Boundary Integral Equations [2 ed.] 9783030711276, 9783030711269

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Applied Mathematical Sciences

George C. Hsiao Wolfgang L. Wendland

Boundary Integral Equations Second Edition

Applied Mathematical Sciences Volume 164

Series Editors Anthony Bloch, Ann Arbor, USA C. L. Epstein, Philadelphia, USA Alain Goriely, Oxford, UK Leslie Greengard, New York, USA Advisory Editors J. Bell, Berkeley, USA P. Constantin, Princeton, USA R. Durrett, Durham, USA R. Kohn, New York, USA R. Pego, Pittsburgh, USA L. Ryzhik, Stanford, USA A. Singer, Princeton, USA A. Stevens, Münster, Germany S. Wright, Madison, USA Founding Editors F. John, New York, USA J. P. LaSalle, Providence, USA L. Sirovich, Providence, USA

The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of computer modeling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will provide an outlet for topics of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. These books should be accessible to readers versed in mathematics or science and engineering, and will feature a lively tutorial style, a focus on topics of current interest, and present clear exposition of broad appeal. A compliment to the Applied Mathematical Sciences series is the Texts in Applied Mathematics series, which publishes textbooks suitable for advanced undergraduate and beginning graduate courses. More information about this series at http://www.springer.com/series/34

George C. Hsiao · Wolfgang L. Wendland

Boundary Integral Equations Second Edition

13

George C. Hsiao Department of Mathematical Sciences University of Delaware Newark, USA

Wolfgang L. Wendland IANS Universität Stuttgart Stuttgart, Germany

ISSN 0066-5452 ISSN 2196-968X  (electronic) Applied Mathematical Sciences ISBN 978-3-030-71127-6  (eBook) ISBN 978-3-030-71126-9 https://doi.org/10.1007/978-3-030-71127-6 Mathematics Subject Classification: 31A10, 31B10, 35J05, 35J20, 35J35, 35Q61, 45B05, 47G30, 74B05, 76D07, 78A25, 78A45 © Springer Nature Switzerland AG 2008, 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Responsible Editor: Martin Peters This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our families for their love and understanding

Preface to the Second Edition

The first edition of this book was published in August, 2008. Since then, various theoretical and computational techniques have been developed for boundary integral equation methods. In particular, fundamental results for solutions of the boundary integral equations in electromagnetics have matured substantially. The widespread scattering of these results throughout the literature has motivated us to update our monograph and collect these important developments and fundamental results to facilitate accessibility. In this second edition, in addition to making minor corrections, correcting typographical errors, updating references, as well as improving or simplifying some of the presentations, we have added two new chapters. One of the new chapters is on Maxwell equations. The other new chapter is on Remarks on Pseudodifferential Operators Related to Time–harmonic Maxwell Equations. The chapter on Maxwell equations includes an introduction to basic electromagnetic theory with emphasis on the reduction of time-harmonic solutions to corresponding boundary integral equations. The chapter on the pseudodifferential operator approach to Maxwell equations discusses, based on the symbol analysis of the electromagnetic boundary integral operators, the coerciveness, strong ellipticity and solutions for boundary integral equations in standard Sobolev spaces. We also added a new section on Function Spaces H(div, Ω) and H(curl, Ω) which gives a brief summary of some basic vector function spaces and related topics needed for the two new chapters. Finally, for convenience, we added a new appendix to the second edition that contains Vector Identities, Green’s Formulae, Surface Differential Operators and Stokes Formulae. The second edition of the monograph now contains a total of twelve chapters and two appendices. We are very pleased that this second edition includes theory and applications of boundary integral equations arising in potential flow, acoustics, elasticity, Stokes flow and electromagnetics. Because of the breadth of these applications, we hope that this second edition will also attract a broader readership.

