S k - VALUED MAPS WITH SINGULARITIES

Hgim Brezis D~pa rtemen t de Math~matiques, Universit~ Paris 6

4, pl. Jussieu, 75252 Paris Cedex 05 and

Rutgers University, New Brunswick, NJ 08903

The purpose of these notes is to present a survey of some recent results and open

problems dealing with the "energy" of S k - valued maps. The original motivation comes

from the theory of liquid crystals; such materials are composed of rod-like molecules with a

well defined orientation, except at isolated points (the "defects") which are observed by the

physicists. The optic axis ~ is a vector of unit length (in tR 3) defined in the domain

t2 C IR 3 (the container of the liquid crystal); so that ~ is a map from ~ into S 2.

Associated with a configuration

be

~o is a deformation energy which we shall usually take to

= / IV I e ax. (0.1)

Physicists consider more general energies, such as,

t~(W ) = S12kl(div~v)2 + k2(~" curl~v)2 + k31 ~o/,,curl~, 12 + a[tr(V~)2 (div~)2]dx

(0.2)

where kl, k2, k 3 and a are positive constants. In the special case where

k 1 = k 2 = k 3 = a = 1, then it is easy to see that t~ = E. While much progress has been

achieved for the energy E, little is known so far for t~. Stable equilibrium configurations

correspond to minima of E (or I~) and therefore it is essential to study the properties of

minimizers. For a detailed discussion of the physical background we refer e.g. to [9], [10],

[13], [16], [17], [18] and [33]. However we feel that the mathematical questions involved in

this field are of great interest for their own sake, an interest which goes much beyond the

original motivation. In fact, it is remarkable that progress has been achieved through the

joint efforts of experts in Nonlinear Partial Differential Equations, Functional Analysis,

Differential Geometry, Geometric Measure Theory, Topology, Numerical Analysis, Graph

Theory, etc.

The plan is the following:

I. The problem of prescribed singularities.

1.1. Point singularities in R 3.

L2. Various generalizations:

1) A domain ~ ¢ [R 3 with constant boundary condition.

2) Holes in ~3.

3) An example related to minimal surfaces.

1.3. Some open problems.

II. The problem of free singularities.

II.1. x / Ix I is aminimizer,

II.2. The analysis of point singularities.

II.3. Energy estimates for maps which are odd on the boundary.

II.4. The gap phenomenon. Density and nondensity of smooth maps between

manifolds. Traces.

IL5. Some open problems.

I. The problem of prescribed singularities.

1.1

be N

admissible maps $ to be

1 N $ = {~eC (~3\i~l{ai}; $2); fN3iV~] 2 < oo and deg(~,ai) = d i

Point singularities in ~3.

We start with a simple question (originally raised by J. Ericksen). Let al, a2,...a N

points given in IR 3 (the desired location of the singularities). Define the class of

Vi}

(1.1)

where the di's are given integers, dieZ with d i ¢ 0. Here deg(~,ai) denotes the Brouwer

degree of ~ restricted to any small sphere around a i. [Stable singularities observed by

the physicists have always degree +1 and the reason why this is so will be given in

Section II.2. However it makes sense to formulate the mathematical question with general

degrees].

The problem is to study the quantity

= In f I 31V~l 2dx (1.2) E

i.e. the least deformation energy needed to produce singularities at a given location with a

given degree. Such a question may seem unrealistic because the container ~ is all of IR 3

and also because one cannot prescribe physically the location of the singularities.

Nevertheless this model problem has interesting features; it has led to the development of

new tools which are useful in more realistic questions.

Surprisingly, there is a simple formula for E:

Theorem 1 ([8]). We have

E = 8~rL (1.3)

where L is the length of a minimal connection (in a sense to be defined below).

So far, we have made no restriction about the di's. However, we must assume that

N E d i = 0 (1.4)

i=l because of the following:

Lemma 1. $ is nonempty if and only if (1.4) holds.

Sketch of the proof. Suppose first that $ is nonempty and let ~c$. We claim that

restricted to a large sphere S R of radius R has degree zero. Intuitively, this is clear

because ~[R3]V<pl 2 < oo implies that, roughly speaking, ~ goes to a constant at infinity.

More precisely, we recall (see e.g. [36]) that if S is a closed two dimensional surface in [R 3

and ¢ is a C 1 map from S into S 2 then

deg ¢ = ~ ISJ ¢ d~ (1.5)

where J ¢ denotes the Jacobian determinant of ¢. A useful way to write J¢ is

J¢ = ¢. CxACy (1.6)

where (x,y) are normal coordinates on S. This follows from the fact that

¢ .¢x = ¢ .¢y = 0, and thus CxACy = (J¢) ¢. We deduce from (1.5) and (1.6)that

Ideg¢l < ~ J'sIVT¢I2d~

where IVT¢[2= I¢xl2÷ ICyl 2 We now return to ~ and choose R 1 large enough so that

singularities a i. By continuity, deg(~lSr) is constant for

R 2 > R 1,

BR1 contains all the

r > R 1. We have for any

]V~12= fR2dr f IV~t2df_> 8zrldeg~OtSrl(R2-R1). f RI< Ixl <R 2 R 1 S r

Letting R2~ oo we see that deg ~]Sr 0 for r > R 1.

From the additivity of the degree we conclude that (1.4) holds. The converse is more

delicate and follows from an explicit construction sketched in the proof of Theorem 1.

Definition of L~ the length of a minimal connection.

It is convenient to start with simple cases:

Case 1: There are only two singularities, a 1 with degree +1 and a 2 with degree -1.

We shall call this a dipole. Here

L = l a l - a 2 ]

is the (Euclidean) distance between the two points. Note that it was easy to guess, from

dimensional analysis, that E is proportional to a length.

Case 2: All the degrees d i are equal +1. Because of (1.5) there are as many + signs as

signs. We rename the points (ai) as positive points Pl ' P2"'"Pk and negative points

n 1, n2,...,n k. Then k

L = M i n E [P i -na( i ) ] (1.7) a i=l

where the minimum is taken over all permutations a of the integers 1 to k.

Case 3: In the general case, proceed as above except that in the list (Pi' ni) points are

repeated according to their multiplicity t dil-

Sketch of the proof of Theorem 1. The proof consists of two independent steps:

A) E _< 8zrL,

B) E > 8rL.

Step A. The main ingredient is the following basic dipole construction:

Lemma 2. Let (al, a2) be a dipole. Given any e > 0 there is a map

on R3, except at (al, a2), such that

deg(~e, al) = +1, deg(~E, a2) = -1,

SlV~eI 2 < 8r la l -a2 l + e,

and moreover

!p E is constant outside an e-neighborhood of the line segment [al, a2].

