Hsiao g wendland w boundary integral equations

and the potential energies of their solution fields, allowing us to considerthe boundary integral equations in the form of variational problems on theboundary manifold of the domain.On th

S S Antman J E Marsden L Sirovich

Advisors

J K Hale P Holmes J Keener

J Keller B J Matkowsky A Mielke

C S Peskin K R Sreenivasan

Volume 164

Applied Mathematical Sciences

1 John: Partial Differential Equations, 4th ed.

2 Sirovich: Techniques of Asymptotic Analysis

3 Hale: Theory of Functional Differential Equations,

2nd ed.

4 Percus: Combinatorial Methods

5 von Mises/Friedrichs: Fluid Dynamics

6 Freiberger/Grenander: A Short Course in

Computational Probability and Statistics.

7 Pipkin: Lectures on Viscoelasticity Theory

8 Giacaglia: Perturbation Methods in Non-linear

Systems

9 Friedrichs: Spectral Theory of Operators in

Hilbert Space

10 Stroud: Numerical Quadrature and Solution of

Ordinary Differential Equations

11 Wolovich: Linear Multivariable Systems

12 Berkovitz: Optimal Control Theory

13 Bluman/Cole: Similarity Methods for Differential

Equations

14 Yoshizawa: Stability Theory and the Existence of

Periodic Solution and Almost Periodic Solutions

15 Braun: Differential Equations and Their

Applications, 4th ed.

16 Lefschetz: Applications of Algebraic Topology

17 Collatz/Wetterling: Optimization Problems

18 Grenander: Pattern Synthesis: Lectures in Pattern

Theory, Vol I

19 Marsden/McCracken: Hopf Bifurcation and Its

Applications

20 Driver: Ordinary and Delay Differential Equations

21 Courant/Friedrichs: Supersonic Flow and Shock

Waves

22 Rouche/Habets/Laloy: Stability Theory by

Liapunov’s Direct Method

23 Lamperti: Stochastic Processes: A Survey of the

26 Kushner/Clark: Stochastic Approximation Methods

for Constrained and Unconstrained Systems

27 de Boor: A Practical Guide to Splines, Revised

Edition

28 Keilson: Markov Chain Models-Rarity and

Exponentiality

29 de Veubeke: A Course in Elasticity

30 Sniatycki: Geometric Quantization and Quantum

Mechanics

31 Reid: Sturmian Theory for Ordinary Differential

32 Meis/Markowitz: Numerical Solution of Partial

Differential Equations

33 Grenander: Regular Structures: Lectures in Pattern

Theory, Vol III

34 Kevorkian/Cole: Perturbation Methods in Applied

Mathematics

35 Carr: Applications of Centre Manifold Theory

36 Bengtsson/Ghil/Källén: Dynamic Meteorology:

Data Assimilation Methods

37 Saperstone: Semidynamical Systems in Infinite

40 Naylor/Sell: Linear Operator Theory in

Engineering and Science

41 Sparrow: The Lorenz Equations: Bifurcations,

Chaos, and Strange Attractors

42 Guckenheimer/Holmes: Nonlinear Oscillations,

Dynamical Systems, and Bifurcations of Vector Fields

43 Ockendon/Taylor: Inviscid Fluid Flows

44 Pazy: Semigroups of Linear Operators and

Applications to Partial Differential Equations

45 Glashoff/Gustafson: Linear Operations and

Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs

46 Wilcox: Scattering Theory for Diffraction Gratings

47 Hale et al.: Dynamics in Infinite Dimensions,

2nd ed.

48 Murray: Asymptotic Analysis

49 Ladyzhenskaya: The Boundary-Value Problems of

Mathematical Physics

50 Wilcox: Sound Propagation in Stratified Fluids

51 Golubitsky/Schaeffer: Bifurcation and Groups in

Bifurcation Theory, Vol I

52 Chipot: Variational Inequalities and Flow in

Porous Media

53 Majda: Compressible Fluid Flow and System of

Conservation Laws in Several Space Variables

54 Wasow: Linear Turning Point Theory

55 Yosida: Operational Calculus: A Theory of

Hyperfunctions

56 Chang/Howes: Nonlinear Singular Perturbation

Phenomena: Theory and Applications

57 Reinhardt: Analysis of Approximation Methods for

Differential and Integral Equations

58 Dwoyer/Hussaini/Voigt (eds): Theoretical

of Technology Pasadena, CA 91125 USA

marsden@cds.caltech.edu

L Sirovich Laboratory of Applied Mathematics Department of Biomathematical Science Mount Sinai School

of Medicine New York, NY 10029-6574 USA

528 Ewing Hall

Institut f r Angewandte Analysis und Numerische Simulation

Mathematics Subject Classification (2001): 47G10-30, 35J55, 45A05, 31A10, 73C02, 76D07

Applied Mathematical Sciences ISSN 0066-5452

ISBN 978-3-540-15284-2

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This book is devoted to the mathematical foundation of boundary integralequations The combination of finite element analysis on the boundary withthese equations has led to very efficient computational tools, the boundaryelement methods (see e g., the authors [139] and Schanz and Steinbach (eds.)[267]) Although we do not deal with the boundary element discretizations

in this book, the material presented here gives the mathematical foundation

of these methods In order to avoid over generalization we have confinedourselves to the treatment of elliptic boundary value problems

The central idea of eliminating the field equations in the domain and ducing boundary value problems to equivalent equations only on the bound-ary requires the knowledge of corresponding fundamental solutions, and thisidea has a long history dating back to the work of Green [107] and Gauss[95, 96] Today the resulting boundary integral equations still serve as amajor tool for the analysis and construction of solutions to boundary valueproblems

re-As is well known, the reduction to equivalent boundary integral equations

is by no means unique, and there are primarily two procedures for this duction, the ‘direct’ and ‘indirect’ approaches The direct procedure is based

re-on Green’s representatire-on formula for solutire-ons of the boundary value lem, whereas the indirect approach rests on an appropriate layer ansatz Inour presentation we concentrate on the direct approach although the corre-sponding analysis and basic properties of the boundary integral operatorsremain the same for the indirect approaches Roughly speaking, one obtainstwo kinds of boundary integral equations with both procedures, those of thefirst kind and those of the second kind

prob-The basic mathematical properties that guarantee existence of solutions

to the boundary integral equations and also stability and convergence sis of corresponding numerical procedures hinge on G˚arding inequalities forthe boundary integral operators on appropriate function spaces In addition,contraction properties allow the application of Carl Neumann’s classical suc-cessive iteration procedure to a class of boundary integral equations of thesecond kind It turns out that these basic features are intimately related

analy-to the variational forms of the underlying elliptic boundary value problems

and the potential energies of their solution fields, allowing us to considerthe boundary integral equations in the form of variational problems on theboundary manifold of the domain.