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We would like to take this opportunity to express our deepest gratitude to the ”Mathematisches Forschungsinstitut Oberwolfach” (MFO), which supported us twice through the Research in Pairs Program (RiP 1634p, and RiP 1837q). Thanks go to Greg Silber for proof reading of the manuscript of the two added chapters. Heartfelt thanks are also due to Gisela Wendland for her highly skilled LaTex typesetting and for preparing the manuscript. Newark, Delaware Stuttgart, Germany 2020

George C. Hsiao Wolfgang L. Wendland

Preface to the First Edition

This book is devoted to the mathematical foundation of boundary integral equations. The combination of finite element analysis on the boundary with these equations has led to very efficient computational tools, the boundary element methods (see e. g., the authors [202] and Schanz and Steinbach (eds.) [373]). 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 confined ourselves to the treatment of elliptic boundary value problems. The central idea of eliminating the field equations in the domain and reducing boundary value problems to equivalent equations only on the boundary requires the knowledge of corresponding fundamental solutions, and this idea has a long history dating back to the work of Green [155] and Gauss [141, 142]. Today the resulting boundary integral equations still serve as a major tool for the analysis and construction of solutions to boundary value problems. As is well known, the reduction to equivalent boundary integral equations is by no means unique, and there are primarily two procedures for this reduction, the ’direct’ and ’indirect’ approaches. The direct procedure is based on Green’s representation formula for solutions of the boundary value problem, whereas the indirect approach rests on an appropriate layer ansatz. In our presentation we concentrate on the direct approach although the corresponding analysis and basic properties of the boundary integral operators remain the same for the indirect approaches. Roughly speaking, one obtains two kinds of boundary integral equations with both procedures, those of the first kind and those of the second kind. The basic mathematical properties that guarantee existence of solutions to the boundary integral equations and also stability and convergence analysis of corresponding numerical procedures hinge on G˚ arding inequalities for the boundary integral operators on appropriate function spaces. In addition, contraction properties allow the application of Carl Neumann’s classical successive iteration procedure to a class of boundary integral equations of the second kind. It turns out that these basic features are intimately related to the variational forms of the underlying elliptic boundary value problems and the potential energies of their solution fields, allowing us to consider

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the boundary integral equations in the form of variational problems on the boundary manifold of the domain. On the other hand, the Newton potentials as the inverses of the elliptic partial differential operators are particular pseudodifferential operators on the domain or in the Euclidean space. The boundary potentials (or Poisson operators) are just Newton potentials of distributions with support on the boundary manifold and the boundary integral operators are their traces there. Therefore, it is rather natural to consider the boundary integral operators as pseudodifferential operators on the boundary manifold. Indeed, most of the boundary integral operators in applications can be recast as such pseudodifferential operators provided that the boundary manifold is smooth enough. With the application of boundary element methods in mind, where strong ellipticity is the basic concept for stability, convergence and error analysis of corresponding discretization methods for the boundary integral equations, we are most interested in establishing strong ellipticity in terms of G˚ arding’s inequality for the variational formulation as well as strong ellipticity of the pseudodifferential operators generated by the boundary integral equations. The combination of both, namely the variational properties of the elliptic boundary value and transmission problems as well as the strongly elliptic pseudodifferential operators provides us with an efficient means to analyze a large class of elliptic boundary value problems. This book contains 10 chapters and an appendix. For the reader’s benefit, Figure 0.0.1 gives a sketch of the topics contained in this book. Chapters 1 through 4 present various examples and background information relevant to the premises of this book. In Chapter 5, we discuss the variational formulation of boundary integral equations and their connection to the variational solution of associated boundary value or transmission problems. In particular, continuity and coerciveness properties of a rather large class of boundary integral equations are obtained, including those discussed in the first and second chapters. In Chapter 4, we collect basic properties of Sobolev spaces in the domain and their traces on the boundary, which are needed for the variational formulations in Chapter 5. Chapter 7 presents an introduction to the basic theory of classical pseudodifferential 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 arbitrary order for general elliptic systems of partial differential equations. If the fundamental solution exists then Levi’s method based on Levi functions allows its construction via an appropriate integral equation. In Chapter 8, we show that every pseudodifferential operator is an Hadamard’s finite part integral operator with integrable or nonintegrable kernel plus possibly a differential operator of the same order as that of the

