cm,$(!).

Embedding Theorems

Let (X, % ·% X), (Y, % · %Y ) be Banach spaces. X is (continuously) embeddedinto Y (denoted X %( Y ) if there exists an injective linear map i: X ( Y anda constant C such that

%i(x)%Y # C%x%X , for all x ! X.

In this case we will often simply identify X with the subspace i(X) ' Y .X is compactly embedded into Y if i maps bounded subsets of X into

relatively compact subsets of Y .

For the spaces that we are primarily interested in we have the following results.First, from Holder’s inequality we obtain:

Appendix A 265

A.3 Theorem. For ! ' IRn with Lebesgue measure Ln(!) < $, 1 # p < q #$, we have Lq(!) %( Lp(!). This ceases to be true if Ln(!) = $.

For Holder spaces, by the theorem of Arzela-Ascoli we have the following com-pactness result; see Adams [1; Theorems 1.30, 1.31].

A.4 Theorem. Suppose ! is a relatively compact domain in IRn, and letm ! IN0, 0 # " < $ # 1. Then Cm,$(!) %( Cm,!(!) compactly.

Finally, for Sobolev spaces we have (see Adams [1; Theorem 5.4]):

A.5 Theorem (Sobolev embedding theorem). Let ! ' IRn be a bounded domainwith Lipschitz boundary, k ! IN, 1 # p # $. Then the following holds:(1&) If k p < n, we have W k,p(!) %( Lq(!) for 1 # q # n p

n#k p ; the embeddingis compact, if q < n p

n#k p .(2&) If 0 # m < k " n

p < m + 1, m ! IN0, we have W k,p(!) %( Cm,!(!) for0 # " # k " m " n

p ; the embedding is compact, if " < k " m " np .

Compactness of the embedding W k,p(!) %( Lq(!) for q < n pn#k p is a conse-

quence of the Rellich-Kondrakov theorem; see Adams [1; Theorem 6.2].Theorem A.5 is valid for W k,p

0 (!)-spaces on arbitrary bounded domains !.

Density Theorem

By Theorem A.2, Sobolev functions can be approximated by functions enjoyingany degree of smoothness in the interior of !. Some regularity condition onthe boundary &! is necessary if smoothness up to the boundary is required:

A.6 Theorem. Let ! ' IRn be a bounded domain of class C1, and let k !IN, 1 # p < $. Then C!(!) is dense in W k,p(!).

More generally, it su!ces that ! has the segment property; see for instanceAdams [1; Theorem 3.18].

Trace and Extension Theorems

For a domain ! '' IRn with Ck-boundary &! = ' , k ! IN, 1 < p < $,denote as W k# 1

p ,p(' ) the space of “traces” u|% of functions u ! W k,p(!).If k = 1 we think of W 1# 1

p ,p(' ) as the set of equivalence classes#{u} +

W 1,p0 (!); u ! W 1,p(!)

$, endowed with the trace norm

%u|%%W

1" 1p

,p(% )

= inf#%v%W 1,p(#); u " v ! W 1,p

0 (!)$.

By this definition, W 1# 1p ,p(' ) is a Banach space.

In particular, if k = 1 and p = 2, the trace operator u )( u|"# is a linearisometry of the (closed) orthogonal complement of H1,2

0 (!) in H1,2(!) ontoH

12 ,2(' ). By the open mapping theorem this provides a bounded “extension

operator” H12 ,2(' ) ( H1,2(!). For general p > 1 (and k = 1) we have:

266 Appendix A

A.7 Theorem. For any ! '' IRn with C1-boundary ' , 1 < p < $ thereexists a continuous linear extension operator ext: W 1# 1

p ,p(' ) ( W 1,p(!) suchthat

&ext(u)

'((%= u, for all u ! W 1# 1

p ,p(' ).