In fact, given any positive integer d

except at

~Ewhich is smooth

(I.8)

(1.9)

(i.I0)

there is a map ~e which is smooth on ~3,

(al, a2) , such that

deg(~ E, al) = d,

J'iV~pEI 2 _< 8~r]al-a21 d + E,

deg(~E, a2) = - i , (1.8")

(1.9,)

and (1.10) holds. Such a map is constructed explicitly in [8] (see also [7]). Putting

together these basic dipoles over a minimal connection it is easy to prove that E < 8~rL.

Clearly, this construction also shows that $ is nonempty when (1.4) holds.

Step B. There are two different methods for proving the lower bound E > 8~rL. Each one

has its own flavor and I will describe both of them.

Proof of (BI Yi~ ~he D-field approach. This is the original method introduced in [8].

To every map ~ we associate the vector field D defined as follows

D = (~o" ~yA~z, ~. ~zA~x, ~. ~xA~Oy) (1.11)

where ~Ox, ~y, ~z denote partial derivatives of ~ with respect to x, y, z. A more

intrinsic way to define D is to say that D is the pull-back under ~ of the canonical

2-form on S 2. The main properties of D are the following:

IDI _<½ Iv l 2 on N U {ai} (1.12) i=1

and

Proof of (1.12).

Since I ~ ] 2 = 1 we have ~.~x

N div D = 4~r i=lE d i 6ai in .~ '(IR3). (1.13)

Changing coordinates at a given point we may always assume that

~ = (0, 0, 1).

= ~" ~y = ~" ~z = 0 and thus we may write

We see that

with a = ( a 1,a 2,a3)

It follows that

W x = ( a 1,b 1 , 0 ) , ~ y = ( a 2 , b 2 , 0 ) and ~ 3 = ( a 3 ' b 3 ' 0 ) '

D = nab

and b = ( b 1,b 2,b3).

IDI _< lallbl <½(lal 2+ Ibl 2) =½1V~I 2. N

Proof of (1.13). On R3\ U _ _{a i} we have i=l

div D = 3 ~x. ~y/~Wz = 0

since ~x' ~y and Wz are in the same plane (perpendicular to ~). In view of a celebrated

Theorem of L. Schwartz we find

div D = N.cc0~Sa. in .~'(N3). ~ ' , l 1

On the other hand, since DeLI(~3), we must have

N " 3 d ivD=i=~ ~ai 1el in .~ (IR). (1.14)

Integrating (1.14) over a small bM1 B around a i we see that

.[ D . n = c i S

where S = OB and n is the outward normal to S. On the other hand, it follows from the

definition of D that D.n = J where ~ is considered as amap restricted to S and J

denotes its 2×2 Jacobian determinant. Applying (1.5) we find that c i = 4~r deg(9,ai).

The proof of (B) then proceeds as follows. Let ~: ~3_~ ~ be any function such that

tt~'[ILip= Sup I~(x)-((y) l < I, x#y I x - y l -

so that IIV~HL~ < 1. We have

N ~lV~l 2 > 2 ~IDI > - 2 ~D.V¢ = 2 ~ 47rdi¢(ai). (1.15)

i=1

Relabelling the points (ai) as positive and negative points (Pi' ni) and taking into

account their multiplicity we have

N k k

i=~I di~(ai) = i~1 ;(pi) - i=l ~ ¢(ni)-

The conclusion of Step B is a direct consequence of the following:

Lemma3. Let M be ametricspace and let Pl' P 2 " " P k and nl, n2 , . . .n k be 2k

points in M. Then

k k M a x { Y~ ~(pi ) - ~ ~(ni) } = L

i=l i= l : M-~ II~llL i p-< 1

k where II~llLip= Sup I~(x)-(($ )l and L = Min i Eld(Pi,na(i)). xCy d(x,y) a -

A quick proof of Lemma 3 relies on the Kantorovich min-max principle (see [32] or

[37]) and the Birkhoff Theorem which asserts that the extreme points of the doubly

stochastic matrices are the permutation matrices (see [8] for details). Another self

contained proof of Lemma 3 is given in [7].

Proof of (B) via the coarea formula. This new proof discovered by F. Almgren -

W. Browder - E. Lieb (see [2]) relies heavily on Federer's coarea formula (see [20], [24] or

[41]), which we recall for the convenience of the reader. Suppose ~ is a C 1 map from a

domain g/c IR n into a manifold N of dimension p g n. (Think, for example, of N as

being a sphere). The differential of ~, D~, is a (p×n) matrix. Set

Jp~ = / det(D~. (D~) t)

where det denotes the determinant of the p×p matrix D~.(D~)t; Jp~ is called the

p--Jacobian of ~. We have

I12Jp~ dx = INc,~n-P(~-I(~)) d~ (1.16)

where ¢~n-p is the (n-p)---dimensional Hausdorff measure in ]~n. In the special case

where N = IR n, then Jn~ is the usual Jacobian determinant of ~ and (1.16) becomes

I12J~ dx = 1~gt)card(~"-l(~)) d~. (1.17)

In the case of interest to us we take ~e$, 12 = IR3\ i~l{ai} U and N = S 2. Therefore we

find

IIR3J2~ dx = / S 2 J g l ( ~ - l ( ~ ) ) d~. (1.18)

First, we claim that

J2 ~ -< ½t V~I 2. (1.19)

With the same notations as in the proof of (1.12) we have

J2~o = ~/ l a l21b l2 - ( a .b )2= laAb I = IDI _< ½17~ol 2.

Next, we claim that, for a.e. ~eS 2,

~ v l ( ~ I ( ~ ) ) > L. (1.20)

This will complete the proof of (B) via the coarea formula.

Proof of (1.20). By Sard's Theorem we know that a.e. ~eS 2 is a regular value of ~o.

When ~ is regular value, the Implicit Function Theorm implies that ~--1(~) is a

collection of curves which either connect the points (ai), or go to infinity, or are closed

loop~. Here ~dl(~--l(~)) is the total length of these curves. In view of (1.18), (1.19) and

since J" [Y~I 2 < co, the total length is finite and hence there is no curve going to infinity.

Furthermore, we shall disregard the closed loops (since they only increase the total length).