On the other hand, the Newton potentials as the inverses of the ellipticpartial differential operators are particular pseudodifferential operators onthe domain or in the Euclidean space The boundary potentials (or Poissonoperators) are just Newton potentials of distributions with support on theboundary manifold and the boundary integral operators are their tracesthere Therefore, it is rather natural to consider the boundary integral op-erators as pseudodifferential operators on the boundary manifold Indeed,most of the boundary integral operators in applications can be recast as suchpseudodifferential operators provided that the boundary manifold is smoothenough

With the application of boundary element methods in mind, where strongellipticity is the basic concept for stability, convergence and error analysis ofcorresponding discretization methods for the boundary integral equations,

we are most interested in establishing strong ellipticity in terms of G˚arding’sinequality for the variational formulation as well as strong ellipticity of thepseudodifferential operators generated by the boundary integral equations.The combination of both, namely the variational properties of the ellipticboundary value and transmission problems as well as the strongly ellipticpseudodifferential operators provides us with an efficient means to analyze alarge class of elliptic boundary value problems

This book contains 10 chapters and an appendix For the reader’s benefit,Figure 0.1 gives a sketch of the topics contained in this book Chapters 1through 4 present various examples and background information relevant tothe premises of this book

In Chapter 5, we discuss the variational formulation of boundary gral equations and their connection to the variational solution of associatedboundary value or transmission problems In particular, continuity and co-erciveness properties of a rather large class of boundary integral equationsare obtained, including those discussed in the first and second chapters InChapter 4, we collect basic properties of Sobolev spaces in the domain andtheir traces on the boundary, which are needed for the variational formula-tions in Chapter 5

inte-Chapter 6 presents an introduction to the basic theory of classicalpseudodifferential operators In particular, we present the construction of

a parametrix for elliptic pseudodifferential operators in subdomains of IRn.Moreover, we give an iterative procedure to find Levi functions of arbitraryorder for general elliptic systems of partial differential equations If the fun-damental solution exists then Levi’s method based on Levi functions allowsits construction via an appropriate integral equation

Preface IX

In Chapter 7, we show that every pseudodifferential operator is anHadamard’s finite part integral operator with integrable or nonintegrablekernel plus possibly a differential operator of the same order as that of thepseudodifferential operator in case of nonnegative integer order In addition,

we formulate the necessary and sufficient Tricomi conditions for the gral operator kernels to define pseudodifferential operators in the domain byusing the asymptotic expansions of the symbols and those of pseudohomoge-neous kernels We close Chapter 7 with a presentation of the transformationformulae and invariance properties under the change of coordinates

inte-Chapter 8 is devoted to the relation between the classical tial operators and boundary integral operators For smooth boundaries, all ofour examples in Chapters 1 and 2 of boundary integral operators belong tothe class of classical pseudodifferential operators on compact manifolds hav-ing symbols of the rational type If the corresponding class of pseudodifferen-tial operators in the form of Newton potentials is applied to tensor productdistributions with support on the boundary manifold, then they generate, in anatural way, boundary integral operators which again are classical pseudodif-ferential operators on the boundary manifold Moreover, for these operatorsassociated with regular elliptic boundary value problems, it turns out thatthe corresponding Hadamard’s finite part integral operators are invariant un-der the change of coordinates, as considered in Chapter 3 This approach alsoprovides the jump relations of the potentials We obtain these properties byusing only the Schwartz kernels of the boundary integral operators However,these are covered by Boutet de Monvel’s work in the 1960’s on regular ellipticproblems involving the transmission properties

pseudodifferen-The last two chapters, 9 and 10, contain concrete examples of ary integral equations in the framework of pseudodifferential operators onthe boundary manifold In Chapter 9, we provide explicit calculations of thesymbols corresponding to typical boundary integral operators on closed sur-faces in IR3 If the fundamental solution is not available then the boundaryvalue problem can still be reduced to a coupled system of domain and bound-ary integral equations As an illustration we show that these coupled systemscan be considered as some particular Green operators of the Boutet de Mon-vel algebra In Chapter 10, the special features of Fourier series expansions

bound-of boundary integral operators on closed curves are exploited

We conclude the book with a short Appendix on differential operators inlocal coordinates with minimal differentiability Here, we avoid the explicituse of the normal vector field as employed in Hadamard’s coordinates inChapter 3 These local coordinates may also serve for a more detailed analysisfor Lipschitz domains

Classical model problems

(Chap 1 and Chap 2)

Fourier representation of

BIOs and ψdOs on Γ ⊂ R2

(Chap 10)

BIEs and ψdOs on Γ

(Chap 8 and Chap 9)

Classical ψdOs and IOs

BBBBBBBBBBBBBBBBBBBB

Fig 0.1.A schematic sketch of the topics and their relations

Our original plan was to finish this book project about 10 years ago.However, many new ideas and developments in boundary integral equationmethods appeared during these years which we have attempted to incorpo-rate Nevertheless, we regret to say that the present book is by no meanscomplete For instance, we only slightly touch on the boundary integral op-erator methods involving Lipschitz boundaries which have recently becomemore important in engineering applications We do hope that we have made

a small step forward to bridge the gap between the theory of boundary gral equation methods and their applications We further hope that this bookwill lead to better understanding of the underlying mathematical structure ofthese methods and will serve as a mathematical foundation of the boundaryelement methods