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pseudodifferential operator in case of nonnegative integer order. In addition, we formulate the necessary and sufficient Tricomi conditions for the integral operator kernels to define pseudodifferential operators in the domain by using the asymptotic expansions of the symbols and those of pseudohomogeneous kernels. We close Chapter 8 with a presentation of the transformation formulae and invariance properties under the change of coordinates. Chapter 9 is devoted to the relation between the classical pseudodifferential operators and boundary integral operators. For smooth boundaries, all of our examples in Chapters 1 and 2 of boundary integral operators belong to the class of classical pseudodifferential operators on compact manifolds having symbols of the rational type. If the corresponding class of pseudodifferential operators in the form of Newton potentials is applied to tensor product distributions with support on the boundary manifold, then they generate, in a natural way, boundary integral operators which again are classical pseudodifferential operators on the boundary manifold. Moreover, for these operators associated with regular elliptic boundary value problems, it turns out that the corresponding Hadamard’s finite part integral operators are invariant under the change of coordinates, as considered in Chapter 3. This approach also provides the jump relations of the potentials. We obtain these properties by using 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 elliptic problems involving the transmission properties. The last two chapters, 10 and 11, contain concrete examples of boundary integral equations in the framework of pseudodifferential operators on the boundary manifold. In Chapter 10, we provide explicit calculations of the symbols corresponding to typical boundary integral operators on closed surfaces in IR3 . If the fundamental solution is not available then the boundary value problem can still be reduced to a coupled system of domain and boundary integral equations. As an illustration we show that these coupled systems can be considered as some particular Green operators of the Boutet de Monvel algebra. In Chapter 11, the special features of Fourier series expansions of boundary integral operators on closed curves are exploited. We conclude the book with a short Appendix on differential operators in local coordinates with minimal differentiability. Here, we avoid the explicit use of the normal vector field as employed in Hadamard’s coordinates in Chapter 3. These local coordinates may also serve for a more detailed analysis for Lipschitz domains.

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Classical model problems (Chap. 1 and Chap. 2)

ψdOs in Ω ⊂ Rn (Chap. 7)

(Chap. 5 and Chap. 6)

(Chap. 11)

@ BA

BA@
BA @
BA @
B A @
? ? B A
@ Generalized multilayer potentials B A
@ Classical ψdOs and IOs in Ω ⊂ Rn @ on Γ = ∂Ω B
A (Chap. 8) @ (Chap. 3) B
A @ B A @
B A
@ B A
@ ? ? B A
B@ A
Sobolev spaces and trace theorems
BIEs and ψdOs on Γ B @@ AA B (Chap. 4) (Chap. 9, Chap. 10 and Chap. 12) B B B B ? ? B B Fourier representation of BIOs Variational formulations of B BVPs and BIEs and ψdOs on Γ ⊂ R2

Abbreviations: Ω ⊂ IRn BVPs BIEs ψdOs IOs BIOs

– – – – – –

A given domain with compact boundary Γ Boundary value problems Boundary integral equations Pseudodifferential operators Integral operators Boundary integral operators

