The Mathematical Theory of Maxwell's Equations

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The Mathematical Theory of Maxwells Equations

Andreas Kirsch and Frank HettlichDepartment of Mathematics

Karlsruhe Institute of Technology (KIT)Karlsruhe, Germany

c

May 24, 2014


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Preface

This book arose from lectures on Maxwells equations given by the authors between 2007 and2013. Graduate students from pure and applied mathematics, physics including geophysics and engineering attended these courses. We observed that the expectations of these groupsof students were quite different: In geophysics expansions of the electromagnetic fields intospherical (vector-) harmonics inside and outside of balls are of particular interest. Graduate

students from numerical analysis wanted to learn about the variational treatments of interiorboundary value problems including an introduction to Sobolev spaces. A classical approachin scattering theory which can be considered as a boundary value problem in the unboundedexterior of a domain uses boundary integral equation methods which are particularlyhelpful for deriving properties of the far field behaviour of the solution. This approach is, forpolygonal domains or, more generally, Lipschitz domains, also of increasing relevance fromthe numerical point of view because the dimension of the region to be discretized is reducedby one. In our courses we wanted to satisfy all of these wishes and designed an introductionto Maxwells equations which coveres all of these concepts but restricted ourselves almostcompletely (except Section 4.3) to the time-harmonic case or, in other words, to the frequencydomain, and to a number of model problems.

The Helmholtz equation is closely related to the Maxwell system (for time-harmonic fields).As we will see, solutions of the scalar Helmholtz equation are used to generate solutions ofthe Maxwell system (Hertz potentials), and every component of the electric and magneticfield satisfies an equation of Helmholtz type. Therefore, and also for didactical reasons, wewill consider in each of our approaches first the simpler scalar Helmholtz equation before weturn to the technically more complicated Maxwell system. In this way one clearly sees theanalogies and differences between the models.

In Chapter 1 we begin by formulating the Maxwell system in differential and integral form.We derive special cases as the E mode and the Hmode and, in particular, the timeharmonic case. Boundary conditions and radiation conditions complement the models.In Chapter 2 we study the particular case where the domain D is a ball. In this case we canexpand the fields inside and outside ofD into spherical wave functiuons. First we study thescalar stationary case; that is, the Laplace equation. We introduce the expansion into scalarspherical harmonics as the analogon to the Fourier expansion on circles in R2. This leadsdirectly to series expansions of solutions of the Laplace equation in spherical coordinates.The extension to the Helmholtz equation requires the introduction of (spherical) Bessel- andHankel functiuons. We derive the most important properties of these special functions indetail. After these preparations for the scalar Helmholtz equation we extend the analysis

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to the expansion of solutions of Maxwells equations with respect to vector wave functions.None of the results of this chapter are new, of course, but we have not been able to find sucha presentation of both, the scalar and the vectorial case, in the literature. We emphasize thatthis chapter is completely self-contained and does not refer to any other chapter except forthe proof that the series solution of the exterior problem satisfies the radiation condition.

Chapter 3 deals with a particular scattering problem. The scattering object is of arbitraryshape but, in this chapter, with sufficiently smooth boundary D. We present the classicalboundary integral equation method and follow very closely the fundamental monographs[5, 6] by David Colton and Rainer Kress. In contrast to their approach we restrict ourselvesto the case of smooth boundary data (as it is the usual case of the scattering by incidentwaves) which allows us to study the setting completely in Holder spaces and avoids thenotions of parallel surfaces and weak forms of the normal derivatives. For the scalarproblem we restrict ourselves to the Neumann boundary condition because our main goalis the treatment for Maxwell equations. Here we believe as also done in [6] that thecanonical spaces on the boundary are Holder spaces where also the surface divergence is

Holder continuous. In order to prove the necessary properties of the scalar and vectorpotentials a careful investigation of the differential geometric properties of the surface Disneeded. Parts of the technical details are moved to the Appendix 6. We emphasize that thischapter is self countained and does not need any results from other chapters (except fromthe appendix).

As an alternative approach for studying boundary value problems for the Helmholtz equationor the Maxwell system we will study the weak or variational solution concept in Chapter 4.We restrict ourselves to the interior boundary value problem with a general source term andthe homogeneous boundary condition of an ideal conductor. This makes it possible to work(almost) solely in the Sobolev spaces H10 (D) and H0(curl, D) of functions with vanishing

boundary traces or tangential boundary traces, respectively. In Section 4.1 we derive the ba-sic properties of these special Sobolev spaces. The characteristic feature is that no regularityof the boundary and no trace theorems are needed. Probably the biggest difference betweenthe scalar case of the Helmholtz equation and the vectorial case of Maxwells equations isthe fact that H0(curl, D) is not compactly imbedded in L

2(D,C3) in contrast to the spaceH10 (D) for the scalar problem. This makes it necessary to introduce the Helmholtz decompo-sition. The only proof which is beyond the scope of this elementary chapter is the proof thatthe subspace ofH0(curl, D) consisting of divergence-free vector fields is compactly imbeddedin L2(D,C3). For this part some regularity of the boundary (e.g. Lipschitz regularity) isneeded. Since the proof of this fact requires more advanced properties of Sobolev spaces it

it transponed to Chapter 5. We note that also this chapter is self contained except of thebeforementioned compactness property.

The final Chapter 5 presents the boundary integral equation method for Lipschitz domains.The investigation requires more advanced properties of Sobolev spaces than those presentedin Section 4.1. In particular, Sobolev spaces on the boundaryDhave to be introduced andthe correponding trace operators. Perhaps different from most of the traditional approacheswe first consider the case of the cube (, )3 R3 and introduce Sobolev spaces of periodicfunctions by the proper decay of the Fourier coefficients. The proofs of imbedding and tracetheorems are quite elementary. Then we use, as it is quite common, the partion of unity and


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local maps to define the Sobolev spaces on the boundary and transfer the trace theorem togeneral Lipschitz domains.We define the scalar and vector potentials in Section 5.2 analogously to the classical caseas in Chapter 3 but have to interpret the boundary integrals as certain dual forms. Theboundary operators are then defined as traces of these potentials. In this way we follow the

classical approach as closely as possible. Our approach is similar but a bit more explicitthan in [16], see also [10]. Once the properties of the potentials and corresponding boundaryoperators are known the introduction and investigation of the boundary integral equations isalmost classical. For example, for Lipschitz domains the Dirichlet boundary value problemfor the scalar Helmholtz equation is solved by a (properly modified) single layer ansatz.This is preferable to a double layer ansatz because the corresponding double layer boundaryoperator fails to be compact (in contrast to the case of smooth boundaries). Also, the singlelayer boundary operator satisfies a Gardings inequality; that is, can be decomposed into acoercive and a compart part. Analogously. the Neumann boundary value problem and theelectromagnetic case are treated. In our presentation we try to show the close connection

between the scalar and the vector cases.Starting perhaps with the pioneering work of Costabel [7] many important contributions tothe study of boundary integral operators in Sobolev spaces for Lipschitz boundaries havebeen published. It is impossible for the authors to give an overview on this subject butinstead refer to the monograph [10] and the survey article [4] from which we have learned alot. As mentioned above, our approach to introduce the Sobolev spaces, however, is differentfrom those in, e.g. [1, 2, 3].

In the Appendix 6 we collect results from vector calculus and differential geometry, in par-ticular various forms of Greens theorem and the surface gradient and surface divergence for(smooth) functions on (smooth) surfaces.

We want to emphasize that it was not our intention to present a comprehensive work onMaxwells equations, not even for the time harmonic case or any of the beforementioned sub-areas. As said before this book arose from and is intended to be material for designinggraduate courses on Maxwells equations. The students should have some knowledge on vec-tor analysis (curves, surfaces, divergence theorem) and functional analysis (normed spaces,Hilbert spaces, linear and bounded operators, dual space). The union of the topics coveredin this monograph is certainly far too much for a single course. But it is very well possibleto choose parts of it because the chapters are all independent of each other. For example, inthe summer term 2012 (8 credit points; that is in our place, 16 weeks with 4 hours per weekplus exercises) one of the authors (A.K.) covered Sections 2.12.6 of Chapter 2 (only interior

cases), Chapter 3 without all of the proofs of the differential geometric properties of thesurface and all of the jump properties of the potentials, and Chapter 4 without Section 4.3.Perhaps these notes can also be useful for designing courses on Special Functions (sphericalharmonics, Bessel functions) or on Sobolev spaces.

One of the authors wants to dedicate this book to his father, Arnold Kirsch (19222013), whotaught him about simplification of problems without falsification (as a concept of teachingmathematics in high schools). In Chapter 4 of this monograph we have picked up thisconcept by presenting the ideas for a special case only rather than trying to treat the most


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general cases. Nevertheless, we admit that other parts of the monograph (in particular ofChapters 3 and 5) are technically rather involved.