inte-Preface XI

In closing, we would also like to mention some other relevant books related

to boundary integral methods such as the classical books on potential ory by Kellogg [155] and G¨unter [113], the mathematical books on boundaryintegral equations by Hackbusch [116], Jaswon and Symm [148], Kupradze[175, 176, 177], Schatz, Thom´ee and Wendland [268], Mikhlin [211, 212, 213],Nedelec [231, 234], Colton and Kress [47, 48], Mikhlin, Morozov and Paukshto[214], Mikhlin and Pr¨ossdorf [215], Dautray and Lions [60], Chen and Zhou[40], Gatica and Hsiao [93], Kress [172], McLean [203], Yu, De–hao [324],Steinbach [290], Freeden and Michel [83], Kohr and Pop [163], Sauter andSchwab [266], as well as the Encyclopedia articles by Maz’ya [202], Pr¨ossdorf[253], Agranovich [4] and the authors [141] For engineering books on bound-ary integral equations, we suggest the books by Brebbia [23], Crouch andStarfield [57], Brebbia, Telles and Wrobel [24], Manolis and Beskos [197],Balaˇs, Sladek and Sladek [11], Pozrikidis [252], Power and Wrobel [251], Bon-net [18], Gaul, K¨ogel and Wagner [94]

the-We are very grateful for the continuous support and encouragement by ourstudents, colleagues, and friends During the course of preparing this book wehave benefitted from countless discussions with so many excellent individualsincluding Martin Costabel, Gabriel Gatica, Olaf Steinbach, Ernst Stephanand the late Siegfried Pr¨ossdorf; to name a few Moreover, we are indebted

to our reviewers of the first draft of the book manuscript for their criticalreviews and helpful suggestions which helped us to improve our presentation

We would like to extend our thanks to Greg Silber, Clemens F¨orster andG¨ulnihal Meral for careful and critical proof reading of the manuscript

We take this opportunity to acknowledge our gratitude to our ties, the University of Delaware at Newark, DE U.S.A and the University

universi-of Stuttgart in Germany; the Alexander von Humboldt Foundation, the bright Foundation, and the German Research Foundation DFG for repeatedsupport within the Priority Research Program on Boundary Element Meth-ods and within the Collaborative Research Center on Multifield Problems,SFB 404 at the University of Stuttgart, the MURI program of AFOSR atthe University of Delaware, and the Neˇcas Center in Prague We express inparticular our gratitude to the Oberwolfach Research Institute in Germanywhich supported us three times through the Research in Pairs Program, where

Ful-we enjoyed the excellent research environment and atmosphere It is also apleasure to acknowledge the generous attitude, the unfailing courtesy, andthe ready cooperation of the publisher

Last, but by no means least, we are gratefully indebted to Gisela land for her highly skilled hands in the LATEX typing and preparation of thismanuscript

Table of Contents

Preface VII

1. Introduction 1

1.1 The Green Representation Formula 1

1.2 Boundary Potentials and Calder´on’s Projector 3

1.3 Boundary Integral Equations 10

1.3.1 The Dirichlet Problem 11

1.3.2 The Neumann Problem 12

1.4 Exterior Problems 13

1.4.1 The Exterior Dirichlet Problem 13

1.4.2 The Exterior Neumann Problem 15

1.5 Remarks 19

2. Boundary Integral Equations 25

2.1 The Helmholtz Equation 25

2.1.1 Low Frequency Behaviour 31

2.2 The Lam´e System 45

2.2.1 The Interior Displacement Problem 47

2.2.2 The Interior Traction Problem 55

2.2.3 Some Exterior Fundamental Problems 56

2.2.4 The Incompressible Material 61

2.3 The Stokes Equations 62

2.3.1 Hydrodynamic Potentials 65

2.3.2 The Stokes Boundary Value Problems 66

2.3.3 The Incompressible Material — Revisited 75

2.4 The Biharmonic Equation 79

2.4.1 Calder´on’s Projector 83

2.4.2 Boundary Value Problems and Boundary Integral Equations 85

2.5 Remarks 91

3. Representation Formulae 95

3.1 Classical Function Spaces and Distributions 95

3.2 Hadamard’s Finite Part Integrals 101

3.3 Local Coordinates 108

3.4 Short Excursion to Elementary Differential Geometry 111

3.4.1 Second Order Differential Operators in Divergence Form 119

3.5 Distributional Derivatives and Abstract Green’s Second Formula 126

3.6 The Green Representation Formula 130

3.7 Green’s Representation Formulae in Local Coordinates 135

3.8 Multilayer Potentials 139

3.9 Direct Boundary Integral Equations 145

3.9.1 Boundary Value Problems 145

3.9.2 Transmission Problems 155

3.10 Remarks 157

4. Sobolev Spaces 159

4.1 The Spaces H s (Ω) 159

4.2 The Trace Spaces H s (Γ ) 169

4.2.1 Trace Spaces for Periodic Functions on a Smooth Curve in IR2 181

4.2.2 Trace Spaces on Curved Polygons in IR2 185

4.3 The Trace Spaces on an Open Surface 189

4.4 Weighted Sobolev Spaces 191

5. Variational Formulations 195

5.1 Partial Differential Equations of Second Order 195

5.1.1 Interior Problems 199

5.1.2 Exterior Problems 204

5.1.3 Transmission Problems 215

5.2 Abstract Existence Theorems for Variational Problems 218

5.2.1 The Lax–Milgram Theorem 219

5.3 The Fredholm–Nikolski Theorems 226

5.3.1 Fredholm’s Alternative 226

5.3.2 The Riesz–Schauder and the Nikolski Theorems 235

5.3.3 Fredholm’s Alternative for Sesquilinear Forms 240

5.3.4 Fredholm Operators 241

5.4 G˚arding’s Inequality for Boundary Value Problems 243

5.4.1 G˚arding’s Inequality for Second Order Strongly Elliptic Equations in Ω 243

5.4.2 The Stokes System 250

5.4.3 G˚arding’s Inequality for Exterior Second Order Problems 254

5.4.4 G˚arding’s Inequality for Second Order Transmission Problems 259

5.5 Existence of Solutions to Boundary Value Problems 259

5.5.1 Interior Boundary Value Problems 260

Table of Contents XVII

5.5.2 Exterior Boundary Value Problems 264

5.5.3 Transmission Problems 264

5.6 Solution of Integral Equations via Boundary Value Problems 265 5.6.1 The Generalized Representation Formula for Second Order Systems 265