We are left with a finite collection of curves connecting the points (ai). Since

deg(~,ai) = die 0, at least one curve emanates from each ai, but there could be more

than one. The simplest situation is the case where each positive point Pi is connected by

one of the curves to a negative point na(i) , for some permutation a. Then, clearly

¢~,1(~---1(~)) > L. Unfortunately, the general situation could be more complicated. For

example, a bad configuration would be if we have 4 points Pl ' P2' nl ' n2 and ~--1(~)

consists only of two curves: one joining Pl to P2 and the other n 1 to n 2. We could

not conclude, because I pl-P21 + I n l -n 21 might be smaller than L! We shall see that

such a configuration is excluded. For this purpose it is convenient to introduce an arrow

(i.e. an orientation) on each curve C.

Let x be any point on C and let (el, e2, e3) be a direct basis with e 1 tangent

to C at x. Consider to restricted to the plane (e2, e3) and its (2×2) Jacobian

determinant J~(x). Note that J~(x)~ 0 since ~ is a regular value. If J ~ ( x ) > 0 the

orientation of C is given by el, and if Jto(x) < 0 take the orientation opposite to e 1.

With this convention, and using the properties of the degree, one can see that at

every point ai, one has the basic relation:

di=deg(~o, ai)=(#outgoing arrows)-(#incoming arrows). (1.21)

For example, an admissible configuration is given by the following figure

+1 ~ +1

<- __~ -1 -1.

Finally, we claim that any collection of curves running between the points (ai) and

satisfying (1.21) contains a connection connecting each Pi to na(i) for some permutation

a. Indeed, start with Pl and take a walk in the direction of the arrow as far as possible.

The last point which has been reached is a negative point (because of (1.21)); call it ha(l).

Erase this path and note that (1.21) persists. So, we may start again from P2' etc.

1.2 Various generaliza,~ions.

1) A domain f~ c I1t 3 with constant boundary condition.

Let ft c N3 be a smooth bounded domain and let a 1. a2,. • .a N be N points given

in ft. Define the class $1 to be

1 N I V~,l 2< ®, deg(~o, ai)= di and ~o is constant on 0 f~}. $1 = { ~ C (~\iUl{ai}; s2);Ia Again, we assume that E di= 0, which is consistent with the condition that ~o is constant

on 0ft. Set

El= Inf / IV l 2 ~c$1

Theorem 2. ([8]). We have

E 1 = 8rL 1

where L 1 is the length of a minimal connection computed using the geodesic distance

between points in fk

Once more, the proof consists of two steps:

A) E 1 < 8~rL1,

B) E 1 ~ 8~rL 1.

10

For the proof of (A) the main ingredient is the following extension of Lemma 2:

Lemma 2". Let C be a simple curve in IR 3 with end points (al, a2). Given any

there is a map ~o E which is smooth on ~3 except at (al,a2), such that

deg(~oE,al) = +1, deg(~oe,a2) = - 1 ,

and

S]v~oel 2 < 8~r(length C) + E

~o E is constant outside an E-neighborhood of C.

e>O

Proof of (B) via the D-f ie ld approach. Consider once more D defined by (1.11). We

N have, since D-n = 0 in 0 g~, ~fl]V~o]2>_ 2~sqD.V~ = 8a'iEldi{(ai)= for any function

¢: ~ -~ ~ such that IlY~llLoo(~)< 1,

denotes the geodesic distance in ft.

with the geodesic distance d~.

Proof of (B) via the coarea formula.

connecting the various points (ai).

= Sup ] . ((x)-((y)] < 1 and dfl(x,y) i.e. II~'ltLi p x,y¢~/ d~(x,y) -

x ty

We may then apply Lemma 3 in M = F/ equipped

Once more ~--1(~) consists of curves in Y~

These curves do not go to O £/ since ~o is constant on

0 fl and we may always assume that ~ is different from that constant. We may then

proceed as above.

2) Holes in {R 3 with prescribed degree.

Assume H1, H2 , . . .H N are disjoint compact sets in {R 3. For simplicity, assume each H i

is the closure of a smooth domain. Set

N Ft = IR3\( U Hi).

i=l Consider

$2 = {~oecl(~;S2); S iV~o 12< ~ and deg(~o, Hi) = d i Vi}.

Here deg(~o, Hi) denotes the degree of ~o restricted to 0H i. The integers dieZ are given

(possibly zero) with the restriction that ~ i = 0, consistent with )~£/tV~ot 2< on. Set

11

E2= Inf ~ 17~12 ~e$ 2 gl

Theorem 3 ([S]). We have

E2= 8~L 2

where L 2 is the length of a minimal connection connecting the holes and computed using

the rule explained below.

Definition of L2._ Given a, b ~ 3 let

and

~(a,b) = I 0 if a and b belong to the same hole H i

L ]a-b I otherwise,

n

D(a,b) = Inf E ~(Xi_l,Xi) i= l

the Infimum being taken over the set of all finite chains x0, X l . . . x n joining a to b (i.e.

x 0 = a and x n = b). (Note that D is a semi-metric; D(a,b) = 0 iff a,b belong to the

same hole.) Given two sets A, B, let

D(A,B) = In f D(x,y). x~A yeB

Once this reduced distance is defined, throw away the holes of degree zero and relabel the

remaining holes as positive holes PI ' P 2 " ' 'Pk and negative holes N1, N2, . . .N k

including multiplicity. Set

k L2= M i n E D(Pi,Na(i) ).

a i = l

Again, the proof consists of two steps:

A) E 1 ~ 8~rL 1

B) E 2 >_. 8~L 2.

Proof of (A). A minimal connection connecting the holes can be thought of as a finite

collection of directed line segments running between pairs of holes and which carry some

multiplicity d. On each of these segments one can apply Lemma 2, thereby constructing a

map WEc$ 2 such that SI7~oEI 2 < 8rL 2 + E.

]2

Proof of (B) via the D-field approach.

falv~[2

Let D be defined by (1.11). We have

> 219t[D [ > 2 Jf D.V( (1.22)

for any function (: [R 3 -* [R such that 117(llLOO([R3)_< 1. Since div D = 0 in f~ we have

Jf~D'V( = I0 r i D . u ) ( = Ei JOHiJ~( (1.23)

where J~ denotes the Jacobian determinant of ~ restricted to H i. Assume, in addition,

that ( = ((Hi) is constant on each Hi, then we have, by (1.5), (1.22) and (1.23)

N k I IV~l 2 - > 8 ~ i d i ( ( H i) = 87ri=tE ( ( P i ) - ((Ni)

for all functions (: tR 3 -* iR such that IIV(IILOO(tR3 ) < 1 which are constant on each hole H i.

This class of functions ( consists precisely of functions such that I ((x)-((Y)l -< D(x,y)

Vx,ye~ 3. Therefore we may apply Lemma 3 in M = ~3 equipped with the (semi) metric

D and deduce that

J I v~ol 2 _> 8~rL 2.