See Adams [1; Theorem 7.53 and 7.55].Covering &! = ' by coordinate patches and defining the Lebesgue space

Lp(' ) as before via such charts (see Adams [1; 7.51]), an equivalent norm forW s,p(' ), where 0 < s < 1, is given by

%u%W s,p =)%u%p

Lp +"

%

"

%

|u(x) " u(y)|p

|x " y|n#1+spdx dy

*1/p

;

see Adams [1; Theorem 7.48].From this, the following may be deduced:

A.8 Theorem. Suppose ! '' IRn is a domain with C1-boundary ' , 1 < p <

$. Then W 1,p(' ) %( W 1# 1p ,p(' ) %( Lp(' ) and both embeddings are compact.

In particular, we have

(A.2) H1,2(!) %( L2(&!)

compactly, for any bounded domain of class C1.

Poincare Inequality

For a bounded domain ! of diameter d and u ! H1,20 (!) there holds

(A.3)"

#|u|2dx # d2

"

#|*u|2 dx .

It su!ces to consider! ' [0, d]+IRn#1 = S, u ! C!0 (!) ' C!

0 (S). Then (A.3)follows immediately from Holder’s inequality and the mean value theorem.

More generally, we state:

A.9 Theorem. For any bounded domain ! of class C1 there exists a constantc = c(!) such that for any u ! H1,2(!) we have

"

#|u|2 dx # c

"

#|*u|2 dx + c

"

"#|u|2 do .

Proof. The argument is modeled on Necas [1; p. 18 f.]. Suppose by contradictionthat for a sequence (um) in H1,2(!) there holds

(A.4) %um%2L2(#) , m

+%*um%2

L2(#) + %um%2L2("#)

,.

Appendix A 267

By homogeneity, we may normalize %um%2L2(#) = 1. But then (um) is bounded

in H1,2(!), and we may assume that um ( u weakly. Moreover, by TheoremA.5.(1&) , it follows that um ( u strongly in L2(!) and by (A.2) also um |"# (u|"# in L2(&!).

But (A.4) also implies that *um ( 0 in L2(!), and um |"# ( 0 inL2(&!). Hence, u ! H1,2

0 (!); moreover, *u = 0. By (A.3) therefore, u - 0.But %u%L2(#) = limm'! %um%L2(#) = 1. Contradiction.

In the same spirit the following variant of Poincare’s inequality may be derived.

A.10 Theorem. Let AR = B2R(0) \ BR(0) ' IRn denote the annulus of sizeR in IRn. There exists a constant c = c(n, p) such that for any R > 0, anyu ! H1,p(AR) there holds

"

AR

|u " uR|p dx # c Rp

"

AR

|*u|p dx ,

where uR denotes the mean of u over the annulus AR.

Proof. Scaling with R, we may assume that R = 1, AR = A1 =: A. Moreover,it su!ces to consider u = u1 = 0. If for a sequence (um) in H1,p(A) withum = 0 we have

1 ="

A|um|p dx , m

"

A|*um|p dx ,

by Theorem A.5 we conclude that um ( u - const. = u = 0 in Lp(A).Contradiction.

Appendix B

In this appendix we recall some fundamental estimates for elliptic equations.A basic reference is Gilbarg-Trudinger [1].

On a domain ! ' IRn we consider second-order elliptic di"erential opera-tors of the form

(B.1) Lu = "aij&2

&xj&xju + bi

&

&xiu + cu,

or in divergence form

(B.2) L u = " &

&xi

-aij

&

&xju

.+ c u ,

with bounded coe!cients aij = aji, bi, and c satisfying the ellipticity condition

aij(i(j , )|(|2

with a uniform constant ) > 0, for all ( ! IRn. By convention, repeatedindices are summed from 1 to n. The standard example is the Laplace operatorL = "*. If aij ! C1, then any operator of type (B.2) also falls into category(B.1) with bj = " "

"xiaij.

Schauder Estimates

Let us first consider the (classical) C!-setting; see Gilbarg-Trudinger [1; The-orems 6.2, 6.6].