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

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Our original plan was to finish this book project about 10 years ago. However, many new ideas and developments in boundary integral equation methods appeared during these years which we have attempted to incorporate. Nevertheless, we regret to say that the present book is by no means complete. For instance, we only slightly touch on the boundary integral operator methods involving Lipschitz boundaries which have recently become more important in engineering applications. We do hope that we have made a small step forward to bridge the gap between the theory of boundary integral equation methods and their applications. We further hope that this book will lead to better understanding of the underlying mathematical structure of these methods and will serve as a mathematical foundation of the boundary element methods. In closing, we would also like to mention some other relevant books related to boundary integral methods such as the classical books on potential theory by Kellogg [224] and G¨ unter [162], the mathematical books on boundary integral equations by Hackbusch [167], Jaswon and Symm [213], Kupradze [251, 252, 253], Schatz, Thom´ee and Wendland [374], Mikhlin [301, 303, 302], Nedelec [328, 331], Colton and Kress [75, 77], Mikhlin, Morozov and Paukshto [304], Mikhlin and Pr¨ossdorf [305], Dautray and Lions [97], Chen and Zhou [65], Gatica and Hsiao [139], Kress [247], McLean [290], Yu, De–hao [442], Steinbach [399], Freeden and Michel [127], Kohr and Pop [233], Sauter and Schwab [371], as well as the Encyclopedia articles by Maz’ya [289], Pr¨ossdorf [354], Agranovich [6] and the authors [204]. For engineering books on boundary integral equations, we suggest the books by Brebbia [43], Crouch and Starfield [92], Brebbia, Telles and Wrobel [44], Manolis and Beskos [281], Balaˇs, Sladek and Sladek [23], Pozrikidis [353], Power and Wrobel [352], Bonnet [37], Gaul, K¨ogel and Wagner [140].

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Acknowledgements We are very grateful for the continuous support and encouragement by our students, colleagues, and friends. During the course of preparing this book we have benefitted from countless discussions with so many excellent individuals including Martin Costabel, Gabriel Gatica, Olaf Steinbach, Ernst Stephan and 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 critical reviews and helpful suggestions which helped us to improve our presentation. We would like to extend our thanks to Greg Silber, Clemens F¨orster and G¨ ulnihal Meral for careful and critical proof reading of the manuscript. We take this opportunity to acknowledge our gratitude to our universities, the University of Delaware at Newark, DE. U.S.A. and the University of Stuttgart in Germany; the Alexander von Humboldt Foundation, the Fulbright Foundation, and the German Research Foundation DFG for repeated support within the Priority Research Program on Boundary Element Methods and within the Collaborative Research Center on Multifield Problems, SFB 404 at the University of Stuttgart, the MURI program of AFOSR at the University of Delaware, and the Neˇcas Center in Prague. We express in particular our gratitude to the Oberwolfach Research Institute in Germany which supported us three times through the Research in Pairs Program, where we enjoyed the excellent research environment and atmosphere. It is also a pleasure to acknowledge the generous attitude, the unfailing courtesy, and the ready cooperation of the publisher. Last, but by no means least, we are gratefully indebted to Gisela Wendland for her highly skilled hands in the LATEX typing and preparation of this manuscript. Newark, Delaware Stuttgart, Germany 2008

George C. Hsiao Wolfgang L. Wendland

Table of Contents

1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Green Representation Formula . . . . . . . . . . . . . . . . . . . . . . . 1.2 Boundary Potentials and Calder´on’s Projector . . . . . . . . . . . . . 1.3 Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Exterior Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 1.4.2 The Exterior Neumann Problem . . . . . . . . . . . . . . . . . . . 1.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 10 11 12 13 13 15 19

2.

Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Low Frequency Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Lam´e System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Interior Displacement Problem . . . . . . . . . . . . . . . . . 2.2.2 The Interior Traction Problem . . . . . . . . . . . . . . . . . . . . . 2.2.3 Some Exterior Fundamental Problems . . . . . . . . . . . . . . 2.2.4 The Incompressible Material . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Hydrodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Stokes Boundary Value Problems . . . . . . . . . . . . . . . 2.3.3 The Incompressible Material — Revisited . . . . . . . . . . . 2.4 The Biharmonic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Calder´ on’s Projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Boundary Value Problems and Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 31 45 47 55 56 61 62 65 66 75 79 83

Representation Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Classical Function Spaces and Distributions . . . . . . . . . . . . . . . . 3.2 Hadamard’s Finite Part Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Local Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Short Excursion to Elementary Differential Geometry . . . . . . .