5.6.2 Continuity of Some Boundary Integral Operators 267

5.6.3 Continuity Based on Finite Regions 270

5.6.4 Continuity of Hydrodynamic Potentials 272

5.6.5 The Equivalence Between Boundary Value Problems and Integral Equations 274

5.6.6 Variational Formulation of Direct Boundary Integral Equations 277

5.6.7 Positivity and Contraction of Boundary Integral Operators 287

5.6.8 The Solvability of Direct Boundary Integral Equations 291 5.6.9 Positivity of the Boundary Integral Operators of the Stokes System 292

5.7 Partial Differential Equations of Higher Order 293

5.8 Remarks 299

5.8.1 Assumptions on Γ 299

5.8.2 Higher Regularity of Solutions 299

5.8.3 Mixed Boundary Conditions and Crack Problem 300

6. Introduction to Pseudodifferential Operators 303

6.1 Basic Theory of Pseudodifferential Operators 303

6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 326

6.2.1 Systems of Pseudodifferential Operators 328

6.2.2 Parametrix and Fundamental Solution 331

6.2.3 Levi Functions for Scalar Elliptic Equations 334

6.2.4 Levi Functions for Elliptic Systems 341

6.2.5 Strong Ellipticity and G˚arding’s Inequality 343

6.3 Review on Fundamental Solutions 346

6.3.1 Local Fundamental Solutions 347

6.3.2 Fundamental Solutions in IRn for Operators with Constant Coefficients 348

6.3.3 Existing Fundamental Solutions in Applications 352

7. Pseudodifferential Operators as Integral Operators 353

7.1 Pseudohomogeneous Kernels 353

7.1.1 Integral Operators as Pseudodifferential Operators of Negative Order 356

7.1.2 Non–Negative Order Pseudodifferential Operators as Hadamard Finite Part Integral Operators 380

7.1.3 Parity Conditions 389

7.1.4 A Summary of the Relations between Kernels and Symbols 392

7.2 Coordinate Changes and Pseudohomogeneous Kernels 394

7.2.1 The Transformation of General Hadamard Finite Part Integral Operators under Change of Coordinates 397

7.2.2 The Class of Invariant Hadamard Finite Part Integral Operators under Change of Coordinates 404

8. Pseudodifferential and Boundary Integral Operators 413

8.1 Pseudodifferential Operators on Boundary Manifolds 414

8.1.1 Ellipticity on Boundary Manifolds 418

8.1.2 Schwartz Kernels on Boundary Manifolds 420

8.2 Boundary Operators Generated by Domain Pseudodifferential Operators 421

8.3 Surface Potentials on the Plane IRn−1 423

8.4 Pseudodifferential Operators with Symbols of Rational Type 446 8.5 Surface Potentials on the Boundary Manifold Γ 467

8.6 Volume Potentials 476

8.7 Strong Ellipticity and Fredholm Properties 479

8.8 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations 485

8.8.1 The Boundary Value and Transmission Problems 485

8.8.2 The Associated Boundary Integral Equations of the First Kind 488

8.8.3 The Transmission Problem and G˚arding’s inequality 489

8.9 Remarks 491

9. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations 493

9.1 Newton Potential Operators for Elliptic Partial Differential Equations and Systems 499

9.1.1 Generalized Newton Potentials for the Helmholtz Equation 502

9.1.2 The Newton Potential for the Lam´e System 505

9.1.3 The Newton Potential for the Stokes System 506

9.2 Surface Potentials for Second Order Equations 507

9.2.1 Strongly Elliptic Differential Equations 510

9.2.2 Surface Potentials for the Helmholtz Equation 514

9.2.3 Surface Potentials for the Lam´e System 519

9.2.4 Surface Potentials for the Stokes System 524

9.3 Invariance of Boundary Pseudodifferential Operators 524

9.3.1 The Hypersingular Boundary Integral Operators for the Helmholtz Equation 525

Table of Contents XIX

9.3.2 The Hypersingular Operator for the Lam´e System 531

9.3.3 The Hypersingular Operator for the Stokes System 535

9.4 Derivatives of Boundary Potentials 535

9.4.1 Derivatives of the Solution to the Helmholtz Equation 541 9.4.2 Computation of Stress and Strain on the Boundary for the Lam´e System 543

9.5 Remarks 547

10 Boundary Integral Equations on Curves in IR 2 549

10.1 Fourier Series Representation of the Basic Operators 550

10.2 The Fourier Series Representation of Periodic Operators A ∈ L m c
(Γ ) 556

10.3 Ellipticity Conditions for Periodic Operators on Γ 562

10.3.1 Scalar Equations 563

10.3.2 Systems of Equations 568

10.3.3 Multiply Connected Domains 572

10.4 Fourier Series Representation of some Particular Operators 574

10.4.1 The Helmholtz Equation 574

10.4.2 The Lam´e System 578

10.4.3 The Stokes System 581

10.4.4 The Biharmonic Equation 582

10.5 Remarks 591

A Differential Operators in Local Coordinates with Minimal Differentiability 593

References 599

Index 613

This chapter serves as a basic introduction to the reduction of elliptic ary value problems to boundary integral equations We begin with modelproblems for the Laplace equation Our approach is the direct formulationbased on Green’s formula, in contrast to the indirect approach based on alayer ansatz For ease of reading, we begin with the interior and exteriorDirichlet and Neumann problems of the Laplacian and their reduction tovarious forms of boundary integral equations, without detailed analysis (Forthe classical results see e.g G¨unter [113] and Kellogg [155].) The Laplace

bound-equation, and more generally, the Poisson bound-equation,

already models many problems in engineering, physics and other disciplines(Dautray and Lions [59] and Tychonoff and Samarski [308]) This equationappears, for instance, in conformal mapping (Gaier [88, 89]), electrostatics(Gauss [95], Martensen [199] and Stratton [298]), stationary heat conduction(G¨unter [113]), in plane elasticity as the membrane state and the torsionproblem (Szabo [300]), in Darcy flow through porous media (Bear [12] andLiggett and Liu [188]) and in potential flow (Glauert [102], Hess and Smith[124], Jameson [147] and Lamb [181]), to mention a few