Karlsruhe, March 2014 Andreas KirschFrank Hettlich


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Contents

Preface 2

1 Introduction 9

1.1 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 The Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Boundary and Radiation Conditions . . . . . . . . . . . . . . . . . . . . . . 18

1.5 The Reference Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Expansion into Wave Functions 27

2.1 Separation in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Expansion into Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Laplaces Equation in the Interior and Exterior of a Ball . . . . . . . . . . . 50

2.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6 The Helmholtz Equation in the Interior and Exterior of a Ball . . . . . . . . 62

2.7 Expansion of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . 72

2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Scattering From a Perfect Conductor 913.1 A Scattering Problem for the Helmholtz Equation . . . . . . . . . . . . . . . 91

3.1.1 Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . 92

3.1.2 Volume and Surface Potentials . . . . . . . . . . . . . . . . . . . . . . 99

3.1.3 Boundary Integral Operators . . . . . . . . . . . . . . . . . . . . . . 112

3.1.4 Uniqueness and Existence . . . . . . . . . . . . . . . . . . . . . . . . 116

3.2 A Scattering Problem for the Maxwell System . . . . . . . . . . . . . . . . . 124

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8 CONTENTS

3.2.1 Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . 125

3.2.2 Vector Potentials and Boundary Integral Operators . . . . . . . . . . 135

3.2.3 Uniqueness and Existence . . . . . . . . . . . . . . . . . . . . . . . . 141

3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4 The Variational Approach to the Cavity Problem 147

4.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.1.1 Basic Properties of Sobolev Spaces of Scalar Functions . . . . . . . . 148

4.1.2 Basic Properties of Sobolev Spaces of Vector Valued Functions . . . . 157

4.1.3 The Helmholtz Decomposition . . . . . . . . . . . . . . . . . . . . . . 160

4.2 The Cavity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.2.1 The Variational Formulation and Existence . . . . . . . . . . . . . . . 163

4.2.2 Uniqueness and Unique Continuation . . . . . . . . . . . . . . . . . . 173

4.3 The TimeDependent Cavity Problem . . . . . . . . . . . . . . . . . . . . . 183

4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5 Boundary Integral Equation Methods for Lipschitz Domains 201

5.1 Advanced Properties of Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . 201

5.1.1 Sobolev Spaces of Scalar Functions . . . . . . . . . . . . . . . . . . . 203

5.1.2 Sobolev Spaces of Vector-Valued Functions . . . . . . . . . . . . . . . 216

5.1.3 The Case of a Ball Revisited . . . . . . . . . . . . . . . . . . . . . . . 235

5.2 Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.3 Boundary Integral Equation Methods . . . . . . . . . . . . . . . . . . . . . . 263

5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

6 Appendix 277

6.1 Table of Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 277

6.2 Results from Linear Functional Analysis . . . . . . . . . . . . . . . . . . . . 279

6.3 Elementary Facts from Differential Geometry . . . . . . . . . . . . . . . . . 281

6.4 Integral Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

6.5 Surface Gradient and Surface Divergence . . . . . . . . . . . . . . . . . . . . 288

Bibliography 292

Index 294


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Chapter 1

Introduction

In this introductory chapter we will explain the physical model and derive the boundary valueproblems which we will investigate in this monograph. We begin by formulating Maxwellsequations in differential form as our starting point. In this monograph we consider exclusivelylinear media; that is, the constitutive relations are linear. The restriction to special casesleads to the analogous equations in electrostatics or magnetostatics and, assuming periodictime dependence when going into the frequency domain, to time harmonic fields. In thepresence of media the fields have to satisfy certain continuity and boundary conditions and,if the region is unbounded, a radiation condition at infinity. We finish this chapter byintroducing two model problems which we will treat in detail in Chapters 3 and 4.

1.1 Maxwells Equations

Electromagnetic wave propagation is described by four particular equations, the Maxwellequations, which relate five vector fields E, D, H, B, Jand the scalar field. In differentialform these read as follows

B

t + curlx E = 0 (Faradays Law of Induction)

D

t curlx H = J (Amperes Law)

divx D = (Gauss Electric Law)

divx B = 0 (Gauss Magnetic Law) .

The fields Eand D denote the electric field (inV /m) andelectric displacement (inAs/m2)respectively, while H and B denote the magnetic field (in A/m) and magnetic flux density(in V s/m2 = T =Tesla). Likewise, J and denote the current density (in A/m2) andcharge density (inAs/m3) of the medium.

Here and throughout we use the rationalized MKS-system, i.e. the fields are given withrespect to the units Volt (V), Ampere (A), meter (m), and second (s). All fields depend both

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10 CHAPTER 1. INTRODUCTION

on the space variable x R3 and on the time variable tR. We note that the differentialoperators are always taken with respect to the spacial variable x without indicating this.The definition of the differential operators div and curl, e.g., in cartesian coordinates andbasic identities are listed in Appendix 6.1 .

The actual equations that govern the behavior of the electromagnetic field were first com-pletely formulated by James Clark Maxwell (18311879) inTreatise on Electricity and Mag-netism in 1873. It was the ingeneous idea of Maxwell to modify Amperes Law which wasknown up to that time in the form curl H = J for stationary currents. Furthermore, hecollected the four equations as a consistent theory to describe electromagnetic phenomena.

As a first observation we note that in domains where the equations are satisfied one derivesfrom the identity div curlH = 0 the well known equation of continuity which combines thecharge density and the current density.

Conclusion 1.1 Gauss Electric Law and Amperes Law imply theequation of continuity

t

= divDt

= div

curlHJ = divJ .Historically, and more closely connected to the physical situation, the integral forms ofMaxwells equations should be the starting point. In order to derive these integral relations,we begin by letting Sbe a connected smooth surface with boundary S in the interior of aregion 0 in R

3 where electromagnetic waves propagate. In particular, we require that theunit normal vector (x) for x S is continuous and directed always into one side ofS,which we call the positive side ofS. By(x) we denote a unit vector tangent to the boundaryofS at x S. This vector, lying in the tangent plane ofStogether with a second vectorn(x) in the tangent plane at x

Sand normal to S is oriented such that

{,n,,

}form

a mathematically positive system; that is, is directed counterclockwise when we look at Sfrom the positive side, and n(x) is directed to the outside ofS. Furthermore, let R3 bean open set with boundary and outer unit normal vector (x) at x .Then Amperes law describing the effect of the external and the induced current on themagnetic field is of the form

S

H d = ddt

S

D ds +S

J ds . (1.1)

It is named by Andre Marie Ampere (17751836).

Next,Faradays law of induction(Michael Faraday, 17911867), which isS

E d = ddt

S

B ds , (1.2)

describes how a time-varying magnetic field effects the electric field.

Finally, the equations include Gauss Electric Law

D ds =

dx (1.3)


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1.2. THE CONSTITUTIVE EQUATIONS 11

descibing the sources of the electric displacement, andGauss Magnetic Law

B ds = 0 (1.4)

which ensures that there are no magnetic currents. Both named after Carl Friedrich Gauss(17771855).

In regions where the vector fields are smooth functions and and are at least continuouswe can apply the following integral identities due to Stokes and Gauss for surfaces S andsolids lying completely inD.

S

curl F ds =

S

F d (Stokes), (1.5)

div F dx =

F

ds (Gauss), (1.6)

see Appendix 6.3. To derive the Maxwells equations in differential form we choose F to beone of the fields H, E, B or D. With these formulas we can eliminate the boundary integralsin (1.1)(1.4). We then use the fact that we can vary the surface Sand the solid in Darbitrarily. By equating the integrands we are led to Maxwells equations in differential formas presented in the begining.

With Maxwells equations many electromagnetic phenomena became explainable. For in-stance they predicted the existence of electromagnetic waves as light or X-rays in vacuum.It took about 20 years after Maxwells work when Heinrich Rudolf Hertz (18571894) could

show experimentally the existence of electromagenetic waves, in Karlsruhe, Germany. Formore details on the physical background of Maxwells equations we refer to text books asJ.D. Jackson,Classical Electrodynamics[11].

1.2 The Constitutive Equations

In the general setting the equations are not yet complete. Obviously, there are more un-knowns than equations. The Constitutive Equationscouple them:

D = D(E,H) and B = B(E,H) .

The electric properties of the material, which give these relationships are complicated. Ingeneral, they not only depend on the molecular character but also on macroscopic quantitiesas density and temperature of the material. Also, there are time-dependent dependenciesas, e.g., the hysteresis effect, i.e. the fields at time tdepend also on the past.

As a first approximation one starts with representations of the form

D = E+ 4P and B = H 4M


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where Pdenotes the electric polarization vector and M the magnetization of the material.These can be interpreted as mean values of microscopic effects in the material. Analogously,and Jare macroscopic mean values of the free charge and current densities in the medium.

If we ignore ferro-electric and ferro-magnetic media and if the fields are relativly small one

can model the dependencies by linear equations of the formD = E and B = H

with matrix-valued functions : R3 R33, the dielectric tensor, and : R3 R33, thepermeability tensor. In this case we call a medium linear.

The special case of an isotropic mediummeans that polarization and magnetization do notdepend on the directions. Otherwise a medium is calledanisotropic. In the isotropic casedielectricity and permeability can be modeled as just real valued functions, and we have

D = E and B = H

with scalar functions , : R3 R.In the simplest case these functions and are constant and we call such a medium homo-geneous. It is the case, e.g., in vacuum.

We indicated already that alsoand Jcan depend on the material and the fields. Therefore,we need a further relation. In conducting media the electric field induces a current. In alinear approximation this is described by Ohms Law:

J = E + Je

where Je is the external current density. For isotropic media the function : R3

R is

called theconductivity . If = 0 then the material is called a dielectric. In vacuum we have = 0 and = 08.854 1012AS/V m, = 0 = 4 107V s/Am. In anisotropic media,also the function is matrix valued.

1.3 Special Cases

Under specific physical assumptions the Maxwell system can be reduced to elliptic secondorder partial differential equations. They serve often as simpler models for electromagneticwave propagation. Also in this monograph we will always explain the approaches first for

the simpler scalar wave equation.

Vacuum

Vacuum is a homogeneous, dielectric medium with = 0, = 0, and = 0, and no chargedistributions and no external currents; that is, = 0 and Je= 0. The law of induction takesthe form

0H

t + curlE = 0 .


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In magnetostatics one considers H being constant in time. For the magnetic field thesituation is different because, by Amperes law we have curlH = J. Thus in generalcurlH does not vanish. However, according to Gauss magnetic law we have

divB = 0 .

From this identity we conclude the existence of a vector potential A : R3 R3 withB = curlA inD. Substituting this into Amperes Law yields (for homogeneous media )after multiplication with0 the equation

0J = curl curlA = divA A.Since curl = 0 we can add gradientsu to A without changing B. We will see laterthat we can choose u such that the resulting potential A satisfies divA= 0. This choice ofnormalization is calledCoulomb gaugenamed by Charles Augustin de Coulomb (17361806).With this normalization we get also in magnetostatics the Poisson equation

A = 0J.We note that in this case the Laplacian is vector valued and has to be taken componentwise.