Proof of (B) via the coarea formula,. As above, ~--1({) consists of curves in f~

connecting the various holes (Hi). Each curve carries an orientation. For every hole H i

one has the basic relation

d i = deg(GHi) = (# outgoing arrows) - (# incoming arrows).

Therefore ~--1(~) contains aconnection connecting each Pi to Nail) for some

permutation ¢.

3) An example related to minimal surfaces.

Suppose we change the energy and use instead

E (~) = f 31V~ldx

then such an expression has the dimension of an a r e a . We are led, by analogy with the

previous problem, to pose the following question. Let F be an oriented smooth Jordan

curve in ~3 and let $ '-- {~cC1(~3\F;81); 117~1 < ~ and deg(~,P) = 1} where

deg(~,F) = 1 means that ~ restricted to any small circle which links P once has

]3

degree 1. Assume $ is nonempty and set

E ' = I n f l 3]V~ldx.

The following result, which was conjectured in [8] (and established there for planar curves)

has been proved in [2].

Th~0rem 4. We have

E =2~rA

where A is the area of an area-minimizing surface spanned by F. t

Sketch of the proof. The inequality E _< 2~rA is shown by constructing explicitly a map

~e$ "concentrated" in a neighborhood of a surface spanned by F with nearly

minimizing area (by analogy with Lemma 2). The reverse inequality E _ 2~rA follows

from the c o , c a formula which takes the form

I 31V~Idx = ItR 3 J1 ~Ix = IS1 ¢'~'2(~"-1(~))d~ •

For a.e. ~eS 1, ~---1(~) is a surface spanning F and therefore Jb2(~--l(~)) > A.

1.3 $om¢ open problems.

Problem 1. Consider the general energy functional

]~(~) = SlR3kl(div ~o)2 + k2(~o.curl ~)2 + k31 ~Acur 1 ~I 2dx.

Is there a simple formula (analogous to (1.3)) for Inf I~(~)? The answer is not known

even in the dipole case.

Problem 2. Let ~ c IR 3 be a smooth bounded domain. Let g: o~ -~ S 2 be a smooth map

of degree zero. Let al, a2 , . . . a N be N points given in f'/ and let

$ {~ec l (~ \ N {ai};S2); -- U S t l t ~ ~ V ~ p , 2 < o o , deg~ ,a i~=di Vi and ~ = g on o~a} i=1

where the di's are given integers such that ~d i = 0. Set

E = Inf llV~J 2

14

and

N = =4~rlEld i and D . n = J g on Of/}. F 2Inf{J ' f / [D[ ;d ivD '= 5ai.

Is it true (or when is it true) that E = F? We a~ways have E >_. F; equality is known

when g is a constant (see the proof of Theorem 2 via the D-field).

Problem 3. We return to the Example related to minimal surfaces discussed above. The

analogue of the vector field D for this problem is the vector field H defined as follows:

H = (~A~x, ~ , ~ y , ~ / -~z ) The main properties of H are

IHt = tV~l (this is the counterpart of (1.12))

and

(1.24)

curl H = 27rDF(this is the counterpart of (1.13)) (1.25)

where D F is a divergence-free measure, supported by P, defined by

1 < Dp,~ > = f0~(X(t)) .X'(t)dt V~C0(a3;gl3 )

and X(t) is any parametrization of F. In view of the results above it is natural to ask the

question whether

Inf J~ IV~l=27rInf{l [Hl ;cur lH 27rA 7 ~e$" DI3 ~3 = Dr} = "

(The answer is positive when r is a planar curve see [8].)

data.

II. The problem of free singularities.

Let f / c IN 3 be a smooth bounded domain and let g: 012 ~ S 2 be a given boundary

Consider now the problem of minimizing the energy on the class of maps

g = {~oeHl(f/;S2); ~ = g on O f~}.

This class allows, of course, point singularities. Clearly,

E = Inf flv~I 2 (2.1)

is achieved and moreover every minimizer satisfies the equation of harmonic maps

- A ~ , = ~]v~] 2 on a (2.2)

15

which is a system (and not a scalar equation). It is well-known through the work of

DeGiorgi [14], Giusti-Miranda [22], Morrey [35], Almgren [1] and others (see a detailed

exposition in the book of Giaquinta [21]) that weak solutions of elliptic systems need not be

smooth. In our case a result of Schoen-Uhlenbeck [39][40] asserts that every minimizer ~v

for (2.1) is smooth except at a finite number of points. In contrast with Section I, the

number and location of the singularities is not prescribed. If deg(g,O f~) ¢ 0 there is a

topological obstruction to regularity since g cannot be extended smoothly inside f~; every

map in the class $ must have at least one singularity. If deg(g,0 F/) = 0, there is no

topological obstruction to regularity since g can be extended smoothly inside fL

Nevertheless, we shall see (in Section II.4) that singularities sometimes appear in order to

lower the energy of the system.

II.1 x/ Ix I is aminimizer.

We start with a simple question. Let f~ = B 3 be the unit ball in R3 and let

g(x) = x be the identity map on O fk The main result is the following:

Theorem 5 ([8]). The map ~(x) = x / Ix I is a minimizer for (2.1). In fact it is the unique

minimizer.

I will present two proofs. The first is the original proof of [8] and is based on the D-field

approach. The second proof is a direct and very elegant argument due to F. H. Lin [34]. A

third method, using the coarea formula, is described in a more general setting in Section

I1.3.

Proof of Theorem 5 via the

and therefore

D-field approaeK We have

I V(T~F) 12 2

1 Ja lv( l -~) 12 dx = 8~ f0

Therefore it suffices to prove that

dr = 8~. (2.3)

]6

IV~,l 2 ~2

In fact, we may assume that

have only to prove (2.4) for minimizers, we may invoke the regularity result of

Schoen-Uhlenbeck). Let D be definedby (1.11). We have

~12 [V~o[2dx > 2J12 [D[dx > 2SflD.V~ dx

-~ ~ such that llV¢[I < L L ®-

for any function {:

We write

But

(the points

that

dx > 81r, V~eHl(yl;S2), ~o(x) = x on 0 12. (2.4)

W is smooth except at a finite number of points (since we

(2.5)

] D.V{dx = / (D.n){d~- ]" (div D) (dx (2.6) 12 0¢l

N (D.n) = Jg = 1 on 0f i (since g is the identity on 0fl) and div D = 4riEldi~ai__

a i are the singularities of ~, with the corresponding degrees dieZ ). Note

N E d i = 1 (2.7)

i=l since g has degree one on 0 f/.