The approach here is based on the relation between the Cauchy data

of solutions via the Calder´on projector As will be seen, the correspondingboundary integral equations may have eigensolutions in spite of the unique-ness of the solutions of the original boundary value problems By appropriatemodifications of the boundary integral equations in terms of these eigenso-lutions, the uniquness of the boundary integral equations can be achieved.Although these simple, classical model problems are well known, the conceptsand procedures outlined here will be applied in the same manner for moregeneral cases

1.1 The Green Representation Formula

For the sake of simplicity, let us first consider, as a model problem, the

Laplacian in two and three dimensions As usual, we use x = (x , , x )∈

2 1 Introduction

IRn (n = 2 or 3) to denote the Cartesian co–ordinates of the points in the

Euclidean space IRn Furthermore, for x, y ∈ IR n, we set

for the inner product and the Euclidean norm, respectively We want to find

the solution u satisfying the differential equation

Here Ω ⊂ IR n is a bounded, simply connected domain, and its boundary

(Later this assumption will be reduced.) As is known from classical analysis,

a classical solution v ∈ C2(Ω) ∩ C1(Ω) can be represented by boundary potentials via the Green representation formula and the fundamental solution

E of (1.1.1) For the Laplacian, E(x, y) is given by

(1.1.3)

for x ∈ Ω ( see Mikhlin [213, p 220ff.]) where n ydenotes the exterior normal

to Γ at y ∈ Γ , ds y the surface element or the arclength element for n = 3

or 2, respectively, and

∂v

∂n (y) :=˜→y∈Γ,˜limy ∈Ω gradv(˜ y) · n y (1.1.4)

The notation ∂/∂n y will be used if there could be misunderstanding due tomore variables

In the case when f ≡ 0 in (1.1.1), one may also use the decomposition in

the following form:



R n

where u now solves the Laplace equation

Here v p denotes a particular solution of (1.1.1) in Ω or Ω c and f has been extended from Ω (or Ω c ) to the entire R n Moreover, for the extended f

we assume that the integral defined in (1.1.5) exists for all x ∈ Ω (or Ω c)

Clearly, with this particular solution, the boundary conditions for u are to

be modified accordingly

Now, without loss of generality, we restrict our considerations to the

so-lution u of the Laplacian (1.1.6) which now can be represented in the form:

For given boundary data u | Γ and ∂u ∂n | Γ, the representation formula (1.1.7)

defines the solution of (1.1.6) everywhere in Ω Therefore, the pair of ary functions belonging to a solution u of (1.1.6) is called the Cauchy data,

In solid mechanics, the representation formula (1.1.7) can also be derived

by the principle of virtual work in terms of the so–called weighted ual formulation The Laplacian (1.1.6) corresponds to the equation of theequilibrium state of the membrane without external body forces and vertical

resid-displacement u Then, for fixed x ∈ Ω, the terms

equal to the virtual work of the resulting boundary forces∂u ∂n | Γ acting against

the displacement E(x, y), i.e.



y∈Γ

∂n (y)ds y .

This equality is known as Betti’s principle (see e.g Ciarlet [42], Fichera [75]

and Hartmann [121, p 159]) Corresponding formulas can also be obtainedfor more general elliptic partial differential equations than (1.1.6), as will bediscussed in Chapter 2

The representation formula (1.1.7) contains two boundary potentials, the

simple layer potential

Here, σ and ϕ are referred to as the densities of the corresponding potentials.

In (1.1.7), for the solution of (1.1.6), these are the Cauchy data which are notboth given for boundary value problems For their complete determination

we consider the Cauchy data of the left– and the right–hand sides of (1.1.7)

on Γ ; this requires the limits of the potentials for x approaching Γ and their

normal derivatives This leads us to the following definitions of boundaryintegral operators, provided the corresponding limits exist For the potentialequation (1.1.6), this is well known from classical analysis (Mikhlin [213,

p 360] and G¨unter [113, Chap II]):

To be more explicit, we quote the following standard results without proof

is called weakly singular if there exist constants c and λ < n − 1 such that

For the Laplacian, for Γ ∈ C2 and E(x, y) given by (1.1.2), one even has

In case n = 2, both kernels in (1.2.12), (1.2.13) are continuously extendable

to a C0-function for y → x (see Mikhlin [213]), in case n = 3 they are weakly

singular with λ = 1 (see G¨unter [113, Sections II.3 and II.6]) For otherdifferential equations, as e.g for elasticity problems, the boundary integrals

in (1.2.7)–(1.2.9) are strongly singular and need to be defined in terms ofCauchy principal value integrals or even as finite part integrals in the sense

of Hadamard In the classical approach, the corresponding function spaces

are the H¨ older spaces which are defined as follows:

a composition of tangential derivatives and the simple layer potential tor V :

dϕ

ds ) (x) for n = 2 (1.2.14)and

6 1 Introduction

For the classical proof see Maue [200] and G¨unter [113, p 73ff]

Note that the differential operator (n y × ∇ y )ϕ defines the tangential derivatives of ϕ(y) within Γ which are H¨ older–continuous functions on Γ Often this operator is also called the surface curl (see Giroire and Nedelec

[101, 232]) In the following, we give a brief derivation for these formulaebased on classical results of potential theory with H¨older continuous densi-ties dϕ ds (y) and (n y × ∇ y )ϕ(y), respectively Note that ds d

x and (n x × ∇ x)

in (1.2.14) and (1.2.15), respectively, are not interchanged with integration over Γ Later on we will discuss the connection of such an interchange with the concept of Hadamard’s finite part integrals For n = 2, note that, for

Since y = z and ∆ y E(z, y) = 0, the second term on the right takes the form

First note that∇ z V ( dϕ ds )(z) is a H¨ older continuous function for z ∈ Ω which

admits a H¨older continuous extension up to Γ (G¨unter [113, p 68]) Thedefinition of derivatives at the boundary gives us

and with the formulae of vector analysis

Since (n y × ∇ y )ϕ(y) defines tangential derivatives of ϕ,

defines a H¨older continuous function for z ∈ Ω which admits a H¨older

con-tinuous limit for z → x ∈ Γ due to G¨unter [113, p 68] implying (1.2.15).