B.1 Theorem. Let L be an elliptic operator of type (B.1), with coe!cients ofclass C!, and let u ! C2(!). Suppose Lu = f ! C!(!). Then u ! C2,!(!),and for any !( '' ! we have

(B.3) %u%C2,!(##) # C&%u%L!(#) + %f%C!(#)

'.

If in addition ! is of class C2+!, and if u ! Co(!) coincides with a functionuo ! C2+!(!) on &!, then u ! C2,!(!) and

(B.4) %u%C2,!(#) # C&%u%L!(#) + %f%C!(#) + %uo%C2,!(!)

'

with constants C possibly depending on L,!, n,", and – in case of (B.3) – on!(.

Lp-theory

For solutions in Sobolev spaces the Calderon-Zygmund inequality is the coun-terpart of the Schauder estimates for classical solutions; see Gilbarg-Trudinger[1; Theorems 9.11, 9.13].

Appendix B 269

B.2 Theorem. Let L be elliptic of type (B.1) with continuous coe!cients aij.Suppose u ! W 2,p

loc (!) satisfies L u = f in ! with f ! Lp(!), 1 < p < $.Then for any !( '' ! we have

(B.5) %u%W 2,p(##) # C&%u%Lp(#) + %f%Lp(#)

'.

If in addition ! is of Class C1,1, and if there exists a function uo ! W 2,p(!)such that u " uo ! H1,p

o (!), then

(B.6) %u%W 2,p(#) # C&%u%Lp(#) + %f%Lp(#) + %uo%W 2,p(#)

'.

The constants C may depend on L,!, n, p, and – in case of (B.5) – on !(.

Weak Solutions

Let L be elliptic of divergence type (B.2), f ! H#1(!). A function u ! H1,20 (!)

weakly solves the equation L u = f if"

#

-aij

&

&xiu&

&xj#+ c u#

.dx "

"

#f#dx = 0, for all # ! C!

o (!).

The integral

L(u,#) ="

#

-aij

&

&xiu&

&xj#+ c u#

.dx

continuously extends to a symmetric bilinear form L on H1,20 (!), the Dirichlet

form associated with the operator L.

A Regularity Result

As an application we consider the equation

(B.7) "*u = g(·, u) in !,

on a domain ! ' IRn, with a Caratheodory function g:! + IR ( IR; that is,assuming g(x, u) is measurable in x ! ! and continuous in u ! IR. Moreover,we will assume that g satisfies the growth condition

(B.8) |g(x, u)| # C&1 + |u|p

',

where p # n+2n#2 , if n , 3. By (B.8) and Theorem A.5, for any u ! H1,2(!) the

composed function g&·, u(·)

'! H#1(!); see also Theorem C.2. The following

estimate is essentially due to Brezis-Kato [1], based on Moser’s [1] iterationtechnique.

270 Appendix B

B.3 Lemma. Let ! be a domain in IRn and let g : ! + IR ( IR be aCaratheodory function such that for almost every x ! ! there holds

(1&) |g(x, u)| # a(x)&1 + |u|

'

with a function 0 # a ! Ln/2loc (!). Also let u ! H1,2

loc (!) be a weak solutionof equation (B.7). Then u ! Lq

loc(!) for any q < $. If u ! H1,20 (!), and

a ! Ln/2(!), then u ! Lq(!) for any q < $.

Proof. Choose + ! C!0 (!) and for s , 0, L , 0 let # = #s,L =

u min#|u|2s, L2

$+2 ! H1,2

0 (!), with supp # '' !. Testing (B.7) with #,we obtain

"

#|*u|2 min

#|u|2s, L2

$+2 dx +

s

2

"

{x%#; |u(x)|s"L}

((*&|u|2

'((2|u|2s#2+2 dx

# "2"

#*u u min

#|u|2s, L2

$*++ dx

+"

#a&1 + 2|u|2

'min

#|u|2s, L2

$+2 dx

# 12

"

#|*u|2 min

#|u|2s, L2

$+2 dx + c

"

#|u|2 min

#|u|2s, L2

$|*+|2 dx

+ 3"

#a|u|2 min

#|u|2s, L2

$+2 dx +

"

#|a|+2 dx .