Time Harmonic Fields

For our purpose of considering wave phenomena the most important situtation are timeharmonic fields. Under the assumptions that the fields allow a Fourier transformation intime we set

E(x; ) = (FtE)(x; ) = RE(x, t) e

it

dt ,

H(x; ) = (FtH)(x; ) =R

H(x, t) eit dt ,

etc. We note that the fields E, H, etc are now complex valued; that is, E(; ), H(; ) :R

3 C3 and also all other Fourier transformed fields. Although they are vector fieldswe denote them by capital Latin letters only. According toFt(u) =iFtu Maxwellsequations transform into the time harmonic Maxwells equations

iB + curl E = 0 ,iD + curl H = E + Je ,

div D = ,

div B = 0 .

Remark: The time harmonic Maxwell system can also be derived from the assumptionthat all fields behave periodically with respect to time with the same frequency. Then thecomplex valued functions E(x, t) =eitE(x), H(x, t) =eitH(x), etc, and their real andimaginary parts satisfy the time harmonic Maxwell system.


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1.3. SPECIAL CASES 15

With the constitutive equations D = EandB= Hwe arrive at

curl E iH = 0 , (1.7a)curl H+ (i )E = Je , (1.7b)

div(E) = , (1.7c)

div(H) = 0 . (1.7d)

Assuming (for simplicity only) additionally an isotropic medium we can eliminate H or Efrom (1.7a) and (1.7b) which yields

curl

1

icurl E

+ (i ) E = Je . (1.8)

and

curl 1i curl H + iH = curl 1i Je , (1.9)respectively. Usually, one writes these equations in a slightly different way by introducing theconstant values0 > 0 and0 > 0 in vacuum and dimensionless, relative valuesr(x), r(x) R and c(x) C, defined by

r =

0, r =

0, c = r + i

0.

Then equations (1.8) and (1.9) take the form

curl 1r

curl E k2c E = i0Je ,curl

1

ccurl H

k2rH = curl

1

cJe

,

with thewave numberk=

00. We conclude from the second equation that div(rH) =

0 because div curl vanishes. For the electric field we obtain that div(cE) =i0k2 div Je = i0

div Je.

In vacuum we have c= 1, r = 1 and therefore the equations reduce to

curl curl E k2 E = i0Je , (1.10)curl curl H k2 H = curl Je . (1.11)

Without external current density, Je = 0, we obtain from curl curl = div the vectorHelmholtz equations

E+k2E= 0 and H+k2H= 0 .

Obviuously the reduced problems considering E or Hare symmetric and we conclude thefollowing important lemma.


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Lemma 1.3 A vector fieldEC2(,C3) combined withH := 1i0

curl Eprovides a solu-

tion of the time harmonic Maxwell system (1.7a)(1.7d) forJe= 0 in vacuum if and only ifEis a divergence free solution of the vector Helmholtz equation; that is,

E+k2E= 0 and div E= 0 in .

Analougously, a divergence free solution of the vector Helmholtz equation H C2(,C3)combined withE := 1

i0curl H leads to a solution of Maxwells equations in vacuum.

The relationship between the Maxwell system and the vector Helmholtz equation remainstrue if we consider time harmonic waves in any homogeneous medium because we only haveto substitute 0 and 0 by complex valued constants and , respectively. In this complexvalued case the wave number k =

is chosen such that Im k 0. As an example for

solutions of the Maxwell system in a homogeneous medium we consider plane waves.

Example 1.4 In the case of vacuum with Je= 0 a short calculation shows that the fields

E(x) = p eik dx and H(x) = (p d) eik dx

are solutions of the homogeneous time harmonic Maxwell equations (1.10), (1.11) providedd is a unit vector in R3 andp C3 withp d=3j=1pjdj = 0. Such fields are called planetime harmonic fields with polarization vector p C3 and direction d, since its wave frontsare planes perpendicular tod.

Additionally the following observation will be useful.

Lemma 1.5 LetEbe a divergence free solution of the vector Helmholtz equation in a domainD. Thenx x E(x) is a solution of the scalar Helmholtz equation.

Proof: By the vector Helmholtz equation and div E= 0 we obtain

x E(x) = 3j=1

div xj Ej(x) = 3j=1

div

Ej(x) e(j) + xj Ej(x)

=3

j=1

2Ejxj

(x) + xjEj (x) = 2div E(x) k2x E(x)

= k2x E(x)

wheree(j) denotes the j th cartesian coordinate unit vector.

As in the stationary situation also the time harmonic Maxwell equations in homogeneousmedia can be treated with methods from potential theory. We make the assumptionc= 1,r = 1 and consider (1.10) and (1.11). Taking the divergence of these equations yields


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1.3. SPECIAL CASES 17

div H= 0 and k 2 div E= i0div Je; that is, div E= (i/0)div Je. Comparing this to(1.7c) yields the time harmonic version of the equation of continuity

div Je = i.

With the vector identity curl curl = + div equations (1.10) and (1.11) can be writtenas

E + k2 E = i0Je + 10

, (1.12)

H + k2 H = curl Je . (1.13)Let us consider the magnetic field first and introduce the magnetic Hertz potential: Theequation div H= 0 implies the existence of a vector potential A with H = curl A. Thus(1.13) takes the form

curl(A + k2 A) = curl Jeand we obtain

A + k2 A = Je + (1.14)for some scalar field . On the other hand, ifA and satisfy (1.14) then

H = curl A and E = 1i0

(curl H Je) = i0A 1i0

(div A )

satisfies the Maxwell system (1.7a)(1.7d).

Analogously, we can introduce electric Hertz potentials ifJe = 0. Because div E= 0 thereexists a vector potential A with E= curl A. Substituting this into (1.12) yields

curl(A + k2 A) = 0

and we obtainA + k2 A = (1.15)

for some scalar field . On the other hand, ifA and satisfy (1.15) then

E = curl A and H = 1

i0curl E = i0A + 1

i0(div A )

satifies the Maxwell system (1.7a)(1.7d). In any case we end up with an inhomogeneous

vector Helmholtz equation.As a particular example we may take a magnetic Hertz vector A of the form A(x) =u(x) zwith a scalar solution u of the twodimensional Helmholtz equation and the unit vectorz= (0, 0, 1) R3. Then

H = curl(uz) =

u

x2, u

x1, 0

,

E = i0z + 1

i0 (u/x3) .


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Ifuis independent ofx3thenEhas only ax3component, and the vector Helmholtz equationfor A reduces to a scalar Helmholtz equation for the potential u. The situation that E hasonly one non zero component is called electric mode, E-mode, or transverse-magnetic mode,TM-mode. Analogously, also the H-mode or TE-mode is consdidered ifHconsists of onlyone non zero component satisfying the scalar Helmholtz equation.

1.4 Boundary and Radiation Conditions

Maxwells equations hold only in regions with smooth parameter functions r,rand. If weconsider a situation in which a surface Sseparates two homogeneous media from each other,the constitutive parameters , and are no longer continuous but piecewise continuouswith finite jumps on S. While on both sides ofS Maxwells equations (1.7a)(1.7d) hold,

the presence of these jumps implies that the fields satisfy certain conditions on the surface.

To derive the mathematical form of this behaviour, the transmission and boundary condi-tions, we apply the law of induction (1.2) to a narrow rectangle-like surface R, containingthe normaln to the surface Sand whose long sides C+ andC are parallel to Sand are onthe opposite sides of it, see the following figure.

When we let the height of the narrow sides, AA and BB , approach zero then C+ and Capproach a curve C on S, the surface integral t

RB ds will vanish in the limit because

the field remains finite. Note, that the normal is the normal to R lying in the tangentialplane ofS. Hence, the line integrals

CE+ d and

CE d must be equal. Since the


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1.4. BOUNDARY AND RADIATION CONDITIONS 19

curve C is arbitrary the integrands E+ and E coincide on every arc C; that is,n E+ n E = 0 on S . (1.16)

A similar argument holds for the magnetic field in (1.1) if the current distribution J =

E+Je remains finite. In this case, the same arguments lead to the boundary condition

n H+ n H = 0 on S . (1.17)If, however, the external current distribution is a surface current; that is, ifJe is of the formJe(x + n(x)) = Js(x)() for small andx Sand with tangential surface field Js andis finite, then the surface integral

RJe ds will tend to

CJs d, and so the boundary

condition isn H+ n H = Js on S . (1.18)

We will call (1.16) and (1.17) or (1.18) transmission boundary conditions.

A special and very important case is that of a perfectly conducting mediumwith boundaryS. Such a medium is characterized by the fact that the electric field vanishes inside thismedium, and (1.16) reduces to

n E = 0 onS .In realistic situations, of course, the exact form of this equation never occurs. Nevertheless,it is a common model for the case of a very large conductivity.

Another important case is the impedance- or Leontovich boundary condition

n H = n (E n) onS

for some non-negative impedance function which, under appropriate conditions, may beconsidered as an approximation of the transmission conditions. Of course these boundaryconditions occur also in the time harmonic case for the fields which we denote by capitalLatin letters.

The situation is different for the normal components E n and H n. We consider GaussElectric and Magnetic Laws and choose to be a box which is separated by a surface S intotwo parts 1 and 2. We apply (1.3) first to all of and then to 1 and 2 separately. Theaddition of the last two formulas and the comparison with the first yields that the normalcomponent D nhas to be continuous. Analogously we obtain B nto be continuous at S,if we consider (1.4). With the constitutive equations one gets

n (r,1E1 r,2E2) = 0 onS and n (r,1H1 r,2H2) = 0 on S .

Conclusion 1.6 The normal components ofE and/orH are not continuous at interfaceswherec and/orr have jumps.