From (1.2.15) we see that the hypersingular integral operator (1.2.6) can

be expressed in terms of a composition of differentiation and a weakly singularoperator This, in fact, is a regularization of the hypersingular distribution,which will also be useful for the variational formulation and related compu-tational procedures

A more elementary, but different regularization can be obtained as follows(see Giroire and Nedelec [101]) From the definition (1.2.6), we see that

If we apply the representation formula (1.1.7) to u ≡ 1, then we obtain Gauss’

well known formula

∂n y (z, y)ϕ(x)ds y = 0 for all z ∈ Ω ,

hence, we find the simple regularization

Since the boundary values for the various potentials are now characterized,

we are in a position to discuss the relations between the Cauchy data on Γ by taking the limit x → Γ and the normal derivative of the left– and right–hand

sides in the representation formula (1.1.7) For any solution of (1.1.6), thisleads to the following relations between the Cauchy data:

10 1 Introduction

spaces,

K, K  : C α (Γ ) → C α (Γ ), C 1+α (Γ ) → C 1+α (Γ ) , (1.2.22)

For the proofs see Mikhlin and Pr¨ossdorf [215, Sections IX, 4 and 7]

the Laplacian have even stronger continuity properties than those in (1.2.22),

namely K, K  map continuously C α (Γ ) → C 1+α (Γ ) and C 1+α (Γ ) →

C 1+β (Γ ) for any α ≤ β < 1 (Mikhlin and Pr¨ossdorf [215, Sections IX, 4 and 7]

and Colton and Kress [47, Chap 2]) Because of the compact imbeddings

C 1+α (Γ ) → C α (Γ ) and C 1+β (Γ ) → C 1+α (Γ ), K and K are compact These

smoothing properties of K and K  do not hold anymore, if K and K  spond to more general elliptic partial differential equations than (1.1.6) This

corre-is e.g the case in linear elasticity However, the continuity properties (1.2.22)remain valid

With Theorem 1.2.3, we now are in a position to show that C Ω indeed is

a projection More precisely, there holds:

These relations will show their usefulness in our variational formulation later

on, and as will be seen in the next section, the Calder´on projector leads in adirect manner to boundary integral equations for boundary value problems

1.3 Boundary Integral Equations

As we have seen from (1.2.18) and (1.2.19), the Cauchy data of a solution of the differential equation in Ω are related to each other by these two equa-

tions As is well known, for regular elliptic boundary value problems, only

half of the Cauchy data on Γ is given For the remaining part, the equations

(1.2.18), (1.2.19) define an overdetermined system of boundary integral tions which may be used for determining the complete Cauchy data In gen-eral, any combination of (1.2.18) and (1.2.19) can serve as a boundary integralequation for the missing part of the Cauchy data Hence, the boundary inte-gral equations associated with one particular boundary condition are by nomeans uniquely determined The ‘direct’ approach for formulating boundaryintegral equations becomes particularly simple if one considers the Dirichletproblem or the Neumann problem In what follows, we will always prefer thedirect formulation

equa-1.3.1 The Dirichlet Problem

In the Dirichlet problem for (1.1.6), the boundary values

are given Hence,

is the missing Cauchy datum required to satisfy (1.2.18) and (1.2.19) for

any solution u of (1.1.6) In the direct formulation, if we take the first

equa-tion (1.2.18) of the Calder´on projection then σ is to be determined by the

boundary integral equation

first kind In the case n = 2 and a boundary curve Γ with conformal radius

equal to 1, the integral equation (1.3.4) has exactly one eigensolution, the so–

called natural charge e(y) (Plemelj [248]) However, the modified equation

12 1 Introduction

is always solvable for σ and the constant ω for given f and given constant Σ

[136] Later on we will come back to this modification

For Σ = 0 it can be shown that ω = 0 Hence, with Σ = 0, this modified

formulation can also be used for solving the interior Dirichlet problem.Alternatively to (1.3.4), if we take the second equation (1.2.19) of theCalder´on projector, we arrive at

x (x, y) is weakly singular due to (1.2.13), provided Γ is smooth.

g = 2Dϕ is defined by the right–hand side of (1.3.7) Therefore, in contrast

to (1.3.4), this is a Fredholm integral equation of the second kind.

This simple example shows that for the same problem we may employdifferent boundary integral equations In fact, (1.3.8) is one of the cele-brated integral equations of classical potential theory — the adjoint to theNeumann–Fredholm integral equation of the second kind with the doublelayer potential — which can be obtained by using the double layer ansatz

in the indirect approach In the classical framework, the analysis of gral equation (1.3.8) has been studied intensively for centuries, includingits numerical solution For more details and references, see, e.g., Atkinson[8], Bruhn et al [26], Dautray and Lions [59, 60], Jeggle [149, 150], Kellogg[155], Kral et al [167, 168, 169, 170, 171], Martensen [198, 199], Maz‘ya [202],Neumann [238, 239, 240], Radon [259] and [316] In recent years, increasingefforts have also been devoted to the integral equation of the first kind (1.3.4)which — contrary to conventional belief — became a very rewarding and fun-damental formulation theoretically as well as computationally It will be seenthat this equation is particularly suitable for the variational analysis

inte-1.3.2 The Neumann Problem

In the Neumann problem for (1.1.6), the boundary condition reads as

∂u

with given ψ For the interior problem (1.1.6) in Ω, the normal derivative ψ

needs to satisfy the necessary compatibility condition



for any solution of (1.1.6), (1.3.9) to exist Here, u |Γ is the missing Cauchy

datum required to satisfy (1.2.18) and (1.2.19) for any solution u of the

Neumann problem (1.1.6), (1.3.9) If we take the first equation (1.2.18) of theCalder´on projector, then u |Γ is determined by the solution of the boundaryintegral equation

operator in (1.3.12) is continuous for n = 2 and weakly singular for n = 3.