Suppose u ! L2s+2(supp(+)). Then for any K , 1 with constants c dependingon the L2s+2-norm of u, restricted to supp(+), there holds

"

#

((*&u min

#|u|s, L

$+'((2 dx # c + c ·

"

#a|u|2 min

#|u|2s, L2

$+2 dx

# c + cK

"

#|u|2 min

#|u|2s, L2

$+2 dx

+ c

"

{x%#; a(x))K}a|u|2 min

#|u|2s, L2

$+2 dx

# c(1 + K) +-

c ·"

{x%#; a(x))K}an/2 dx

.2/n

+-"

#

((u min#|u|s, L

$+(( 2n

n"2 dx

.n"2n

# c(1 + K) + c1,(K) ·"

#

((*&u min

#|u|s, L

$+'((2 dx ,

where

,(K) =-"

{x%#; a(x))K}an/2 dx

.2/n

( 0 (K ( $) .

Appendix B 271

Fix K such that c1,(K) = 12 and observe that for this choice of K (and s as

above) we now may conclude that

"

#

((*&u min

#|u|s, L

$+'((2 dx # c(1 + K)

remains uniformly bounded in L. Hence we may let L ( $ to derive that

|u|s+1+ ! H1,20 (!) %( L2$

(!) .

That is, whenever u ! L2s+2loc (!) we find that u ! L

(2s+2)nn"2

loc (!). Nowiterate, letting s0 = 0, si +1 = (si#1+1) n

n#2 , if i , 1, to obtain the conclusionof the lemma. If u ! H1,2

0 (!), we may let + = 1 to obtain that u ! Lq(!) forall q < $.

To apply Lemma B.3, note that, if u ! H1,2loc (!) weakly solves (B.7) with a

Caratheodory function g with polynomial growth

|g(x, u)| # C&1 + |u|p

',

and if p # n+2n#2 for n > 2, then assumption (1&) of Lemma B.3 is satisfied with

a(x) =|g

&x, u(x)

'|

1 + |u(x)|! Ln/2

loc (!) .

By Lemma B.3, therefore, u ! Lqloc(!), for any q < $. In view of our growth

condition for g this implies that "*u = g(u) ! Lqloc(!) for any q < $. Thus,

by the Calderon-Zygmund inequality, Theorem B.2, u ! W 2,qloc (!), for any

q < $, whence also u ! C1,!loc (!) by the Sobolev embedding theorem, Theorem

A.5, for any " < 1. Moreover, if u ! H1,20 (!), and if &! ! C2, by the same

token it follows that u ! W 2,q & H1,20 (!) %( C1,!(!). Now we may proceed

using Schauder theory. In particular, if g is Holder continuous, then u ! C2(!)and is a non-constant, classical C2-solution of Equation (B.7). Finally, if g and&! are smooth, higher regularity (up to the boundary) can be obtained byiterating the Schauder estimates.

Maximum Principle

A basic tool for proving existence of solutions to elliptic boundary value prob-lems in Holder spaces is the maximum principle.

We state this in a form due to Walter [1; Theorem 2], allowing for moregeneral coe!cients c in the operator L than in classical versions.

272 Appendix B

B.4 Theorem. Suppose L is elliptic of type (B.1) on a domain ! and supposeu ! C2(!) & C1(!) satisfies

L u , 0 in !, and u , 0 on &!.

Moreover, suppose there exists h ! C2(!) & C0(!) such that

Lh , 0 in !, and h > 0 on !.

Then either u > 0 in !, or u = $h for some $ # 0.