Finally, we specify the boundary conditions to the E- and H-modes defined above (see page18). We assume that the surface S is an infinite cylinder in x3direction with constantcross section. Furthermore, we assume that the volume current density Jvanishes near the


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boundary Sand that the surface current densities take the form Js = jsz for the E-modeand Js = js

z for the H-mode. We use the notation [v] := v|+ v| for the jump of

the function v at the boundary. Then in the E-mode we obtain the transmission boundarycondition

[u] = 0 , ( i)u

= js onS ,and in the H-mode we get

[u] =js ,

u

= 0 on S .

In scattering theory the solutions live in the unbounded exterior of a bounded domain D.In these situations the behavior of electromagnetic fields at infinity has to be taken intoaccount. As an example we consider the fields in the case of a Hertz-dipolat the origin.

Example 1.7 In the case of plane waves (see example 1.4) the wave fronts are planes. Now

we look for waves with spherical wave fronts. A direct computation shows that

(x) = 1

4

eik|x|

|x| , x R3 \ {0} ,

is a solution of the Helmholtz equation in R3 \ {0}. It is called the fundamental solutionofthe Helmholtz equation at the origin (see Definition 3.1) which is essential as we will see inChapter 3. Physically it can be interpretated as the solution generated by a point source atthe origin similar to the electrostatic case (see Example 1.2).

Furthermore, defining

H(x) = curl(x, 0)p = 14

curleik|x||x| p= 14eik

|x

||x| p

for some a constant vector p C3, we obtain from equation (6.5) and the fact that (, 0)solves the Helmholtz equation,

curl H(x) = div(x, 0)p (x, 0)p = div(x, 0)p +k2 (x, 0)pand thus by (6.3) curl curl H=k2 H. ThereforeHand E= i0 curl Hconstitute a solutionof the time harmonic Maxwell equations in vacuum (see Lemma 1.3).

Computing the gradient

(x, 0) = 14

ik 1|x|

eik|x||x|

x|x| = ik (x, 0)

x|x|

eik|x|

4|x|2x|x| (1.19)

we obtain by recalling the frequency > 0 and wave number k =

00 that the time-dependent magnetic field has the form

H(x, t) = H(x) eit = 1

4

x

|x|p

ik

|x| 1

|x|2

ei(k|x|t)

of theHertz-dipolcentered at the origin with dipol moment p eit.


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1.4. BOUNDARY AND RADIATION CONDITIONS 21

We observe that all of the functions E,H, of this example decay as 1/|x| as|x| tendsto infinity. This asymptotic is not sufficient in describing a scattered field, since we alsoobtain solutions of the Helmholtz equation and the Maxwell equations, respectively, withthis asymptotic behaviour if we replace in the last example the wave number k byk.

To distinguish these fields we must consider the factor e

i(k

|x

|t)

ande

i(

k

|x

|t)

. In the firstcase we obtain outgoing wave fronts while in the second case where the wave number isnegative we obtain ingoing wave fronts. For the scattering of electromagnetic waves thescattered waves have to be outgoing waves. Thus, it is required to exclude the second onesby additional conditions which are called radiation conditions.

From (1.19) we observe that the two cases for the Helmholtz equation can be distinguishedby subtractingik. This motivates a general characterization of radiating solutions u of theHelmholtz equation by the Sommerfeld radiation condition, which is

lim

|x

|

|x|x

|x

| u iku = 0 .

Similarly, we find a condition for radiating electromagnetic fields from the behavior of theHertz-dipol. Computing

E(x) = i

0curl H(x)

= i

40

k2

x

|x|p

x|x| +

1

|x|2 ik

|x|

3x p|x|

x

|x|p

eik|x|

|x|we conclude

lim|x|0

0E(x) x+ 0|x|H(x) = lim|x|0 04 2 + 1ik|x| x|x|peik|x||x| = 0 .Analogously, we obtain lim|x|0

0H(x)x0|x|E(x)

= 0. We note that both

conditions are not satisfied if we replace k byk. The radition condition

lim|x|

0 H(x) x |x|0 E(x)

= 0 (1.20a)

orlim|x|

0 E(x) x +|x|0 H(x)

= 0 , (1.20b)

are called Silver-Muller radiation condition for time harmonic electromagnetic fields.

Later we will show that these radiation conditions are sufficient for the existence of uniquesolutions of scattering problems. Additionally from the representation theorem in Chapter 3we will prove equivalent formulations and the close relationship of the Sommerfeld and theSilver-Muller radiation condition. Additionally, the limiting absorption principle will giveanother justification for this definition of radiating solutions.

Finally we discuss the energy of scattered waves. In general the energy density of electro-magnetic fields in a linear medium is given by 1

2(E D + H B). Thus, from Maxwells


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equations, the identity (6.9), and the divergence theorem in a region we obtain

t

1

2

E D +H B dx

=

E (curlHJ) H curlEdx

= div(EH) E Jdx=

(EH) ds

E Jdx .

This conservation law for the energy of electromagnetic fields is called Poyntings Theorem.Physically, the right hand side is read as the sum of the energy flux through the surface given by the Poynting vector, EH, and the electrical work of the fields with the electricalpower J E.Let us consider the Poynting theorem in case of time harmonic fields with frequency >0 invacuum, i.e. J = 0, = 0, = 0. Substituting E(x, t) = Re E(x)eit= 12E(x)eit +E(x)eit and H(x, t) = 12H(x)eit +H(x)eit into the left hand side of Poyntingstheorem lead to

t

1

2

0 E(x, t)2 +0H(x, t)

2 dx

=

t

1

8

0

Re (E(x)2e2it) + 2|E(x)|2 +0 Re (H(x)2e2it) + 2|H|2 dx

= i4

0 Re (E(x)2e2it) +0 Re (H(x)2e2it) dx .

On the other hand we compute the flux term as

(EH) ds = 12

Re

E H ds + 12

Re

E He2it ds .By the Poynting Theorem the two integrals coincide for all t. Thus we conclude for the timeindependent term

Re

(E H) ds = 0 .

The vector fieldE His called thecomplex Poynting vector . If has the form = R =B(0, R) \ D for a bounded domain D with sufficiently smooth boundary contained in theballB(0, R) of radius R we observe conservation of energy in the form

Re

|x|=R

(E H) ds = Re

D

(E H) ds .

Furthermore, from this identity we obtain

2

00 Re

D

(E H) ds

=

|x|=R

0 |E|2 +0 |H |2 ds|x|=R

|0 H 0 E|2 dx .


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1.5. THE REFERENCE PROBLEMS 23

If we additionally assume radiating fields, the Silver-Muller condition implies

limR

|x|=R

0 H 0 E2 ds = 0which yields boundedness of |x|=R |E|2 ds and, analogously, |x|=R |H|2 ds as R tends toinfinity.

1.5 The Reference Problems

After this introduction into the mathematical description of electromagnetic waves the aim ofthe textbook becomes more obvious. In general we can distinguish (at least) three commonapproaches which lead to existence results of boundary value problems for linear partialdifferential equations: expanding solutions into spherical wave functions by separation of

variables techniques, reformulation and treatment of a given boundary value problem interms of integral equations in Banach spaces of functions, and the reformulation of theboundary value problem as a variational equation in Hilbert spaces of functions. It is theaim of this monograph to discuss all of these common methods in the case of time harmonicMaxwells equations.

We already observed the close connection of the Maxwell system with the scalar Helmholtzequation. Therefore, before we treat the more complicated situation of the Maxwell systemwe investigate the methods for this scalar elliptic partial differential equation in detail.Thus, the structure of all following chapters will be similar: we first discuss the technique inthe case of the Helmholtz equation and then we extend it to boundary value problems forelectromagnetic fields.

As we have mentioned already in the preface it is not our aim to present a collection ofallor at least some interesting boundary value problems. Instead, we present the ideas for twoclassical reference problems only which we introduce next.

Scattering by a perfect conductor

The first one is the scattering of electromagnetic waves in vacuum by a perfect conductor:Given a bounded region Dand some solution Einc and Hinc of the unperturbed time harmonicMaxwell system

curl Einc i0Hinc = 0 in R3 , curl Hinc +i0Einc = 0 in R3 ,

the problem is to determineE, Hof the Maxwell system

curl E i0H= 0 in R3 \ D , curl H+ i0E= 0 in R3 \ D ,

such that Esatisfies the boundary condition

E= 0 on D


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and both, Eand H, can be decomposed into E=Es +Einc and H=Hs +Hinc in R3 \ Dwith some scattered field Es, Hs which satisfy the Silver-Muller radiation condition

lim|x| |

x

|

0 Hs(x)

x

|x|

0 Es(x) = 0

lim|x|

|x|

0 Es(x) x|x| +

0 H

s(x)

= 0

uniformly with respect to all directions x/|x|.

D E= 0

Einc,Hinc

Es, Hs

A perfectly conducting cavity

For the second reference problem we consider D R3 to be a bounded domain with suffi-ciently smooth boundaryDand exterior unit normal vector (x) atx D. Furthermore,functions,, and are given on D and some source Je: D C3. Then the problem is todetermine a solution (E, H) of the time harmonic Maxwell system

curl E iH = 0 inD , (1.21a)curl H+ (i )E = Je inD , (1.21b)

with the boundary condition

E= 0 on D . (1.21c)


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1.5. THE REFERENCE PROBLEMS 25

-

D

, ,

E= 0

Of course, for general , , L(R3) we have to give first a correct interpretation of thedifferential equations by a so called weak formulation, which will be presented in detail inChapter 4.

Throughout, we will have these two reference problems in mind for the whole presentation.We will start with constant electric parameters inside or outside a ball. Then we can expectradially symmetric solutions which can be computed by separation of variables in termsof spherical coordinates. This approach will be worked out in Chapter 2. It will lead usto a better understanding of electromagnetic waves from its expansion into spherical wavefunctions. In particular, we can see explicitely in which way the boundary data are attained.