95 95 101 108 111

3.

86 91

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3.4.1 Second Order Differential Operators in Divergence Form119 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.

5.

Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Spaces H s (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Trace Spaces H s (Γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Trace Spaces for Periodic Functions on a Smooth Curve in IR2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Trace Spaces on Curved Polygons in IR2 . . . . . . . . . . . . . 4.3 The Trace Spaces on an Open Surface . . . . . . . . . . . . . . . . . . . . . 4.4 Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Function Spaces H( div , Ω) and H( curl , Ω) . . . . . . . . . . . . . . . Variational Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Partial Differential Equations of Second Order . . . . . . . . . . . . . 5.1.1 Interior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Abstract Existence Theorems for Variational Problems . . . . . . 5.2.1 The Lax–Milgram Theorem . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Fredholm–Nikolski Theorems . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Fredholm’s Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The Riesz–Schauder and the Nikolski Theorems . . . . . . 5.3.3 Fredholm’s Alternative for Sesquilinear Forms . . . . . . . . 5.3.4 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 G˚ arding’s Inequality for Boundary Value Problems . . . . . . . . . 5.4.1 G˚ arding’s Inequality for Second Order Strongly Elliptic Equations in Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 The Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 G˚ arding’s Inequality for Exterior Second Order Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 G˚ arding’s Inequality for Second Order Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Existence of Solutions to Boundary Value Problems . . . . . . . . . 5.5.1 Interior Boundary Value Problems . . . . . . . . . . . . . . . . . . 5.5.2 Exterior Boundary Value Problems . . . . . . . . . . . . . . . . . 5.5.3 Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 169 181 185 189 191 194 207 207 212 216 227 231 232 238 238 248 252 254 255 255 263 266 271 272 272 276 277

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5.6 Solution of Integral Equations via Boundary Value Problems . 5.6.1 The Generalized Representation Formula for Second Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Continuity of Some Boundary Integral Operators . . . . . 5.6.3 Continuity Based on Finite Regions . . . . . . . . . . . . . . . . . 5.6.4 Continuity of Hydrodynamic Potentials . . . . . . . . . . . . . 5.6.5 The Equivalence Between Boundary Value Problems and Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.6 Variational Formulation of Direct Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.7 Positivity and Contraction of Boundary Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.8 The Solvability of Direct Boundary Integral Equations 5.6.9 Positivity of the Boundary Integral Operators of the Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Partial Differential Equations of Higher Order . . . . . . . . . . . . . . 5.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Assumptions on Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Higher Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . 5.8.3 Mixed Boundary Conditions and Crack Problem . . . . . 6.

277 277 280 282 284 287 289 300 304 305 307 312 312 313 314

Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 6.2 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 6.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 6.4 Time Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 6.4.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 6.5 Electromagnetic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 6.6 Transmission and Boundary Conditions . . . . . . . . . . . . . . . . . . . 328 6.7 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 6.7.1 Scattering problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 6.7.2 Eddy current problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 6.8 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 6.8.1 The cavity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 6.8.2 Exterior problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.8.3 The transmission problem . . . . . . . . . . . . . . . . . . . . . . . . . 339 6.9 Representation Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 6.10 Boundary Integral Equations for Electromagnetic fields . . . . . . 344 6.10.1 The Calderon projector and the capacity operators . . . 349 6.10.2 Weak solutions for a fundamental problem . . . . . . . . . . . 351 6.11 Application of the Electromagnetic Potentials to Eddy Current Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 6.11.1 The ’(A, ϕ) − (A) − (ψ)’ formulation in the bounded domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 6.11.2 The  (A, ϕ) − (ψ) formulation in an unbounded domain369

XVIII Table of Contents

6.11.3 Electric field in the dielectric domain Ω D . . . . . . . . . . . . 6.11.4 Vector potentials — revisited . . . . . . . . . . . . . . . . . . . . . . 6.12 Applications of boundary integral equations to scattering problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12.1 Scattering by a perfect electric conductor, EFIE and MFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12.2 Scattering by a dielectric body . . . . . . . . . . . . . . . . . . . . . 6.12.3 Scattering by objects with impedance boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.