In particular, let L be given by (B.2) with coe!cients aij ! C1,!(!), c !C!(!). Then L is self-adjoint and possesses a complete set of eigenfunctions(#j) in H1,2

0 (!) &C2,!(!) with eigenvalues )1 < )2 # )3 # . . .. Moreover, #1

has constant sign, say, #1 > 0 in !. Suppose that the first Dirichlet eigenvalue

)1 = infu$=0

(Lu, u)L2

(u, u)L2, 0.

Then in Theorem B.4 we may choose h = #1, and the theorem implies thatany solution u ! C2(!) & C1(!) of Lu , 0 in !, and such that u , 0 on !either is positive throughout ! or vanishes identically.

The strong maximum principle is based on the Hopf boundary maximumprinciple; see Walter [1; p. 294]:

B.5 Theorem. Let L be elliptic of type (B.1) on the ball B = BR(0) ' IRn,with c , 0. Suppose u ! C2(B) & C0(B) satisfies Lu , 0 in B, u , 0 on&B, and u , - > 0 in B&(0) for some . < R, - > 0. Then there exists/ = /(L, -, ., R) > 0 such that

u(x) , /&R " |x|

'in B .

In particular, if u ! C2(B) & C1(B) and if u(x0) = 0 for some x0 ! &B, thenthe interior normal derivative of u at the point x0 is strictly positive.

Weak Maximum Principle

For weak solutions of elliptic equations we have the following analogue of The-orem B.4.

B.6 Theorem. Suppose L is elliptic of type (B.2) and suppose the Dirichletform of L is positive definite on H1,2

0 (!) in the sense that

L(u, u) > 0 for all u ! H1,20 (!), u .= 0.

Then, if u ! H1,2(!) weakly satisfies Lu , 0 in the sense that

L(u,#) , 0 for all non-negative # ! H1,20 (!),

Appendix B 273

and u , 0 on &!, it follows that u , 0 in !.

Proof. Choose # = u# = max {"u, 0} ! H1,20 (!). Then

0 # L(u, u#) = "L(u#, u#) # 0

with equality if and only if u# - 0; that is u , 0.

Theorem B.6 can be used to strengthen the boundary maximum principle The-orem B.5:

B.7 Theorem. Let L satisfy the hypotheses of Theorem B.6 in ! = BR(0) =B ' IRn with coe!cients aij ! C1. Suppose u ! C2(B) & C1(B) satisfiesLu , 0 in B, u , 0 on &B and u , - > 0 in B&(0) for some . < R, - > 0.Then there exists / = /(L, -, ., R) > 0 such that u(x) , /

&R " |x|

'in B.

Proof. We adapt the proof of Walter [1; p. 294]. For large C > 0 the functionv = exp

&C

&R2 " |x|2

''" 1 satisfies Lv # 0 in B \ B&(0). Moreover, for small

, > 0 the function w = ,v satisfies w # u for |x| # . and |x| = R. Hence,Theorem B.6 – applied to u"w on B \B&(0) – shows that u , w in B \B&(0).

Application

As an application, consider the operator L = "* " /, where / < )1, thefirst Dirichlet eigenvalue of "* on !. Let u ! H1,2

0 (!) or u ! C2(!) &Co(!)weakly satisfy L u # Co in !, u # 0 on &!, and choose v(x) = C

&C" |x"x0|2

'

with x0 ! ! and C su!ciently large to achieve that v > 0 on ! and L v , C0.Then w = v " u satisfies

Lw , 0 in !, w > 0 on &! ,

and hence w is non-negative throughout !. Thus

u # v in ! .

More generally, results like Theorem B.4 or B.5 can be used to obtain L!- oreven Lipschitz a-priori bounds of solution to elliptic boundary value problemsby comparing with suitably constructed “barriers”.

Appendix C

In this appendix we discuss the issue of (partial) di"erentiability of variationalintegrals of the type

(C.1) E(u) ="

#F

&x, u(x),*u(x)

'dx,

where u ! H1.2(!), for simplicity. Di"erentiability properties will cruciallydepend on growth conditions for F .