Applying (2.5) and (2.6) we find

/ [v~'l 2dx >- 8~- Sup

I}~]]Lip- <1

{ f Cd# - J" Cd~}

N d # = l ~ d ~ (d~ is the surface measure on S 2) and d~,= Y, where

i=l

no information so far about the location of the points a i and the integers di, except for

di~ai. Since we have

Sup { l e d # - SCdv} (2.8) ¢

114[[Lip- q

v which are finite sums of Dirac masses with

(2.7), we may as well write

]V~o]2dx > 87r In f fl v e ~

where ~,g denotes the class of measures

integer coefficients (di) and ~ i = 1.

Next, we use the following:

Lemma 4. Let M be a compact metric space and let # be a given probability measure on

M. Then

]7

In f Sup {S ( d # - ( d v } = M i n J d(x,y)d#(x). uecg (:M-~R M yeM M

tI :I1L i p<l

For the proof of Lemma 4 see [8] or [7] (the proof in [7] relies on a new result in Graph

Theory by Hamidoune-Las Vergnas [23]).

We may now complete the proof of Theorem 5 by applying Lemma 4 in M =

(with its Euclidean distance). We obtain

S [V~o[2- >8rMin_ J yefl Off

since

I g-yl = (2.9)

1 ~([-~-Yl + I~-Yl)d~ >/a ]~]d~=4r. S0 fl] ~-y] d~ = ~S o

This establishes that x/Ix I is a minimizer. For the proof of uniqueness, see [8].

Remark 1. The same argument shows that if ~ is any domain in IR 3 and g: O fl ~ S 2

a boundary data such that

Jg>_0 on O~ and deg (g ,0~)= l

then, for every ~eHl(fl;S 2) such that ~ = g on 0 ~, we have

Jfl lV~l 2 >- 2 Miny e~ JOY/d~2(y'~) Jg(~) d~

where d~ denotes the geodesic distance in ~.

Lin~s proof of Theorem 5. It relies on the following pointwise inequality which holds a.e.

for any ~eHl(gt;S2):

[V~] 2 >_ (div ~)2 _ tr(V~)2 (2.10)

where (V~)2 is the square of the 3×3 matrix V~.

Proof of (2.10). Changing coordinates we may always assume that, at a given point,

Oi Set a i j = ~ i ; since ]~12=1 we find

Clearly we have

= ( o , o , 1 ) .

a3j=O for j=1,2,3.

is

t8

2 ~ aii (div ~)2 = (iElaii)2<_ 2i=1 2

3 2 a2" 2 tr(7~) 2 = E a..a..> - Ea . .

i , j= l 1] ]1 - i~1 21 i#j l]

lv 12= a. 2. + i=I la i#j j

which implies (2.10).

On the other hand, a direct computation shows that

(div ~o) 2- tr(7~o) 2 = div((div ~0)~---(7~o)~). (2.11)

Combining (2.10) and (2.11), and integrating over fl leads

I a 17~p[ 2dx _ > l0 a[(div ~a)(~.n)-(7~)~.n]d(. (2.12)

Using the fact that ~(x) = x on 0 fl it is easy to see that the RHS in (2.12) equals 8~.

Remark 2. Lin's proof extends to any dimension. Let fl = B n be the unit ball in sn;

then x/Ix] is a minimizing harmonic map i.e.

fBnlV~o[2 > /BnIV(-[~-)]2 VcpeHl(Bn,sn-1), ¢p(x) = x on 0ft.

In dimension n the counterpart of (2.10) is

I V~o 12 > n~_ [(div 9~)2 _ tr(V~o)2] (2.10")

while (2.11) remains unchanged. The RHS in (2.12) equals (n-1)tsn-11 and

IBnlV( )I2 = Isn-ll.

It was already know (see Jager-Kanl [31]) that x/Ix] is a minimizer for n k 7 but this

was an open problem for 3 < n < 6.

Remark 3. The map ~o(x) = x/I x I is always a solution of the Euler equation associated

to the functional t~ defined in (0.2), for any choice of the constant kl,k2,k 3. A variant of

Lin's proofs shows that x / Ix I is a minimizer if k 1 < min(k2,k3). On the other hand,

H61ein [30] has shown that x/Ix] is no_.At a minimizer if 8(k2-kl) + k 3 < 0. The exact

range for which x/Ix] is a minimizer is ~ known.

II.2 The analysis of point singularities.

t9

The main result of this Section is a first order expansion near a singularity for a

minimizer ~o:

Theorem fi ([8]). Assume fi c R3 is any domain and g: 0 ~ -~ S 2 is any boundary data.

Let ~o be minimizer for (2.1). Then all its singularities have degree *1. Moreover, for

every singularity x o there is a rotation R such that

a ( x - x o) • as × x o

R~mark 4. Prior to this result, Hardt-Kinderlehrer-Lin [26] had obtained a universal

bound for the degree of singularities. Moreover (experimental and) numerical evidence in

[11] indicated that this degree is +1.

The argument relies on a blow-up technique originally introduced in [22] and a

careful analysis of the homogeneous tangent map. More precisely, assume x o = 0; as

-~ 0 W(ex) -~ ¢(x) (see [38] and [42]) where ¢ is a minimizing harmonic map depending

only on the direction x / Ix I, say ¢(x) = h(x/Ix I). The punch line of the proof is the

following:

Theorem 7 ([8]). Assume ~ = B 3 and h: O ~2 -~ S 2 is any boundary data. Then the

homogeneous estension ¢(x) = h(x / ]x l ) is not a minimizer unless h is a constant or

h = ~- Rotation.

The proof of Theorem 7 is quite involved; see [8].

II.3 .E. ~ergy estimates for maps which are odd on the boundary.

The following result conjectured in [8] is due to Coron---Gulliver [12].

The0r~m 8. Assume ~ = B 3 and let g: 0 ~2 -~ S 2 be an odd boundary condition, i.e.

g(-x) =--g(x) on 0 ~. Then

20

S [7~ol2>87r V~oeHl(£/;S2),~x)=g(x) on a ~ . (2.13)

In particular, if we choose g(x) = x we obtain (2.4) and deduce that x/Ixl is a

minimizing harmonic map.

Sketch of the proof. It relies on the coarea formula. Recall that (see Section 1.1)

1 ~- 19t IV~ol2dx >_ l a J2~ tx = IS2O~l(~"l(f))d f.