8.

9.

379 387 395 397 416 423

Introduction to Pseudodifferential Operators . . . . . . . . . . . . . 7.1 Basic Theory of Pseudodifferential Operators . . . . . . . . . . . . . . 7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn . . . . . . . . . . . 7.2.1 Systems of Pseudodifferential Operators . . . . . . . . . . . . . 7.2.2 Parametrix and Fundamental Solution . . . . . . . . . . . . . . 7.2.3 Levi Functions for Scalar Elliptic Equations . . . . . . . . . 7.2.4 Levi Functions for Elliptic Systems . . . . . . . . . . . . . . . . . 7.2.5 Strong Ellipticity and G˚ arding’s Inequality . . . . . . . . . . 7.3 Review on Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Local Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Fundamental Solutions in IRn for Operators with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Existing Fundamental Solutions in Applications . . . . . .

429 429 452 454 457 459 466 468 471 472

Pseudodifferential Operators as Integral Operators . . . . . . . 8.1 Pseudohomogeneous Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Integral Operators as Pseudodifferential Operators of Negative Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Non–Negative Order Pseudodifferential Operators as Hadamard Finite Part Integral Operators . . . . . . . . . . . 8.1.3 Parity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 A Summary of the Relations between Kernels and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Coordinate Changes and Pseudohomogeneous Kernels . . . . . . . 8.2.1 The Transformation of General Hadamard Finite Part Integral Operators under Change of Coordinates . . . . . 8.2.2 The Class of Invariant Hadamard Finite Part Integral Operators under Change of Coordinates . . . . . . . . . . . . .

479 479

Pseudodifferential and Boundary Integral Operators . . . . . . 9.1 Pseudodifferential Operators on Boundary Manifolds . . . . . . . . 9.1.1 Ellipticity on Boundary Manifolds . . . . . . . . . . . . . . . . . . 9.1.2 Schwartz Kernels on Boundary Manifolds . . . . . . . . . . . . 9.2 Boundary Operators Generated by Domain Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

474 477

482 506 515 518 520 523 530 539 540 544 546 547

Table of Contents

XIX

Surface Potentials on the Plane IRn−1 . . . . . . . . . . . . . . . . . . . . . 549 Pseudodifferential Operators with Symbols of Rational Type . 572 Surface Potentials on the Boundary Manifold Γ . . . . . . . . . . . . 593 Volume Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Strong Ellipticity and Fredholm Properties . . . . . . . . . . . . . . . . 605 Strong Ellipticity of Boundary Problems and Integral Equations611 9.8.1 The Boundary Value and Transmission Problems . . . . . 611 9.8.2 The Associated Boundary Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 9.8.3 The Transmission Problem and G˚ arding’s inequality . . 614 9.9 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 9.3 9.4 9.5 9.6 9.7 9.8