Frechet-di"erentiability

A functional E on a Banach space X is Frechet-di"erentiable at a point u ! Xif there exists a bounded linear map DE(u) ! X*, called the di"erential of Eat u, such that

|E(u + v) " E(u) " DE(u)v|%v%X

( 0

as %v%X ( 0. E is of class C1, if the map u )( DE(u) is continuous.

C.1 Theorem. Let ! '' IRn. Suppose F :! + IR+ IRn ( IR is measurable inx ! !, continuously di"erentiable in u ! IR and p ! IRn, with Fu = "

"uF, Fp =""pF , and the following growth conditions are satisfied:(1&) |F (x, u, p)| # C(1 + |u|s1 + |p|2), where s1 # 2n

n#2 , if n , 3,(2&) |Fu(x, u, p)| # C(1 + |u|s2 + |p|t2), where t2 < 2, if n # 2, respectively,where s2 # n+2

n#2 , t2 # n+2n , if n , 3,

(3&) |Fp(x, u, p)| # C(1 + |u|s3 + |p|), where s3 # nn#2 , if n , 3.

Then (C.1) defines a C1-functional E on H1,2(!). Moreover, DE(u) is givenby

< v, DE(u) >="

#

&Fu(x, u,*u)v + Fp(x, u,*u) ·*v

'dx .

Theorem C.1 applies for example to the functional

G(u) ="

#|u|p dx

with p # 2nn#2 , if n , 3, or to Dirichlet’s integral

E(u) =12

"

#|*u|2 dx.

Appendix C 275

Theorem C.1 rests on a result by Krasnoselskii [1; Theorem I.2.1]. For simplic-ity, we state this result for functions g:!+ IRm ( IR. To ensure measurabilityof composed functions g

&x, u(x)

', with u ! Lp, we assume g:! + IRm ( IR is

a Caratheodory function; that is, g is measurable in x ! ! and continuous inu ! IRm.

C.2 Theorem. Suppose g:!+ IRm ( IR is a Caratheodory function satisfyingthe growth condition(1&) |g(x, u)| # C(1 + |u|s) for some s , 1.Then the operator

u )( (g(·, u&·)

'

is continuous from Lsp(!) into Lp(!) for any p, 1 # p < $.

Theorem C.2 asserts that Nemitskii operators – that is, evaluation operatorslike (C.1) – are continuous if they are bounded. For nonlinear operators this isquite remarkable.

Using this result, Theorem C.1 follows quite naturally from the Sobolevembedding theorem, Theorem A.5. To get a flavor of the proof, we establishcontinuity of the derivative of a functional E as in Theorem C.1. For u0, u !H1,2

0 (!), we estimate

%DE(u) " DE(u0)% = supv%H

1,20

&v&H

1,20

'1

| < v, DE(u)" DE(u0) > |

# supv

"

#|Fu(x, u,*u)" Fu(x, u0,*u0)| |v| dx

+ supv

"

#|Fp(x, u,*u) " Fp(x, u0,*u0)| |*v| dx

# supv

-"

#|Fu(x, u,*u)" Fu(x, u0,*u0)|

2nn+2 dx

.n+22n

·

/

0"

#

|v|2n

n"2 dx

1

2

n"22n

+ supv

-"

#|Fp(x, u,*u)" Fp(x, u0,*u0)|2 dx

. 12

·

/

0"

#

|*v|2 dx

1

2

12

if n , 3 – which we will assume from now on for simplicity. Now, by Theo-rem A.5 the integrals involving v are uniformly bounded for v ! H1,2

0 (!) with%v%H1,2

0# 1. By our growth conditions (2&) and (3&), moreover, Fu (respec-

tively Fp) can be estimated like

|Fu(x, u,*u)|2n

n+2 # C+1 + |u|

2nn"2 + |*u|2

,,

and by Theorem C.2 it follows that DE(u) ( DE(u0), if u ( u0, as desired.