In Chapter 3 we will use the fundamental solution of the scalar Helmholtz equation torepresent electromagnetic waves by integrals over the boundary D of the region D. Theseintegral representations by boundary potentials are the basis for deriving integral equationson D. We prefer to choose the indirect approach; that is, to search for the solution interms of potentials with densities which are determined by the boundary data through aboundary integral equation. To solve this we will apply the RieszFredholm theory. In thisway we are to prove the existence of unique solutions of the first reference problem for anyperfectly conducting smooth scattering obstacle.

The cavity problem will be investigated in Chapter 4. The treatment by a variationalapproach requires the introduction of suitable Sobolev spaces. The Helmholtz decomposition

makes it possible to transfer the ideas of the simpler scalar Helmholtz equation to the Maxwellsystem. The LaxMilgram theorem in Hilbert spaces is the essential tool to establish anexistence result for the second reference problem.

The reason for including Chapter 5 into this monograph is different than for Chapters 3 and4. While for the latter ones our motivation was the teaching aspect (these chapters arosefrom graduate courses) the motivation for Chapter 5 is that we were not able to find sucha thorough presentation of boundary integral methods for Maxwells equations on Lipschitzdomains in any textbook. Lipschitz domains, in particular polyedral domains, play obviouslyan important role in praxis. We were encouraged by our collegues from the numerical analysis


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group to include this chapter to have a reference for further studies. The integral equationmethods themselves are not much different from the classical ones on smooth boundaries.The mapping properties of the boundary operators, however, require a detailed study ofSobolev spaces on Lipschitz boundaries. In this Chapter 5 we use and combine methods ofChapters 3 and 4. Therefore, this chapter can not be studied independently of Chapters 3

and 4.


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Chapter 2

Expansion into Wave Functions

This chapter, which is totally independent of the remaining parts of this monograph,1 studiesthe fact that the solutions of the scalar Helmholtz equation or the vectorial Maxwell systemin balls can be expanded into certain special wave functions. We begin by expressing theLaplacian in spherical coordinates and search for solutions of the scalar Laplace equationor Helmholtz equation by separation of the (spherical) variables. It will turn out thatthe spherical parts are eigensolutions of the LaplaceBeltrami operator while the radialpart solves an equation of Euler type for the Laplace equation and the spherical Besseldifferential equation for the case of the Helmholtz equation. The solutions of these differentialequations will lead to spherical harmonics and spherical Bessel and Hankel functions. Wewill investigate these special functions in detail and derive many important properties. Themain goal is to express the solutions of the interior and exterior boundary value problemsas series of these wave functions. As always in this monograph we first present the analysis

for the scalar case of the Laplace equation and the Helmholtz equation before we considerthe more complicated case of Maxwells equations.

2.1 Separation in Spherical Coordinates

The starting point of our investigation is the boundary value problem inside (or outside)the simple geometry of a ball in the case of a homogeneous medium. We are interested in

solutionsu of the Laplace equation or Helmholtz equation and later of the time-harmonicMaxwell system which can be separated into a radial part v : R>0 C and a sphericalpart K :S2 C; that is,

u(x) = v(r) K(x) , r >0 , x S2 ,

1except of the proof of a radiation condition

27


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28 CHAPTER 2. EXPANSION INTO WAVE FUNCTIONS

whereS2 = {x R3 : |x| = 1} denotes the unit sphere in R3. Here,r = |x| =

x21+x22+x

23

and x= x|x| S2 denote the spherical coordinates; that is,

x = r cos sin r sin sin

r cos with r R>0, [0, 2), [0, ] ,and x= (cos sin , sin sin , cos ).

In the previous chapter we have seen already the importance of the differential operator in the modelling of electromagnetic waves. It occurs directly in the stationary cases and alsoin the differential equations for the magnetic and electric Hertz potentials. Also, solutionsof the full Maxwell system solve the vector Helmholtz equation in particular cases. Thus therepresentation of the Laplacian in spherical polar coordinates is of essential importance.

= 1

r2

r r2

r + 1

r2 sin2

2

2 +

1

r2 sin

sin

(2.1)=

2

r2 +

2

r

r +

1

r2 sin2

2

2 +

1

r2 sin

sin

.

Definition 2.1 Functionsu: R which satisfy theLaplace equation u= 0 in arecalled harmonic functions in . Complex valued functions are harmonic if their real andimaginary parts are harmonic.

The Laplace operator separates into a radial part, which is 1r2

r (r2

r ) = 2

r2 + 2r

r , and the

spherical part which is called the Laplace-Beltrami operator, a differential operator on theunit sphere.

Definition 2.2 The differential operatorS2 :C2(S2) C(S2) with representation

S2 = 1

sin2

2

2+

1

sin

sin

.

in spherical coordinates is called spherical Laplace-Beltrami operator onS2.

With the notation of the Beltrami operator and Dr = r

we obtain

= D2r+2

rDr+

1

r2S2 =

1

r2Dr

r2Dr

+

1

r2S2.

Assuming that a potential or a component of a time-harmonic electric or magnetic field canbe separated as u(x) = u(rx) = v(r)K(x) with r =|x| and x= x/|x| S2, a substitutioninto the Helmholtz equation u+k2u= 0 for some k C leads to

0 = u(rx) +k2u(rx) = v(r) K(x) +2

rv(r) K(x) +

1

r2v(r) S2K(x) +k

2v(r) K(x) .


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2.1. SEPARATION IN SPHERICAL COORDINATES 29

Thus we obtain, providedv(r)K(x) = 0,

v(r) + 2r v(r)

v(r) +

1

r2S2K(x)

K(x) +k2 = 0 . (2.2)

If there is a nontrivial solution of the supposed form, it follows the existence of a constant C such that v satisfies the ordinary differential equation

r2v(r) + 2r v(r) +

k2r2 +

v(r) = 0 for r >0 , (2.3)

andKsolves the partial differential equation

S2K(x) = K(x) on S2 . (2.4)

In the functional analytic language the previous equation describes the problem to determineeigenfunctionsKand corresponding eigenvaluesof the spherical Laplace-Beltrami operatorS2. This operator is selfadjoint and non-positive with respect to the L2(S2)norm, seeExercise 2.2. Especially, we observe that R


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with Fourier coefficients

fm = 1

2

f(s)eims ds , m Z .

The convergence of the series must be understood in the L2-sense, that is,

f(t) M

m=Nfm e

imt

2

dt 0 , N, M .

Thus by the previous result we conclude that the possible functions {y1,m : m Z} constitutea complete orthonormal system in L2(, ).Next we consider the dependence on . Using = m2 turns (2.5) into

sin sin y2() m2 + sin2 y2() = 0fory2. With the substitutionz= cos (1, 1] andw(z) =y2() we arrive at theassociatedLegendre differential equation

(1 z2)w(z) + m2

1 z2

w(z) = 0 . (2.6)

An investigation of this equation will be the main task of the next section leading to theso called Legendre polynomials and to a complete orthonormal system of spherical surfaceharmonics. In particular, we will see that only for =n(n+ 1) for n N0 there existsmooth solutions.

Obviously, the radial part of separated solutions given by the ordinary differential equation(2.3) depends on the wave number k. For k = 0 and =n(n+ 1) we obtain the Eulerequation

r2v(r) + 2r v(r) n(n+ 1) v(r) = 0 . (2.7)By the ansatz v(r) =r we compute (+ 1) = n(n+ 1), which leads to the fundamentalset of solutions

v1,n(r) =rn and v2,n(r) =r

(n+1) .

If we are interested in nonsingular solutions of the Laplace equation inside a ball we haveto choose v1,n(r) =r

n. In the exterior of a ballv2,n(r) =r(n+1) will lead to solutions which

decay as|x| tends to infinity.In the case of k= 0 we rewrite the differential equation (2.3) for =n(n+ 1) by thesubstitutionz= kr and v(r) = v(kr) into the spherical Bessel differential equation

z2 v(z) + 2zv(z) +

z2 n(n+ 1) v(z) = 0 . (2.8)In Section 2.5 we will discuss the Bessel functions, which solve this differential equation.Combining spherical surface harmonics and the corresponding Bessel functions will lead tosolutions of the Helmholtz equation and further on also of the Maxwell system.


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2.2. LEGENDRE POLYNOMIALS 31

2.2 Legendre Polynomials

Let us consider the case of harmonic functions u, that is u = 0. In the previous sectionwe saw already that a separation in spherical coordinates leads to solutions of the formu(x) = rn(c1 e

im +c2 ei m ) w(cos ) where w is determined by the associated Legendre

differential equation (2.6) with R


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where k0is chosen such that 2k0(2k0+1)+ >0 and

(2k0+1)/2

/

(2k0+1)(k0+1) 1/2.

Then a2k does not change its sign anymore for k k0. Taking the logarithm of (2.10) forj = 2k yields

ln|a2(k+1)

| = ln 2k(2k+ 1) +(2k+ 1)(2k+ 2) + ln |a2k| = ln1

(2k+ 1) /2(2k+ 1)(k+ 1) + ln |a2k| .

Now we use the elementary estimate ln(1 u) u 2u2 for 0 u 1/2 which yields

ln |a2(k+1)| (2k+ 1) /2(2k+ 1)(k+ 1)

(2k+ 1) /2(2k+ 1)(k+ 1)

2+ ln |a2k|

1k+ 1

ck2

+ ln |a2k|

for some c >0. Therefore, we arrive at the following estimate for ln |a2k|:

ln |a2k| k1

j=k0

1j+ 1

c k1

j=k0

1j2

+ ln |a2k0| fork k0 .

Usingk1

j=k0

1

j+ 1

k1j=k0

j+1j

dt

t =

kk0

dt

t = ln k ln k0

yields

ln

|a2k

| ln k + ln k0

c

j=k01

j

2 + ln

|a2k0

| = c

and thus

|a2k| expck

for allk k0 .We note that a2k/a2k0 is positive for all k k0 and thus

w1(x)

a2k0 a0

a2k0+

k01k=1

a2ka2k0

expck |a2k0|

x2k +

exp c

|a2k0|

k=1

1

kx2k c exp c|a2k0|

ln(1 x2) .