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 10.1 Newton Potential Operators for Elliptic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 10.1.1 Generalized Newton Potentials for the Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 10.1.2 The Newton Potential for the Lam´e System . . . . . . . . . . 631 10.1.3 The Newton Potential for the Stokes System . . . . . . . . . 632 10.2 Surface Potentials for Second Order Equations . . . . . . . . . . . . . 633 10.2.1 Strongly Elliptic Differential Equations . . . . . . . . . . . . . . 636 10.2.2 Surface Potentials for the Helmholtz Equation . . . . . . . 640 10.2.3 Surface Potentials for the Lam´e System . . . . . . . . . . . . . 645 10.2.4 Surface Potentials for the Stokes System . . . . . . . . . . . . 650 10.3 Invariance of Boundary Pseudodifferential Operators . . . . . . . . 650 10.3.1 The Hypersingular Boundary Integral Operators for the Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 10.3.2 The Hypersingular Operator for the Lam´e System . . . . 657 10.3.3 The Hypersingular Operator for the Stokes System . . . 661 10.4 Derivatives of Boundary Potentials . . . . . . . . . . . . . . . . . . . . . . . 661 10.4.1 Derivatives of the Solution to the Helmholtz Equation 667 10.4.2 Computation of Stress and Strain on the Boundary for the Lam´e System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 10.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 11. Boundary Integral Equations on Curves in IR2 . . . . . . . . . . . . 11.1 Fourier Series Representation of the Basic Operators . . . . . . . . 11.2 The Fourier Series Representation of Periodic Operators A ∈ Lm c (Γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Ellipticity Conditions for Periodic Operators on Γ . . . . . . . . . . 11.3.1 Scalar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Multiply Connected Domains . . . . . . . . . . . . . . . . . . . . . . 11.4 Fourier Series Representation of some Particular Operators . .

675 676 682 688 689 694 698 701

XX

Table of Contents

11.4.1 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 The Lam´e System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 The Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.4 The Biharmonic Equation . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

701 705 707 709 718

12. Remarks on Pseudodifferential Operators for Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 12.2 Symbols of P and the corresponding Newton potentials . . . . . 720 12.3 Representation formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 12.4 Symbols of the Electromagnetic Boundary Potentials . . . . . . . . 723 12.5 Symbols of boundary integral operators . . . . . . . . . . . . . . . . . . . 729 12.6 Symbols of the Capacity Operators . . . . . . . . . . . . . . . . . . . . . . . 731 12.7 Boundary Integral Operators for the Fundamental Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 12.8 Coerciveness and Strong Ellipticity . . . . . . . . . . . . . . . . . . . . . . . 734 12.9 G˚ arding’s inequality for the sesquilinear form A in (6.12.23) . 737 12.10Existence Theorem 6.12.6 revisited . . . . . . . . . . . . . . . . . . . . . . . 740 12.11Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 A. Appendix A: Local Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 745 B. Appendix B: Vector Field Identities, Integration Formulae 751 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775

1. Introduction

This chapter serves as a basic introduction to the reduction of elliptic boundary value problems to boundary integral equations. We begin with model problems for the Laplace equation. Our approach is the direct formulation based on Green’s formula, in contrast to the indirect approach based on a layer ansatz. For ease of reading, we begin with the interior and exterior Dirichlet and Neumann problems of the Laplacian and their reduction to various forms of boundary integral equations, without detailed analysis. (For the classical results see e.g. G¨ unter [162] and Kellogg [224].) The Laplace equation, and more generally, the Poisson equation , −Δv = f

in Ω or Ω c

already models many problems in engineering, physics and other disciplines (Dautray and Lions [95] and Tychonoff and Samarski [422]). This equation appears, for instance, in conformal mapping (Gaier [133, 134]), electrostatics (Gauss [141], Martensen [283] and Stratton [408]), stationary heat conduction (G¨ unter [162]), in plane elasticity as the membrane state and the torsion problem (Szabo [411]), in Darcy flow through porous media (Bear [25] and Liggett and Liu [267]) and in potential flow (Glauert [150], Hess and Smith [178], Jameson [212] and Lamb [258]), 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 corresponding boundary integral equations may have eigensolutions in spite of the uniqueness of the solutions of the original boundary value problems. By appropriate modifications of the boundary integral equations in terms of these eigensolutions, the uniquness of the boundary integral equations can be achieved. Although these simple, classical model problems are well known, the concepts and procedures outlined here will be applied in the same manner for more general 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 = (x1 , . . . , xn ) ∈ © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_1

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 ∈ IRn , we set x·y =

n 

1

xj yj and |x| = (x · x) 2

j=1

for the inner product and the Euclidean norm, respectively. We want to find the solution u satisfying the differential equation −Δv := −

n  ∂2v = f in Ω. ∂x2j j=1

(1.1.1)