It is easily shown that u0 = 1 defines an eigensolution of the homogeneous

equation corresponding to (1.3.12); and that f (x) in (1.3.12) satisfies the classical orthogonality condition if and only if ψ satisfies (1.3.10) Classical

potential theory provides that (1.3.12) is always solvable if (1.3.10) holdsand that the null–space of (1.3.12) is one–dimensional, see e.g Mikhlin [212,Chap 17, 11]

Alternatively, if we take the second equation (1.2.19) of the Calder´onprojector, we arrive at the equation

This is a hypersingular boundary integral equation of the first kind for u |Γ which also has the one–dimensional null–space spanned by u0| Γ = 1, as can

easily be seen from (1.2.14) and (1.2.15) for n = 2 and n = 3, respectively.

Although this integral equation (1.3.13) is not one of the standard types, wewill see that, nevertheless, it has advantages for the variational formulationand corresponding numerical treatment

1.4 Exterior Problems

In many applications such as electrostatics and potential flow, one often dealswith exterior problems which we will now consider for our simple modelequation

1.4.1 The Exterior Dirichlet Problem

For boundary value problems exterior to Ω, i.e in Ω c = IRn \ Ω, infinity

belongs to the boundary of Ω c and, therefore, we need additional growth or

radiation conditions for u at infinity Moreover, in electrostatic problems, for

14 1 Introduction

instance, the total charge Σ on Γ will be given This leads to the following

exterior Dirichlet problem, defined by the differential equation

where the direction of n is defined as before and the normal derivative is now

defined as in (1.1.7) but with z ∈ Ω c In the case ω = 0, we may consider the Cauchy data from Ω c on Γ , which leads with the boundary data of (1.4.5)

Here, the boundary integral operators V, K, K  , D are related to the limits

of the boundary potentials from Ω c similar to (1.2.3)–(1.2.6), namely

from those in (1.2.4) and (1.2.5), respectively

For any solution u of (1.4.1) in Ω c with ω = 0, the Cauchy data on Γ

are reproduced by the right–hand side of (1.4.6), which therefore defines the

Ω c Clearly,

whereI denotes the identity matrix operator.

For the solution of (1.4.1), (1.4.2) and (1.4.3), we obtain from (1.4.5) amodified boundary integral equation,

This, again, is a first kind integral equation for σ = ∂n ∂u | Γ, the unknown

Cauchy datum However, in addition, the constant ω is also unknown Hence,

we need an additional constraint, which here is given by

If we take the normal derivative at Γ on both sides of (1.4.5), we arrive

at the following Fredholm integral equation of the second kind for σ, namely

1

This is the classical integral equation associated with the exterior Dirichletproblem which has a one–dimensional space of eigensolutions Here, the spe-cial right–hand side of (1.4.14) always satisfies the orthogonality condition

in the classical Fredholm alternative Hence, (1.4.14) always has a solution,which becomes unique if the additional constraint of (1.4.13) is included

1.4.2 The Exterior Neumann Problem

Here, in addition to (1.4.1), we require the Neumann condition

from (1.4.15), where ω is now an additional parameter, which can be

pre-scribed arbitrarily according to the special situation The representation

for-mula (1.4.5) remains valid Often ω = 0 is chosen in (1.4.3), (1.4.4) and (1.4.5) The direct approach with x → Γ in (1.4.5) now leads to the bound-

ary integral equation

For any given ψ and ω, this is the classical Fredholm integral equation of

16 1 Introduction

Atkinson [8], Dieudonn´e [61], Kral [168, 169], Maz‘ya [202], Mikhlin [211,

212]) (1.4.16) is uniquely solvable for u | Γ

If we apply the normal derivative to both sides of (1.4.5), we find the

hypersingular integral equation of the first kind,

This equation has the constants as an one–dimensional eigenspace The cial right–hand side in (1.4.17) satisfies an orthogonality condition in theclassical Fredholm alternative, which is also valid for (1.4.17), e.g., in thespace of H¨older continuous functions on Γ , as will be shown later Therefore, (1.4.17) always has solutions u | Γ Any solution of (1.4.17) inserted into the

spe-right hand side of (1.4.5) together with any choice of ω will give the desired

unique solution of the exterior Neumann problem

For further illustration, we now consider the historical example of thetwo–dimensional potential flow of an inviscid incompressible fluid around an

airfoil Let q ∞ denote the given traveling velocity of the profile defining a

uniform velocity at infinity and let q denote the velocity field Then we have the following exterior boundary value problem for q = (q1, q2):

for the arc length 0 ≤ s ≤ L with x(0) = x(L) = T E, whose periodic

extension is only piecewise C ∞ With Bernoulli’s law, the condition (1.4.21)

is equivalent to the Kutta–Joukowski condition, which requires bounded andequal pressure at the trailing edge (See also Ciavaldini et al [44]) The origin 0

of the co–ordinate system is chosen within the airfoil with T E on the x1–axis

and the line 0 T E within Ω, as shown in Figure 1.4.1.

Figure 1.4.1:Airfoil in two dimensions

As before, the exterior domain is denoted by Ω c := IR2\Ω Since the

flow is irrotational and divergence–free, q has a potential which allows the

In this formulation, u ∈ C2(Ω c)∩ C0(Ω c) is the unknown disturbance

po-tential, ω0is the unknown circulation around Γ which will be determined by

the additional Kutta–Joukowski condition, that is

lim

Ω c x→T E |∇u(x)| = |∇u| |T E exists (1.4.27)

We remark that condition (1.4.27) is a direct consequence of condition

(1.4.21) By using conformal mapping, the solution u was constructed by

Kirchhoff [157], see also Goldstein [105]

We now reduce this problem to a boundary integral equation As in (1.4.5)and in view of (1.4.26), the solution admits the representation

Note that K is defined by (1.5.2) which is valid at T E, too The right–hand

sides of (1.4.32) and (1.4.33) are both H¨older continuous functions on Γ Due

to the classical results by Carleman [37] and Radon [259], there exist unique

solutions u0 and u1 in the class of continuous functions A more detailedanalysis shows that the derivatives of these solutions possess singularities atthe trailing edge TE More precisely, one finds (e.g in the book by Grisvard[108, Theorem 5.1.1.4 p.255]) that the solutions admit local singular expan-sions of the form

where Θ is the exterior angle of the two tangents at the trailing edge,  denotes the distance from the trailing edge to x, ϑ is the angle from the

lower trailing edge tangent to the vector (x − T E), where ε is any positive

number Consequently, the gradients are of the form

where e ϑ is a unit vector with angle (1− π

Θ )ϑ from the lower trailing edge tangent, for both cases, i = 0, 1.