From this we observe that w1(x) +as x 1 or w1(x) as x 1 dependingon the sign ofa2k0. By the same arguments one shows for positive a2k0+1 thatw2(x) +as x +1 and w2(x) as x 1. For negativea2k0+1 the roles of + andhave to be interchanged. In any case, the sum w(x) = w1(x) +w2(x) is not bounded on[1, 1] which contradicts our requirement on the solution of (2.9). Therefore, this case cannot happen.

Case 2: There ism Nwithaj = 0 for all j m; that is, the series reduces to a finite sum.Let mbe the smallest number with this property. From the recursion formula we concludethat =n(n+ 1) for n= m 2 and, furthermore, that a0 = 0 ifn is odd and a1 = 0 if


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n is even. In particular,w is a polynomial of degree n. We can normalizew byw(1) = 1because w(1) = 0. Indeed, ifw(1) = 0 the differential equation

(1 x2) w(x) 2x w(x) + n(n+ 1) w(x) = 0

for the polynomialw would implyw (1) = 0. Differentiating the differential equation wouldyieldw(k)(1) = 0 for all k N, a contradiction to w= 0.

In this proof of the theorem we have proven more than stated. We collect this as a corollary.

Corollary 2.4 The Legendre polynomialsPn(x) =n

j=0 ajxj have the properties

(a) Pn is even for evenn and odd for oddn.

(b) aj+2= j(j+ 1) n(n+ 1)

(j+ 1)(j+ 2)

aj, j= 0, 1, . . . , n

2.

(c)

+11

Pn(x)Pm(x) dx= 0 forn =m.

Proof: Only part (c) has to be shown. We multiply the differential equation (2.9) forPnbyPm(x), the differential equation (2.9) for Pm byPn(x), take the difference and integrate.This yields

0 =

+1

1

Pm(x) ddx (1 x2) Pn(x) Pn(x) ddx (1 x2) Pm(x) dx+

n(n+ 1) m(m+ 1) +11

Pn(x) Pm(x) dx .

The first integral vanishes by partial integration. This proves part (c).

Before we return to the Laplace equation we prove some further results for the Legendrepolynomials.

Lemma 2.5 For the Legendre polynomials it holds

max1x1

|Pn(x)| = 1 for alln= 0, 1, 2, . . . .

Proof: For fixedn N we define the function

(x) := Pn(x)2 +

1 x2n(n+ 1)

Pn(x)2 for allx [1, +1] .


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We differentiate and have

(x) = 2 Pn(x)

Pn(x) + 1 x2n(n+ 1)

Pn (x) x

n(n+ 1)Pn(x)

=

2 Pn(x)n(n+ 1) n(n+ 1) Pn(x) + ddx{(1 x2) Pn(x)} +x Pn(x) = 2 x P

n(x)

2

n(n+ 1) . = 0 by the differential equation

Therefore >0 on (0, 1] and


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Analogously, we multiply the second part of (2.11) by xj , integrate, and apply partial inte-grationn-times. This yields

Bj = n(n+ 1)

1

1xj

dn

dxn(x2 1)n dx = n(n+ 1) (1)n n!

1

1(x2 1)n dx ifj =n

and zero ifj < n. This proves that the polynomial

d

dx

(1 x2) d

n+1

dxn+1(x2 1)n

+ n(n+ 1)

dn

dxn(x2 1)n

of degree n is orthogonal in L2(1, 1) to all polynomials of degree at most n and, therefore,has to vanish. Furthermore,

dn

dxn(x2 1)n

x=1

= dn

dxn

(x+ 1)n (x 1)nx=1

=n

k=0

nk

dkdxk

(x+ 1)n

x=1

dnk

dxnk(x 1)n

x=1

.

From dnk

dxnk(x 1)n

x=1

= 0 for k 1 we conclude

dn

dxn(x2 1)n

x=1

=

n

0

2n

dn

dxn(x 1)n

x=1

= 2n n!

which proves the theorem.

As a first application of the formula of Rodrigues we can compute the norm ofPninL2(1, 1).Theorem 2.7 The Legendre polynomials satisfy

(a)

+11

Pn(x)2 dx =

2

2n+ 1 for alln= 0, 1, 2, . . ..

(b)

+11

x Pn(x) Pn+1(x) dx = 2(n+ 1)

(2n+ 1)(2n+ 3) for alln= 0, 1, 2, . . ..

Proof: (a) We use the representation ofPn by Rodrigues and n partial integrations,

+11

dn

dxn(x2 1)n d

n

dxn(x2 1)n dx = (1)n

+11

d2n

dx2n

(x2 1)n (x2 1)n dx

= (1)n (2n)!+11

(x2 1)n dx = (2n)!+1

1

(1 x2)n dx .


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It remains to compute In:=

+11

(1 x2)n dx.

We claim: In = 2

2n(2n

2)

2

(2n+ 1)(2n 1) 1 = 2 4n (n!)

2

(2n+ 1)! for alln N

.

The assertion is true for n = 1. Let it be true for n 1,n 2. Then

In = In1 +11

x2 (1 x2)n1 dx = In1 ++11

x d

dx

1

2n(1 x2)n

dx

= In1 12n

In and thus In = 2n

2n+ 1In1 .

This proves the representation ofIn. We arrive at

+11

Pn(x)2 dx =

1

(2n n!)2(2n)! 2 4n (n!)

2

(2n+ 1)! =

2

2n+ 1.

(b) This is proven quite similarily:

+11

x dn

dxn(x2 1)ndx

n+1

dxn+1(x2 1)n+1 dx

= (1)n+1+11

(x2 1)n+1 dn+1dxn+1

x

dn

dxn(x2 1)n

dx

= (1)n+1+11

(x2 1)n+1

x d2n+1dx2n+1 (x2 1)n

=0

+ (n+ 1) d2n

dx2n(x2 1)n

=(2n)!

dx

= (n+ 1) (1)n+1(2n)! In+1 = 2(n+ 1)(2n)! (2n+ 2)(2n)(2n 2) 2(2n+ 3)(2n+ 1)(2n

1)

1

= (n+ 1)2(2nn!)(2n+1(n+ 1)!)

(2n+ 1)(2n+ 3) ,

which yields the assertion.

The formula of Rodrigues is also useful in proving recursion formulas for the Legendrepolynomials. We prove only some of these formulas. For the remaining parts we referto the exercises.


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Theorem 2.8 For allx R andn N, n 0, we have

(a) Pn+1(x) Pn1(x) = ( 2n+ 1) Pn(x),(b) (n+ 1)Pn+1(x) = ( 2n+ 1)x Pn(x)

nPn

1(x),

(c) Pn(x) = nPn1(x) + xP

n1(x),

(d) xPn(x) = nPn(x) + P

n1(x),

(e) (1 x2)Pn(x) = (n+ 1)

x Pn(x) Pn+1(x)

= n x Pn(x) Pn1(x),(f ) Pn+1(x) + P

n1(x) = Pn(x) + 2x P

n(x),

(g) (2n+ 1)(1 x2)Pn(x) = n(n+ 1)

Pn1(x) Pn+1(x)

,

(h) n Pn+1(x)

(2n+ 1)x Pn(x) + (n+ 1)P

n

1(x) = 0.

In these formulas we have setP1= 0.

Proof: (a) The formula is obvious for n = 0. Let now n 1. We calculate, using theformula of Rodrigues,

Pn+1(x) = 1

2n+1 (n+ 1)!

dn+2

dxn+2(x2 1)n+1

= 1

2n+1 (n+ 1)!

dn

dxn 2(n+ 1) d

dxx (x2 1)n

= 1

2n n!

dn

dxn

(x2 1)n + 2n x2(x2 1)n1= Pn(x) +

1

2n1 (n 1)!dn

dxn

(x2 1)n + (x2 1)n1= Pn(x) + 2n Pn(x) + P

n1(x)

which proves formula (a).

(b) The orthogonality of the system{Pn : n = 0, 1, 2, . . .} implies its linear independence.Therefore,{P0, . . . , P n}forms a basis of the spacePn of all polynomials of degree n. Thisyields existence ofn, n R and qn3 Pn3 such that

Pn+1(x) = nxPn(x) + nPn1(x) + qn3 .

The orthogonality condition implies that

+11

qn3 Pn+1 dx= 0 ,

+11

qn3 Pn1 dx= 0 ,

+11

x qn3(x) Pn(x) dx= 0 ,


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thus+11

qn3(x)2dx = 0 ,

and therefore qn

3

0. From 1 = Pn+1(1) = nPn(1) +nPn

1(1) = n+ n we conclude

thatPn+1(x) = nx Pn(x) + ( 1 n)Pn1(x) .

We determine n by Theorem 2.7:

0 =

+11

Pn1(x) Pn+1(x) dx

= n

+1

1x Pn1(x) Pn(x) dx + (1 n)

+1

1Pn1(x)2 dx

= n

2n

(2n 1)(2n+ 1) 2

2n 1

+ 2

2n 1,

thus n=2n+ 1

n+ 1. This proves part (b).

(c) The definition ofPn yields

Pn(x) = 1

2nn!

dn+1

dxn+1(x2 1)n

= 12n1(n 1)! d

n

dxn[x(x2 1)n1]

= x

2n1(n 1)!dn

dxn(x2 1)n1 + n

2n1(n 1)!dn1

dxn1(x2 1)n1

= x Pn1(x) + n Pn1(x) .

(d) Differentiation of (b) and multiplication of (c) for n + 1 instead ofnby n+ 1 yields

(n+ 1) Pn+1(x) = (2n+ 1)x P

n(x) + ( 2n+ 1)Pn(x) nPn1(x),(n+ 1) Pn+1(x) = (n+ 1)

2Pn(x) + (n+ 1)x P

n(x) ,

thus by subtraction

0 =

(2n+ 1) (n+ 1)2Pn(x) + (2n+ 1) (n+ 1)x Pn(x) n Pn1(x)which yields (d).For the proofs of (e)(h) we refer to the exercises.