276 Appendix C

Natural Growth Conditions

Conditions (1&)–(3&) of Theorem C.1 require a special structure of the functionF ; for instance, terms involving |*u|2 cannot involve coe!cients depending onu. Consider for example the functional in Section I.1.5, given by

F (x, u, p) = gij(u)pipj .

Note that Fu has the same growth as F with respect to p. More generally, foranalytic functions F such that

(C.2) |p|2 # F (x, u, p) # C&|u|

'&1 + |p|2

'

one would expect the following growth conditions:

Fu(x, u, p) # C&|u|

'&1 + |p|2

'(C.3)

Fp(x, u, p) # C&|u|

'&1 + |p|

'(C.4)

for x ! !, u ! IR, and p ! IRn.Under these growth assumptions, in general a functional E given by (C.1)

cannot be Frechet-di"erentiable in H1,2(!) any more. However, minimizers(in H1,2

0 (!), say) still may exist, compare Theorem I.1.5. Is it still possible toderive necessary conditions in the form of Euler-Lagrange equations? – Theanswer to this question may be positive, if we consider only a restricted set ofminimizers and a narrower class of “testing functions”, that is of admissiblevariations:

C.3 Theorem. Suppose E is given by (C.1) with a Caratheodory function F ,of class C1 in u and p, satisfying the natural growth conditions (C.2)–(C.4).Then, if u,# ! H1,2 & L!(!), the directional derivative of E at u in direction# exists and is given by:

d

d,E(u + ,#)

(('=0

="

#

&Fu(x, u,*u)#+ Fp(x, u,*u) ·*#

'dx .

In particular, at a minimizer u ! H1,2 & L! of E, with F satisfying (C.2)–(C.4), the Euler-Lagrange equations are weakly satisfied in the sense that

"

#

&Fu(x, u,*u) · #+ Fp(x, u,*u)*#

'dx = 0

holds for all # ! H1,20 & L!(!).

Note that the assumption u ! L! often arises naturally, as in Theorem I.1.5.Sometimes, boundedness of minimizers may also be derived a posteriori. Forfurther details, we refer to Giaquinta [1] or Morrey [4]. The question of dif-ferentiability of functionals in general is quite subtle, as is illustrated by anexample of Ball and Mizel [1].

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Index 301

Index

Approximate solutions 33, 69Area 6, 19, 220

Co-area formula 43Barrier 273

(see also: Sub- and super-solution)Bifurcation 180 !.Calderon-Zygmund inequality 268 f.Caratheodory function 275Category 100

(see also: Index theory)Change in topology 169Characteristic function 6Coercive

functional 4operator 60

Compactnessbounded compactness 2concentration-compactness 37 f., 39 f.,

44 f.compensated compactness 25 !.local compactness 41, 175, 226global compactness in the critical case

184, 231(see also: Palais-Smale condition)

of a sequence of measures 39Concentration-function (of measure) 40Conformal

conformal group of the disk 20 f.conformal invariance 20, 194, 232conformal Laplace operator 197conformality relation 20#-conformality theorem of Morrey 20

Convexity 9, 12 f., 21, 25, 54, 57, 61 f.,103, 105, 162 !.

polyconvex 25quasiconvex 54

Critical point (value) 1at infinity 169in convex sets 164of mountain pass type 143of non-di!erentiable functional 152saddle point 1, 50, 54, 73, 74, 76, 77,

87, 108, 151Deformation lemma 81 !.

for C1-functionals on Banach spaces83

for C1-functionals on Finsler manifolds87

for non-di!erentiable functionals 153 f.,158

on convex sets 165Developing map 197 f.

Dichotomy (of a sequence of measures)40

Dirichlet integral 20Dual variational problem 62, 67Eigenvalue for Dirichlet problem

Courant-Fischer characterization 97Rayleigh-Ritz characterization 14f.Weyl asymptotic formula 118

Elliptic equations 14 !., 16 !., 32 !.,98 !., 110 !., 116 !., 120 !., 128 f.,147 !., 150 !., 166 !., 170 !.

degenerate elliptic equations 4 !., 183on unbounded domains 36 !., 150 !.with critical growth 170 !.