Hence, from equations (1.4.27) and (1.4.29) we obtain, for  → 0, the

condition for ω0,

The solution u1 corresponds to q ∞= 0, i.e the pure circulation flow, which

can easily be found by mapping Ω conformally onto the unit circle in the complex plane The mapping has the local behavior as in (1.4.34) with α1= 0

since T E is mapped onto a point on the unit circle (see Lehman [183]) Consequently, ω0is uniquely determined from (1.4.36) We remark that this

choice of ω0shows that

For applications in engineering, the strong smoothness assumptions for the

boundary Γ need to be relaxed allowing corners and edges Moreover, for crack and screen problems as in elasticity and acoustics, respectively, Γ is

not closed but only a part of a curve or a surface To handle these types ofproblems, the approach in the previous sections needs to be modified accord-ingly

To be more specific, we first consider Lyapounov boundaries Following

Mikhlin [212, Chap 18], a Lyapounov curve in IR2 or Lyapunov surface Γ in

IR3 (G¨unter [113]) satisfies the following two conditions:

1 There exists a normal n x at any point x on Γ

2 There exist positive constants a and κ ≤ 1 such that for any two points

them satisfies

20 1 Introduction

In fact, it can be shown that for 0 < κ < 1, a Lyapounov boundary coincides with a C 1,κ boundary curve or surface [212, Chap 18]

For a Lyapounov boundary, all results in Sections 1.1–1.4 remain valid

if C2 is replaced by C 1,κ accordingly These non–trivial generalizations can

be found in the classical books on potential theory See, e.g., G¨unter [113],Mikhlin [211, 212, 213] and Smirnov [284]

In applications, one often has to deal with boundary curves with corners,

or with boundary surfaces with corners and edges The simplest tion of the previous approach can be obtained for piecewise Lyapounov curves

generaliza-in IR2 with finitely many corners where Γ = ∪ N

j=1 Γ j and each Γ j being an

open arc of a particular closed Lyapounov curve The intersections Γ j ∩Γ j+1

are the corner points where Γ N +1 := Γ1 In this case it easily follows that

there exists a constant C such that



Γ\{x}

∂n y (x, y) |ds ≤ C for all x ∈ IR2. (1.5.1)

This property already ensures that for continuous ϕ on Γ , the operator K

is well defined by (1.2.4) and that it is a continuous mapping in C0(Γ ).

However, (1.2.8) needs to be modified and becomes

a sum of a compact operator and a contraction, provided the corner angles

are not 0 or 2π This decomposition is sufficient for the classical Fredholm alternative to hold for (1.3.11) with continuous u, as was shown by Radon

[259] For the most general two–dimensional case we refer to Kral [168].For the Neumann problem, one needs a generalization of the normal deriv-

ative in terms of the so–called boundary flow, which originally was introduced

by Plemelj [248] and has been generalized by Kral [169] It should be

men-tioned that in this case the adjoint operator K  to K is no longer a bounded

operator on the space of continuous functions (Netuka [237]) The simple layer

potential V σ is still H¨older continuous in IR2 for continuous σ However, its

normal derivative needs to be interpreted in the sense of boundary flow.This situation is even more complicated in the three–dimensional case

because of the presence of edges and corners Here, for continuous ϕ it is still

not clear whether the Fredholm alternative for equation (1.3.11) remains valideven for general piecewise Lyapounov surfaces with finitely many corners and

Fig 1.5.1 Configuration of an arc Γ in IR2

edges; see, e.g., Angell et al [7], Burago et al [31, 32], Kral et al [170, 171],Maz‘ya [202] and [316]

On the other hand, as we will see, in the variational formulation ofthe boundary integral equations, many of these difficulties can be circum-vented for even more general boundaries such as Lipschitz boundaries (seeSection 5.6)

To conclude these remarks, we consider Γ to be an oriented, open part of

a closed curve or surface Γ (see Figure 1.5.1) The Dirichlet problem here is

to find the solution u of (1.4.1) in the domain Ω c = IRn \ Γ subject to the

boundary conditions

u+ = ϕ+ on Γ+ and u = ϕ − on Γ − (1.5.3)

where Γ+ and Γ − are the respective sides of Γ and u+ and u − the

corre-sponding traces of u The functions ϕ+ and ϕ −are given with the additionalrequirement that

at the endpoints of Γ for n = 2, or at the boundary edge of Γ for n = 3.

In the latter case we require ∂Γ to be a C ∞– smooth curve Similar to theregular exterior problem, we again require the growth condition (1.4.3) for

n = 2 and (1.4.4) for n = 3 For a sufficiently smooth solution, the Green

representation formula has the form

with n the normal to Γ pointing in the direction of the side Γ+ If we

sub-stitute the given boundary values ϕ+, ϕ − into (1.5.4), the missing Cauchy

datum σ is now the jump of the normal derivative across Γ Between this unknown datum and the behaviour of u at infinity viz.(1.4.3) we arrive at



Γ

By taking x to Γ+ (or Γ −) we obtain (in both cases) the boundary integral

equation of the first kind for σ on Γ ,

where ψ+and ψ −are given smooth functions By applying the normal

deriv-atives ∂/∂n x to the representation formula (1.5.4) from both sides of Γ , it is not difficult to see that the missing Cauchy datum ϕ =: [u] = u+− u − on Γ

satisfies the hypersingular boundary integral equation of the first kind for ϕ,

2(ψ++ ψ −)− K 

Γ [ψ] on Γ , (1.5.12)

where the operators D Γ and K Γ  again are given by (1.2.6) and (1.2.5) with

in the framework of variational problems, this integral equation (1.5.12) is

uniquely solvable for ϕ with ϕ = 0 at the endpoints of Γ for n = 2 or at the boundary edge ∂Γ of Γ for n = 3, respectively Here, Σ in (1.4.3) or (1.4.4) is already given by (1.5.8) and ω can be chosen arbitrarily For further analysis

of these problems see [146], Stephan et al [294, 297], Costabel et al [49, 52]