Now we go back to the associated Legendre differential equation (2.6) and determine solutionsinC2(1, +1) C[1, +1] for m = 0.


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Theorem 2.9 The functions

Pmn (x) = ( 1 x2)m/2 dm

dxmPn(x) , 1< x


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2.3 Expansion into Spherical Harmonics

Now we return to the Laplace equation u= 0 and collect our arguments. We have shownthat for n N and 0 m n the functions

hmn(r,,) = rn Pmn (cos ) ei m

are harmonic functions in all ofR3. Analogously, the functions

v(r,,) = rn1 Pmn (cos ) eim

are harmonic in R3 \ {0}and decay at infinity.It is not so obvious that the functions hmn are polynomials.

Theorem 2.10 The functionshmn(r,,) = rnPmn (cos ) e

im with0mn are homo-geneous polynomials of degreen. The latter property means thath

mn(x) =

n

hnm(x) for all

x Rn and R.

Proof: First we consider the case m = 0, that is the functions h0n(r,,) = rn Pn(cos ).

Letnbe even, that is, n = 2 for some . Then

Pn(t) =

j=0

ajt2j , t R ,

for some aj. Fromr = |x| and x3= r cos we write h0n in the form

h0n(x) = |x|n Pn

x3|x|

= |x|2

j=0

a2jx2j3|x|2j =

j=0

a2jx2j3|x|2(j) ,

and this is obviously a polynomial of degree 2 = n. The same arguments hold for oddvalues ofn.

Now we show that also the associated functions; that is, for m >0, are polynomials of degreen. We write

hmn(r,,) = rn Pmn (cos ) e

im = rn sinm dm

dxmPn(x)x=cos e

im .

and set Q = dm

dxmPn. Let again n = 2 be even. ThenQ(t) =

j=0 bjt2jm for some bj .

Furthermore, we use the expressions

cos = x3

r , sin =

1

r

x21+x

22 ,

as well as

cos = 1

r

x1sin

= x1

x21+x22

, sin = x2

x21+x22

.


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2.3. EXPANSION INTO SPHERICAL HARMONICS 41

to obtain

hmn(x) = rnm (x21+x

22)

m/2 Qx3

r

(x1+ix2)m(x21+x

22)

m/2 = rnm Q

x3r

(x1+ix2)

m

=

j=0

bjr2(

j)

x2j

m

3 (x1+ix2)m

which proves the assertion for even n. For oddn one argues analogously. It is clear from thedefinition that the polynomialhmn is homogeneous of degree n.

Definition 2.11 Letn N {0}.(a) Homogeneous harmonic polynomials of degreen are calledspherical harmonics of order

n.

(b) The functionsKn: S2 C, defined as restrictions of spherical harmonics of degreen

to the unit sphere are called spherical surface harmonics of order n.

Therefore, the functions hmn(r,,) = rnP

|m|n (cos )eim are spherical harmonics of order n

forn m n. Spherical harmonics which do not depend on are called zonal. Thus, incase ofm = 0 the functions h0n are zonal spherical harmonics.

For any spherical surface harmonic Kn of order nthe function

Hn(x) = |x|nKn

x

|x|

, x R3 ,

is a homogeneous harmonic polynomial; that is, a spherical harmonic. We immediately have

Lemma 2.12 IfKn denotes a surface spherical harmonic of ordern, the following holds.

(a) Kn(x) = (1)nKn(x) for allx S2.(b)

S2

Kn(x) Km(x) ds(x) = 0 for alln =m.

Proof: Part (a) follows immediately sinceHn is homogeneous (set = 1).(b) With Greens second formula in the region{x R3 : |x|


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Theorem 2.13 The set of spherical harmonics of order n is a vector space of dimension2n+ 1. In particular, there exists a system{Kmn :nmn} of spherical harmonics ofordern such that

S2Kmn K

n ds = m, =

1 form= ,0 form

= ;

that is,{Kmn : n m n} is an orthonormal basis of this vector space.

Proof: Every homogeneous polynomial of degree n is necessarily of the form

Hn(x) =n

j=0

Anj(x1, x2) xj3 x R3 , (2.13)

whereAnj are homogeneous polynomials with respect to (x1, x2) of degree n j. SinceHnis harmonic it follows that

0 = Hn(x) =n

j=0

xj32Anj(x1, x2) +n

j=2

j(j 1) xj23 Anj(x1, x2) ,

where 2 = 2

x21+

2

x22denote the two-dimensional Laplace operator. From 2A0 = 2A1= 0

we conclude that

0 =n2j=0

[2Anj(x1, x2) + (j+ 1)(j+ 2) Anj2(x1, x2)] xj3 ,

and thus by comparing the coefficients

Anj2(x1, x2) = 1(j+ 1)(j+ 2)

2Anj(x1, x2)

for all (x1, x2) R2, j = 0, . . . , n 2; that is (replacen j byj )

Aj2 = 1(n j+ 1)(n j+ 2) 2Aj for j = n, n 1, . . . , 2 . (2.14)

Also, one can reverse the arguments: IfAnand An1are homogeneous polynomials of degreen and n

1, respectively, then all of the functions Aj defined by (2.14) are homogeneous

polynomials of degree j , andHn is a homogeneous harmonic polynomial of order n.

Therefore, the space of all spherical harmonics of order nis isomorphic to the space(An, An1) :An, An1 are homogeneous polynomials of degree nand n 1, resp.

.

From the representationAn(x1, x2) =n

i=0

ai xi1 x

ni2 we note that the dimension of the space of

all homogeneous polynomials of degree n is justn +1. Therefore, the dimension of the spaceof all spherical harmonics of order n is (n+ 1) +n= 2n+ 1. Finally, it is well known that


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any basis of this finite dimensional Euclidian space can be orthogonalized by the method ofSchmidt.

Remark: The set{Kmn :n m n} is not uniquely determined. Indeed, for anyorthogonal matrixA

R33; that is,AA= I, also the set

{Kmn(Ax) :

n

m

n

}is an

orthonormal system. Indeed, the substitution x = Ay yieldsS2

Kmn(x) Kmn(x) ds(x) =

S2

Kmn(Ay) Kmn(Ay) ds(y) = n,nm,m.

Now we can state that the spherical harmonics hmn(x) = hmn(, ) =r

nP|m|

n (cos )eim withn m n, which we have determined by separation, constitute such an orthogonal basisof the space of spherical harmonics of order n.

Theorem 2.14 The functionshm

n(r,,) = rn

P|m

|n (cos )eim

,n m n are sphericalharmonics of ordern. They are mutually orthogonal and, therefore, form an orthogonal basisof the(2n+ 1)-dimensional space of all spherical harmonics of ordern.

Proof: We already know that hmn are spherical harmonics of order n N. Thus thetheorem follows from

|x|


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Now we differentiate the Legendre differential equation (2.9) (m 1)-times (see (2.12) formreplaced bym 1); that is,

(1 x2) dm+1

dxm+1Pn(x) 2m x d

m

dxmPn(x) +

n(n+ 1) m(m 1)

dm1

dxm1Pn(x) = 0 ,

thus, after multiplication by (1 x2)m1,d

dx

(1 x2)m d

m

dxmPn(x)

+

n(n+ 1) m(m 1) (1 x2)m1 dm1

dxm1Pn(x) = 0 .

Substituting this into (2.15) yields

+11

Pmn (x)2 dx =

n(n+1)m(m1) +1

1

Pm1n (x)2 dx = (n+m) (nm+1)

+11

Pm1n (x)2 dx .

This is a recursion formula for+1

1 Pmn (x)2 dx with respect to m and yields the assertion byusing the formula for m = 0.

Now we define the normalized spherical surface harmonics Ymn by

Ymn (, ) :=

(2n+ 1) (n |m|)!

4 (n+ |m|)! P|m|

n (cos ) eim, n m n, n= 0, 1, . . . (2.16)

They form an orthonormal system in L2(S2). We will identify Ymn (x) with Ym

n (, ) forx= (sin cos , sin sin , cos ) S2.

From (2.4) with = n(n+ 1) we remember that Ym

n satisfies the differential equation1

sin

sin

Ymn (, )

+

1

sin2

2Ymn (, )

2 + n(n+ 1) Ymn (, ) = 0 ;

that is,S2Y

mn + n(n+ 1) Y

mn = 0 (2.17)

with the Laplace-Beltrami operator S2 of Definition 2.2. Thus, we see that = n(n + 1)for n N are the eigenvalues of S2 with eigenfunctions Ymn ,|m| n. We mentionedalready (see also Exercise 2.2) that this operator S2 is selfadjoint and non-negative. In thesetting of abstract functional analysis we note that is a densely defined and unboundedoperator from L2(S2) into itself which is selfadjoint. This observation allows the use of

general functional analytic tools to prove, e.g., that the eigenfunctions Ymn : |m| n, n=0, 1, . . .

of form a complete orthonormal system ofL2(S2). We are going to prove this

fact directly starting with the following result which is of independent interest.

Theorem 2.16 For anyf L2(1, 1) the followingFunkHecke Formula holds.S2

f(x y) Ymn (y) ds(y) = n Ymn (x) , x S2 ,

for alln N andm= n , . . . , n wheren= 211 f(t) Pn(t) dt.


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Proof: We keep x S2 fixed and choose an orthogonal matrix A which depends on xsuch that x:= A1x= Ax is the north pole; that is, x= (0, 0, 1). The transformationy= Ay yields

S2

f(x

y) Ymn (y) ds(y) =

S2

f(x

Ay) Ymn (Ay

) ds(y) = S2

f(x

y) Ymn (Ay) ds(y) .