Energyenergy functional 25, 71 f., 150, 231energy inequality 199, 204, 238, 243stored energy 25energy surface 61

Epigraph 57 f.Equivariant 82, 84, 86, 94Euler-Lagrange equations 1Finsler manifolds 85 f.Frechet-di!erential 274Functional at infinity 36Genus 94 f.Geodesics 61, 89

closed geodesics on spheres 89 !.Gradient 84

gradient-flow 84, 135(see also: Pseudo-gradient)

Group action 82, 84, 86, 94Hamiltonian systems 60 !., 103 !., 124,

130 !., 137 !., 150Hardy space 35, 224Harmonic map 8, 71 !., 169, 231 !.

evolution problem 238 !.Harmonic sphere 240Index theory 94 !., 99

Benci-index 101 !.Krasnoselskii genus 94 !.Ljusternik-Schnirelman category

100 f.pseudo-index 101

Intersection lemma 113Invariant

under flow 87under group action (see: Equivariant)

Isoperimetric inequality 43, 224Legendre condition 13Legendre-Fenchel transform 58 f., 63 f.Limiting problem 170, 184

302 Index

Linking 125 !.examples of linking sets 125 !., 134

Lower semi-continuity 2 !., 8 !., 25, 51,58, 70

Maximum principle 271 !.Mean curvature equation 169, 220 !.Measure

compactness of a sequence of measures39

concentration function of measure 40dichotomy of a sequence of measures

40vanishing of a sequence of measures 39

Minimal surface 6 f., 19 !., 169parametric minimal surface 19 f.minimal cones 7minimal partitioning surfaces 6 f.

Minimax principle 74, 87 !., 96 f.Courant-Fischer 97Palais 87

Minimizer 1, 51, 70, 144, 166Minimizing sequence 3, 53, 55 !., 70Monotone operator 60Monotonicity (of index) 99Mountain pass lemma 74, 76, 108 !., 112Palais-Smale condition 77 !.

Condition (C) 78Condition (P.-S.) 78Cerami’s variant 80for non-di!erentiable functionals 152local 177on convex sets 164

Palais-Smale sequence 54, 78Penalty method 69 f.Periodic solutions

of Hamiltonian systems 61 !., 103 !.,130 !., 137 !., 150

of semilinear wave equation 65 !., 124,150

with prescribed minimal period 104 f.Perimeter (of a set) 6Perron’s method 16 !.Plateau problem 19 !., 214

boundary condition 19

Pohozaev identity 155 f., 171Poincare inequality 266 f.Pseudo-gradient flow 84

(see also: Deformation lemma)Pseudo-gradient vector field 81, 86

for non-di!erentiable functionals 153on convex sets 164

Pseudo-Laplace operator (p-Laplacian) 5,183

Regular point (value) 1, 164Regularity theory 16, 31, 57, 268 !.

for minimal surfaces 24for the constant mean curvature equation

223in elasticity 31partial regularity for evolution of har-

monic maps 245 !., 248 f., 253 !.Rellich-Kondrakov theorem 265Schauder estimates 268Schwarz-symmetrization 42Separation of spheres 169, 239 !.Sobolev embedding (inequality) 42 !.,

170 !., 242, 265 !.density of smooth maps in Sobolev spaces

235 f.Sub-additivity (of index) 99Sub-di!erential 58Sub-solution 16 f.Super-solution 17Supervariance (of index) 99Support

support hyperplane 58support function 58

Symmetry 16, 36, 169Symmetry group (see: Group action)Symplectic structure 60Technique

Fatou-lemma technique 34hole-filling technique 56

Vanishing (of a sequence of measures) 39Variational inequality 13, 166Volume 220, 223 !.Wave equation 65 !., 124, 150Yamabe problem 18, 194 !.

Ergebnisse der Mathematik Volume 34 · Applications to Nonlinear Partial Differential Equations and...

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