Here, we cannot assert that for a.e. feS 2, ,~gl(---l(f)) > 4~r. However, we may write

Ss2~¢l(~'- l({))df = ½ / 2 'g l (~ ' - l ( f ) U ~ ' l ( - f ) ) d f . S

The conclusion of Theorem 8 follows from:

Claim. For a.e. feS 2, we have

_> 2. (2.14/

As above, we may a~ways assume that ~o is a minimizer. Therefore ~o is smooth except

at a finite number of singularities (ai) where ~o acts like + a rotation. Assuming f and

- f are regular values, the set C = ~- - l ( f )u~- l ( - f ) consists of a collection of curves

running between the points (ai) and O ft. There may also be closed loops - but we

disregard them as in Section 1.1. By contrast with the analysis in Section 1.1 the singular

points (ai) do not play any special role. In fact, since ~o acts like + a rotation near the

singularities, we find that near each a i , C looks like a simple smooth curve passing

through a i.

We claim that

~-1(~) n 0 ~ consists of k points with k odd. (2.15)

Indeed, since g is odd, we know by Borsuk's Theorem (see e.g. [36]) that deg(g,0 ~2) is

odd. On the other hand if we write (Xi)l<ihk = ~---l(f)n 0 £t, then (see [36]) e.g. k

deg(g,O ~) = E sign i=1 Jg(×i) and therefore k must be odd.

Thus the set C fl 0 ~2 consists of k pairs of antipodal points with k odd.

We split C tl 0~2 as P U Q where

21

and

P = xe012 or ~(x) -~ and Jg(x)<

Q={xc012 either ~ x ) = ~ and J g ( X ) < i l

or ~(x) -~ and Jg(X)>

Clearly P and Q are antipodal set (i.e. P = - Q ) , each one containing k points. Write

P = {pl,P2,...,pk } and Q = {qt,q2,...,qk} ,

with qi = -Pi"

It is not difficult to check that C consists of k disjoint curves connecting the P points

to the Q points (for this purpose, it is convenient to orient the curves in C as in Section

1.1). Therefore we find that

k k Length of C _> i=lE [pi-qa(i)[ = i=lE [Pi + Pa(i)["

For some permutation a of {1,2,...,k}. The conclusion follows from

Lemma 5. Assume k is odd and {pl,P2,...,pk} are any k points in IR n. Then k E ]Pi + Pa(i) [ > 2 "

for any permutation a.

Proof. Suppose first that

have

a is the permutation 1 -~ 2, 2 ~ 3,..., (k- l ) ~ k, k -~ 1. Then we

IPl + P2 [ + [P2 + P3 [ + "'" + [Pk-14- Pk[ + ]Pk + Pl[

= [Pl 4- P2 [ 4- [ - P 2 - P 3 [ + "'" 4- l - P k - l - P k [ + [Pk 4-Pl[ -> 2[Pl[

(by the triangle inequality).

In the general case we may decompose a into such elementary cycles. Since k is odd, at / t t

least one of them involves k elements (1 < k < k) with k odd. Then we ~ e reduced

to the previous case.

Remark 5. The same argument shows that if 12 ¢ IR 3 is any symmetric domain (i.e.

- 12 = 12) with 0e12 and g: 9 12 -~ S 2 is odd, then

/ IV l 2-> 8rdist(0,0a) V~penl(fl; S2), ~x) = g(x) on 012. 12

Remark 6. The conclusion of Theorem 8 fails if instead of assuming that g is odd one

22

merely assumes that deg(g,0 ~) is odd. In fact, given any E > 0, it is easy to construct

(using Lemma 2) a map 9cHI(B3;S2) such that ~o I OB 3 has degree one and

SB3[T~pI2 < E.

II.4 The gap phenomenon. Density and nondensitY of smooth maps between manifolds.

Traces.

Theorem 9.

such that

We start with a very interesting phenomenon discovered by Hardt-Lin [27].

Let ~2 = B 3. There exist smooth boundary data g: 0 f -~ S 2 of degree zero

Min SIv~I 2 < Inf SIV~o[ 2. (2.16) ~peH 1 (f ;S 2) ~ eCI(~;S 2)

~v=g on a f ~-=g on O Note that since g has degree zero there are always maps ~cCI(~;S 2) such that

~o=g on 0 ~.

Sketch of the construction. In fact, given any E > 0 we may choose a g such that, in

(2.16), LHS -~ e and RHS z 16~-. For this purpose we place two dipoles (al,a2) , (bl,b2)

along the z-axis with a l= (0,0,t+e), a2= (0,0,1--~), b l= (0,0,--l+E) and

b2= (0,0,-I-E). Using Lemma 2 we obtain a map ~e which is smooth except at the

points {al,a2,bl,b2} and such that 2

StV~E I < 32~E + 2E.

Define g to be the restriction of ~E to Off, so that g is smooth and deg(g,0f) = 0.

Clearly LHS < 32~ + 2E since we may take ~o E as a testing function. In order to obtain

a lower bound for I~HS we use the D-field method. Let D be the D-field associated

with ~o. We have

SflV~p] 2 > 2 S [D[ > D.V¢= f ( D . n ) ¢ = - F~ - 2Sf 2S0 2/0 ~Jg~

for every function ~ such that IiV~II _< 1 (here we use the fact that ~eC 1, so that L ~

div D = 0). Choosing a function ~ such that ~ __-- 0 near (0,0,-1) and ~" ~_ 2--E we

obtain, using (1.5),

23

J~IV~l 2 > 2(2---E)47r.

Theorem 9 implies in particular that smooth S2-valued maps (satisfying a

boundary condition) need no_At be dense in the corresponding Sobolev space.

In the same spirit, it has been pointed out by Schoen-Uhlenbeck [39] that the map

~(x) -- x / tx l cannot be approximated in the H 1 norm by maps ~neCI(B3;S2). Here is

a simple proof of this fact using the D-field. Suppose, by contradiction, that such a

sequence exists and let (Dn) be the corresponding D-fields. Clearly D n -~ D in

LI(Ft;~ 3) and div Dn= 0. It follows that div D = 0 in .~'(gt). On the other hand we

know that div D = 4~r5 o - a contradiction.

Going to a more general setting we may ask, following Eells-Lemaire [15], whether

CI(M;N) is dense in wl'P(M;N) where M and N are two manifolds (M may have a

boundary, but not N). That question has now been completely settled:

The0rem 10. If p > dim M, then CI(M;N) is dense in wI'P(M;N).

The case where p > dim M is easy because the Sobolev embedding Theorem

implies that wl 'P(M;N) ¢ C(M,N) and the usual convolution technique can be applied.

The case p = dim M is slightly more delicate and requires a special argument (see [39]

and [40]).

Theorem 11. Suppose p < dim M; then CI(M;N) is dense in wI'P(M,N) if and only if

the homotopy group 7r[p](N) = 0.