The function Ymn (Ay) is again a spherical surface harmonic of order n, thus

Ymn (Ay) =n

k=nakY

kn(y) , y S2 , (2.18)

whereak =

S2Ymn (Ay) Y

kn (y) ds(y). Using this and polar coordinates

y= (sin cos , sin sin , cos ) S2 yields (note that x y= cos )

S2

f(x y) Ym

n (y) ds(y) =

n

k=n ak

S2

f(x y) Yk

n(y) ds(y)

=n

k=nak

(2n+ 1) (n |k|)!

4 (n+ |k|)!

0

20

f(cos ) P|k|n (cos ) eik d sin d

= a0

2n+ 1

4 2

0

f(cos ) Pn(cos ) sin d = n a0

2n+ 1

4

where we have used the substitution t= cos in the last integral. Now we substitute y = xin (2.18) and have, using Ynk (x) = 0 for k= 0,

Ymn (x) = Ym

n (Ax) =n

k=nakY

kn(x) = a0

2n+ 1

4 Pn(1)

=1

,

thus S2

f(x y) Ymn (y) ds(y) = n Ymn (x)

which proves the theorem.

As a first application of this result we prove the addition formula. A second application willbe the JacobiAnger expansion, see Theorem 2.32.

Theorem 2.17 The Legendre polynomialsPn and the spherical surface harmonics definedin (2.16) satisfy theaddition formula

Pn(x y) = 42n+ 1

nm=n

Ymn (x) Ym

n (y) for allx, y S2 .


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Proof: For fixed y S2 the functionx Pn(x y) is a spherical harmonic of order n and,therefore, has an expansion of the form

Pn(x y) =n

m=n

S2Pn(z y) Ym(z) ds(z) Ymn (x) =

n

m=nn Y

m(y) Ymn (x)

with

n = 2

11

Pn(t) Pn(t) dt = 4

2n+ 1

by Theorem 2.7 where we used the previously proven FunkHecke formula for f=Pn.

We formulate a simple conclusion as a corollary.

Corollary 2.18

(a)n

m=n |Ym

n (x)

|2 =

2n+ 1

4

for allx

S2 andn= 0, 1, . . ..

(b)Ymn (x)

2n+ 1

4 for allx S2 andn= 0, 1, . . ..

Proof: Part (a) follows immediately from the Addition Formula of Theorem 2.17 fory = x.Part (b) follows directly from (a).

After this preparation we are able to prove the completeness of the spherical surface har-monics.

Theorem 2.19The functionsYmn :nmn, nN {0} are complete inL2(S2);that is, every functionf L2(S2) can be expanded into a generalized Fourier series in the

form

f =

n=0

nm=n

(f, Ymn )L2(S2) Ym

n . (2.19a)

The series can also be written as

f(x) = 1

4

n=0

(2n+ 1)

S2

f(y) Pn(y x) ds(y) , x S2 . (2.19b)

The convergence in (2.19a) and (2.19b) has to be understood in theL2sense.

Furthermore, on bounded sets inC1(S2)the series converge even uniformly; that is, for everyM >0 and >0 there existsN0 N, depending only onM and, such that

Nn=0

nm=n

(f, Ymn )L2(S2) Ym

n f

= maxxS2

N

n=0

nm=n

(f, Ymn )L2(S2) Ym

n (x) f(x)


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for all N N0 and all f C1(S2) withf1, = max{f, f} M, and, analo-gously, for (2.19b).

Here, the space C1(S2) consists of those functions f such that (with respect to sphericalcoordinates and) the functiuonsf, f/, and 1sin f/ are continuous and periodic

with respect to .

Proof: First we prove the second part. Therefore, let fC1(S2) withf1,M. Withthe Addition Formula, see Theorem 2.17, we obtain for the partial sum

(SNf)(x) =N

n=0

nm=n

(f, Ymn )L2(S2) Ym

n (x) =

S2

f(y)N

n=0

nm=n

Ymn (x) Ym

n (y) ds(y)

=N

n=0

S2

2n+ 1

4 Pn(x y) f(y) ds(y) .

This yields already the equivalence of (2.19a) and (2.19b). With (2n + 1) Pn= P

n+1 Pn1of Theorem 2.8 (set P1 0) this yields

(SNf)(x) = 1

4

Nn=0

S2

f(y)

Pn+1(x y) Pn1(x y)

ds(y)

= 1

4

S2

f(y)

PN+1(x y) +PN(x y)

ds(y) .

Let againz= (0, 0, 1)be the north pole and choose an orthogonal matrix A (depending onx) such that Ax= z. Then, by the transformation formula,

(SNf)(x) = 1

4

S2

f(Ay)

PN+1(z y) +PN(z y)

ds(y) .

In spherical polar coordinatesy(, ) = (sin cos , sin sin , cos )this is, definingF() =

12

20

f

Ay(, )

d,

(SNf)(x) = 1

2

0 F() P

N+1(cos ) +P

N(cos ) sin d

= 1

2

+11

F(arccos t)

PN+1(t) +P

N(t)

dt

= 1

2F(arccos t)

PN+1(t) +PN(t)

+11

=F(0)

12

+11

d

dtF(arccos t)

PN+1(t) +PN(t)

dt .


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We note that partial integration is allowed because t F(arccos t) is continuously differen-tiable in (1, 1) and continuous in [1, 1]. The value = 0 corresponds to the north poley= z, thus

F(0) = 1

2

2

0

f(Az) d = f(Az) = f(x) ,

and therefore for any (0, 1) we obtain

(SNf)(x) f(x) = 12

+11

d

dtF(arccos t)

PN+1(t) +PN(t)

dt

1

2

1+1

+

11+

+

11

d

dtF(arccos t)

|PN+1(t)| + |PN(t)| dt

= 12

( I1 + I2 + I3) .

We estimate these contributions separately. First we use thatPn(t)1 for all t[1, 1]

andn N (see Lemma 2.5). From ddtarccos t 0 be given. Choose such that I1+ I3 2 . Thendepends only onM and .With this choise ofwe consider I2 and use the inequality of Cauchy-Schwarz.

I2 max1+t1 ddtF(arccos t)

=:c

+11

1 |PN+1(t)| + |PN(t)| dt

2 c PN+1L2+ PNL2 2

2 c 22N+ 1.We estimate c by

c F max1+t1 ddtarccos t

M max1+t1 ddtarccos t

.Now we can choose N0, depending only on and M, such that I2 2 for all N N0.Therefore,|SN(f)(x) f(x)| for all N N0. This proves uniform convergence of theFourier series.


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The first part is proven by an approximation argument. Indeed, we use the general propertyof orthonormal systems thatSNfis the best approximation offin the subspace span{Ymn :|m| n, n= 0, . . . , N }; that is,

SNf fL2(S2) g fL2(S2) for allg span{Ymn : |m| n, n= 0, . . . , N } .Let now f L2(S2) and > 0 be given. Since the spaceC1(S2) is dense in L2(S2) thereexistsh C1(S2) such thath fL2(S2) /2. Therefore,

SNf fL2(S2) SNh fL2(S2) SNh hL2(S2) +h fL2(S2) 4 SNh h +h fL2(S2)

4 SNh h +

2.

SinceSNhconverges uniformly to h we can find N0 N such that the first part is less than/2 for all N N0 which ends the proof.

As a corollary we can prove completeness of the Legendre polynomials.

Corollary 2.20 The polynomials n+ 1/2 Pn : n N0 form a complete orthonormalsystem inL2(1, 1); that is, for anyf L2(1, 1) there holds

f =

n=0

fn Pn with fn =

n+

1

2

11

f(t) Pn(t) dt , n N0 .

Forf C1[1, 1] the series converges uniformly.

Proof: The functiong(x) =f(x3) =f(cos ) can be considered as a function on the spherewhich is independent of, thus it is a zonal function. The expansion (2.19a) yields

f(x3) = n=0

nm=n

amn Ym

n (x)

with

amn = (g, Ym

n )L2(S2) =

2n+ 1

4

(n m)!(n+m)!

0

20

f(cos ) P|m|n (cos ) eim sin dd

=

0 , m = 0 ,

(2n+ 1)

0f(cos ) Pn(cos ) sin d , m= 0 ,

=

0 , m = 0 ,(2n+ 1)

11

f(t) Pn(t) dt , m= 0 .

Thus,

f(x3) =

n=0

4

2n+ 1fn Y

0n (x) =

n=0

fn Pn(x3)

and the proof is complete.


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2.4 Laplaces Equation in the Interior and Exterior of

a Ball

In the previous two sections we constructed explicitely a complete orthonormal system of

functions in L2(S2). They play exactly the role of the normalized exponential functions 12

exp(in) :n Zon the unit circleS1, parametrized byx = (cos , sin ), [0, 2];that is, the classical Fourier expansion functions. It is the aim of this section to expandsolutions of the Laplace equation for balls and solve the corresponding Dirichlet boundaryvalue problems.

Theorem 2.21 Let u C2B(0, R) be harmonic in the ball B(0, R); that is, satisfiesthe Laplace equation u = 0 in B(0, R). Then there exist unique mn C,|m| n,n= 0, 1, 2, . . . with

u(rx) =

n=0

n

m=n

mn

rn Ymn

(x) , 0

r < R , x

S2 . (2.20)

The series converges uniformly with all of its derivatives in every closed ball B[0, R] withR < R.

Proof: For every r (0, R) the function x u(rx) is in C2(S2), and, therefore, can beexpanded into a series by Theorem 2.19; that is,

u(rx) =

n=0

nm=n

umn(r) Ym

n (x) , x S2 .

The coefficients are given by umn(r) = u(r, ), Ymn L2(S2).We show that umn satisfies a differential equation of Euler type. Using the Laplace equa-tion in spherical coordinates for u, the self-adjoint Laplace-Beltrami operator S2, and theeigenvalue equation (2.17), yields

d

dr

r2

d

dr

umn(r) =

S2

r

r2

The Mathematical Theory of Maxwell's Equations

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