If ~r[p](N) ¢ 0, Bethuel and Zheng [6] have constructed a map ~ewl'P(M,N)

which cannot be approximated by smooth maps (their proof uses an earlier result of White

[43]). The converse, namely if ~r[p](N) = 0 then smooth maps are dense, is a deep result

of Bethuel [4].

As a consequence of Theorem 11 we see that CI(B3,S 2) is not dense in Hl(B3,S2)

because ~r2($2 ) ~ 0, but CI(B3,S 3) is dense in HI(B3,S 3) because ~r2($3 ) = 0.

When smooth maps are not dense in wI'P(M,N) it is a very interesting problem to

study the W I'p closure of smooth maps. A special case has been settled by Bethuel [5]:

Theorem 12. Let ~eHI(B3,S2) and let D be its corresponding D-field. Then there

exists a sequence (~n) in CI(B3,S 2) such that ~n -~ ~ in H 1 if and only if ~p satisfies

24

div D = 0 in ~ ' (B3) . (2.17)

Condition (2.17) is clearly necessary but the converse is far from obvious and we

refer to [5].

In [4] and [6] there are also interesting results showing that maps in wl'P(M,N)

can be approximated by maps which are smooth everywhere except on sets of low

dimensions.

Finally, let us mention that Haxdt-Lin [28} have studied the trace on O ~ of maps

in wI'P(~/;N) where ~ ¢ IR n is a smooth domain. Their main result is:

Theorem 13. Assume 1 < p < n and ri(N) = 0 for every o < i < [p]-l, then any map

i _ 1 geW P'P(0 ~;N) is a trace of a map ~oewl'P(M;N).

They also present an example of a map geH1/2(0B3;S I) which is not the trace of

any map ~eHI(B3;S1).

II.5 Some open problems.

Problem 4. Let ~ be a minimizer for the energy I~ defined by (0.2) in the class

g = {~oeHl(~;S2); ~o = g on 0 ~}. Is ~o smooth except at a finite number of points? [A

partial regularity result of Hardt-Kinderlehrer-Lin [25], [26} asserts that ~o is smooth

except on a set Z with ¢~gl(z) = 0 and even Jga(Z) = 0 for some a < 1.]

Problem 5. Let Y~ c ~3 be a bounded smooth domain and let g: 0 9/-~ S 2 be a given

boundary data. Let ,/¢ denote the class of all measures u of the form u = E dihai with

aid2 , dleZ and Zd i = deg(g,0 ~), where the points (ai) are placed arbitrarily in Yr.

Does one have

Min I [ 7~ t2=In f{ l ID[ ;d ivDe~4and D.n=~F~Jg on Of'/}? ~peH 1 (a;S 2) ~ fl

~o=g on 0 f~

[The answer is positive if ~ = B 3 and g(x) = x on 0 a; this follows from the proof of

Theorem 5 via the D-field.]

25

Problem 6. Find a simple proof of Lemma 4. Are there any interesting applications of

Lemma 4 in other fields? What happens to In f Sup{l{d#- ~{dv} if # is a signed veJ t

measure of total mass one (instead of a probability measure)? What happens to

InfSup{ } if Jg is replaced by v ¢

d i = • = with dieZ and FA i = k} ~'k {v,v Z - ~ ~ai

where k > 1 is a given integer?

Problem 7. Assume g~ = B n (n >_ 3) and h: 0 fl -+ S n-1 is a (smooth) boundary data.

When is ¢(x) = h(x/IxJ) a minimizer for

M i n llv~ol2 ? p ¢H 1 ( ~ ; S n - l ) ~o=h on 0 ~q

Problem 8. Let ~ C R3 be a bounded domain and let g: 0 ~2 -~ S 2 be a boundary data

such that deg(g,8 ~2) = 0. Let ~o be a minimizer for

Min S [V~o[ 2 , ~oeH 1 (~,1;S 2)

~o=g on O

Are there simple conditions on g which guarantee that ~o has no singularity? For

example, a condition like

1 0 a l V T g l 2 < 8 r

denotes the tangential gradient. Note that the example described in Theorem 9 where V T

satisfies

S0~IVTg[2 = 16~r+ E

and the corresponding minimizers have singularities. Almgren-Lieb [3] (resp. Hardt-Lin

[29]) have obtained estimates on the numbers of singularities of ~o in terms of [[VTg[[L2

(resp. I[YTg[[LOO) , however there is no control on the size of the constants involved.

Problem 9. Can one prove Theorem 8 using a D-field approach? More precisely, assume

~2 = B 3 and let g: 0 ~ -* S 2 be an odd boundary condition. Let DeLI(~;~ 3) be such

that div De,/g (defined in Problem 5) and D.n = 1 j :~ g on 8f~. Does on have

26

~tIDI >_ 1 ?

Problem I0. Can one prove Theorem 8 via Lin's device? This boils down to determine

whether the RHS of (2.12) is bounded below by 8~ for any map ~o which is odd on 0 ~.

Problem II. Can one extend Theorem 8 to B n, n >. 3? More precisely, let 12 = B n and

assume ~cHI(~;S n-l) is odd on 0 fL Does one have

IalV~12dx >_ ~1217(]~)12dx = ~{sn-iI ?

Problem 12. Let Ft = B 3 and let g: 0 It ~ S 2 be a smooth map of degree zero.

Is Inf{~ft{V~{2 ," {eC1(~9;$2), ~ = g on 0 fi} achieved? What is the corresponding

problem if deg(g,0 fl) # 07

Problem 13. Let ~ c ]{3 be a smooth bounded domain and let g: 0 gt ~ S 2 be a (smooth)

boundary data. Given ~(HI(yt;S 2) with { = g on O~ define

S(~) = (:Su~_~R{~ytD.V¢- I0 flJg¢} (2.18)

117({[LOO_< I

where D is D-field associated with ~ through (IAI).

In the special case where ~ is smooth except at isolated singularities (ai) of degrees (di)

then S(~} = 47rL where L is the length of a minimal connection connecting the

singularities (ai) (computed with the geodesic distance dfl). This is a direct consequence

of (i.13) and Lemma 3. Of course, if ~ has no singularity, then S(~) = 0.

Given ~(Hl(ft;S 2) with ~o : g on O ~ it seems of interest to introduce the

"modified" energy

E#(~.o) = ~ lye{ 2 + 2S(~) (2.19)

which takes into account the "interaction" of the singularities.

In view of the example constructed in the proof of Theorem 9 it seems reasonable to

ask whether

In f I {7~o{ 2 = In f E#(~) ? (2.20) ~oecl(fi;S 2) ~ ~oeHl(fi;S 2)

q~=g on 0 ~ ~o=g on 0 ~ There are some partial results by Bethuel [5] in that direction. Also, is RHS in (2.20)

achieved for some ~o which is not smooth? Study the properties of E # with respect to

strong and weak convergence in H 1.

27

Problem 14. Is CI(M;N) always dense in wI'P(M;N) for the weak topology of wI 'P?

[The answer is positive for M = B 3, N = S 2 and p = 2, see [4].]

Problem 15. Study the density (or nondensity) of CI(M;N) in ws'P(M;N) where s

need not be an integer. Partial results have been obtained by Escobedo [9].

Problem !6, Are the assumptions in Theorem 13 sharp? How does one recognize whether

given map geH1/2(0B3,S1) is the trace of a ~eHI(B3,S1). Same question for general

manifold and 1 < p < oo.

28

References

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

pl]

[12] [13]

[14]

F. Almgren, Existence and regularity almost everywhere of solutions to elliptic

variational problems among surfaces of varying topological type and singularity

structure, Ann. of Math. 27 (1968), P. 321-391.

F. Almgren - W. Browder - E. Lieb, Co-area, liquid crystals, and minimal surfaces

in DD7 - a selection of papers, Springer (1987).

F. Almgren - E. Lieb, Singularities of energy minimizing maps from the ball to the

sphere, Bull. Amer. Math. Soc. 17 (1987), p. 304-306 and detailed paper to appear.

F. Bethuel, Approximation dans des espa~de Sobolev entre deux varie'tgs et

groupes d'homotopie, C.R. Acad. Sc• Paris (1988) and detailed paper to appear.

F. Bethuel, A characterization of maps in Hl(B3,S2), which can be approximated

by smooth maps, to appear.

F. Bethuel - X. Zheng, Sur la densit~ des fanctlons reguheres entre deux vametes

dans les espac~de Sobolev, C.R. Acad. Sc. Paris 303 (1986), p. 447-447 and Density

of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. (to

appear).

H. Brezis, Liquid crystals and energy estimates for S2-valued maps, in [16]. J .

H. Brezls - J. M. Coron - E. Lieb, Harmonic maps with defects, Comm. Math.

Phys. 107 (1986), p. 649-705.

W. Brinkman - P. Cladis, Defects in liquid crystals, Physics Today, May 1982, p.

48-54.

S. Chandrasekhar, LiQuid crystals, Cambridge University Press (1977).

R. Cohen - R. Hardt - D. Kinderlehrer - S. Y. Lin - M. Luskin, Minimum energy

configurations for liquid crystals: computational results in [16].

J. M. Coron - R. Gulliver, Minimizing p-Harmonic maps into spheres (to appear).

P. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford (1974)•

E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di

tipo ellitico, Boll U.M.I. 4 (1968), p. 135-I37.

29

[15] J. Fells - L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10

(1978), p 1-68.

[16] J. Ericksen, Equilibrium theory of liquid crystals, in Advances in Liquid Crystals 2,

(G.Brown ed.), Acad. Press (1976), p. 233-299.

[17] J. Ericksen, Static theory of point defects in nematic liquid crystals (to appear).

[18] J. Ericksen - D. Kinderlehrer ed., Theory and applications of liquid crystals, IMA

Series Vol. 5, Springer (1987).

[19] M. Escobedo, Some remarks on the density of regular mappings in Sobolev classes of

sM-valued functions (to appear).

[20] H. Federer, Geometric measure theory, Springer (1969).

[21] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic

systems, Princton Univ. Press (1983).

[22] E. Guisti - M. Miranda, Sulla regularita delle soluzioni deboli di una classe di

sistemi ellitici quasilineari, Arch. Rat. Mech. Anal. ~ (1968), p. 173-184.

[23] Y. Hamidoune - M. Las Vergnas, Local edge-connecting in regular bipartite graphs

(to appear).

[24] R. Hardt, An introduction to geometric rncasure theory, Lecture Notes, Melbourne

Univ. (1979).

[25] R. Hart - D. Kinderlehrer - F. Lin, Existence and partial regularity of static liquid

crystal configurations, Comm. Math. Phys. !05 (1986), p. 547-570.

[26] R. Hardt - D. Kinderlehrer - F. Lin, Stable defects of minimizers of constrained

variational principles, Ann. I.H.P., Analyse Nonhnealre (to appear).

[27] lt. Hardt - F. Lin, A remark on H I mappings, Manuscripta Math. 56 (1986), p.

1-10.

30

[28] R. Hardt - F. Lin, Mappings minimizing the L p norm of the gradient, Comm.

Pure Appl. Math. 40 (1987), p. 556-588.

[29] R. Hardt - F. Lin, Stability of singularities of minimizing harmonic maps, Math.

Annalen (to appear).

[30] F. H~lein, Minima de la fonctionnelle energie libre des cristanx liquides, C.R. Acad.

Sc. Paris 305 (1987), p. 565-568.

[31] W. Jager - H. Kaul, Rotationally symmetric harmonic maps from a ball into a

sphere and the regularity problem for weak solutions of elliptic systems, J. Reine

Angew. Math. 343 (1983), p. 146-161.

[32] L. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk SSSR 37 (1942), p.

227- 229.

[33] M. Kleman, Points, lines and walls, John-Wiley (1983).

[34] F. Lin, Une remarque sur l'application x / Ix I, C.R. Acad. Sc. Paris (~ paral^tre).

[35] C. Morrey, Partial regularity results for nonlinear elliptic systems, J. Math. Mech.

17 (1968), p. 649-670.

[36] L. Nirenberg, Topics in Nonlinear Functional Analysis, N.Y.U. Lecture Notes, New

York (1974).

[37] S. Rachev, The Monge-Kantorovich mass transference problem and its stochastic

applications, Theory of Prob. and Applic. 29 (1985), p. 647-676.

[38] R. Schoen - K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom.

17 (1982), p. 307-335.

[39] R. Schoen - K. Uhlenbeck, Boundary regularity and the Dirichlet problem for

harmonic maps, J. Diff. Geom. 18 (1983), p. 253-268.

[40] R. Schoen - K. Uhlenbeck, Approximation theorems for Sobolev mappings (to

appear).

[41] L. Simon, Lectures on geometric measure theory, Canberra (1984).

[42] L. Simon, Asymptotics for a class of nonlinear evolution equations with applications

to geometric problems, Ann. of Math. 11~ (1983), p. 525-571.

[43] B. White, Infina of energy functionals in homotopy classes, J. Diff. Geom. 23

(1986), p. 127-142.