Here Ω ⊂ IRn is a bounded, simply connected domain, and its boundary Γ is sufficiently smooth, say twice continuously differentiable, i.e. Γ ∈ C 2 . (Later this assumption will be reduced.) As is known from classical analysis, a classical solution v ∈ C 2 (Ω) ∩ C 1 (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 log |x − y| for n = 2 , E(x, y) = 1 2π 1 (1.1.2) for n =3 . 4π |x−y| The presentation of the solution reads    ∂v ∂E(x, y) E(x, y) (y)dsy − v(y) dsy + E(x, y)f (y)dy v(x) = ∂n ∂ny y∈Γ

y∈Γ

Ω

(1.1.3) for x ∈ Ω ( see Mikhlin [302, p. 220ff.]) where ny denotes the exterior normal to Γ at y ∈ Γ , dsy the surface element or the arclength element for n = 3 or 2, respectively, and ∂v (y) := lim gradv(˜ y ) · ny . y˜→y∈Γ,˜ y ∈Ω ∂n

(1.1.4)

The notation ∂/∂ny will be used if there could be misunderstanding due to more variables. In the case when f ≡ 0 in (1.1.1), one may also use the decomposition in the following form:  E(x, y)f (y)dy + u(x) (1.1.5) v(x) = vp (x) + u(x) := Rn

where u now solves the Laplace equation −Δu = 0 in Ω .

(1.1.6)

1.2 Boundary Potentials and Calder´ on’s Projector

3

Here vp 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 Rn . 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 solution u of the Laplacian (1.1.6) which now can be represented in the form:   ∂u ∂E(x, y) E(x, y) (y) − u(y) dsy . (1.1.7) u(x) = ∂n ∂ny y∈Γ

y∈Γ

∂u |Γ , the representation formula (1.1.7) For given boundary data u|Γ and ∂n defines the solution of (1.1.6) everywhere in Ω. Therefore, the pair of boundary functions belonging to a solution u of (1.1.6) is called the Cauchy data, namely   u|Γ . (1.1.8) Cauchy data of u := ∂u ∂n |Γ

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 residual formulation. The Laplacian (1.1.6) corresponds to the equation of the equilibrium state of the membrane without external body forces and vertical displacement u. Then, for fixed x ∈ Ω, the terms  ∂E(x, y) u(x) + u(y) dsy ∂ny y∈Γ

correspond to the virtual work of the point force at x and of the resulting boundary forces ∂E(x, y)/∂ny against the displacement field u, which are ∂u |Γ acting against equal to the virtual work of the resulting boundary forces ∂n the displacement E(x, y), i.e.  ∂u E(x, y) (y)dsy . ∂n y∈Γ

This equality is known as Betti’s principle (see e.g. Ciarlet [68], Fichera [119] and Hartmann [173, p.159]). Corresponding formulas can also be obtained for more general elliptic partial differential equations than (1.1.6), as will be discussed in Chapter 2.

1.2 Boundary Potentials and Calder´ on’s Projector The representation formula (1.1.7) contains two boundary potentials, the simple layer potential

4

1. Introduction

 V σ(x)

:=

E (x, y)σ(y) dsy ,

x ∈ Ω ∪ Ωc ,

(1.2.1)

y∈Γ

and the double layer potential  ∂ W ϕ(x) := ( E(x, y))ϕ(y)dsy , x ∈ Ω ∪ Ω c . ∂ny

(1.2.2)

y∈Γ

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 not both 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 boundary integral operators, provided the corresponding limits exist. For the potential equation (1.1.6), this is well known from classical analysis (Mikhlin [302, p. 360] and G¨ unter [162, Chap.II]): V σ(x)

:=

Kϕ(x)

:=

K  σ(x)

:=

Dϕ(x)

:=

lim V σ(z)

z→x∈Γ

for x ∈ Γ , (1.2.4)

lim

W ϕ(z) + 12 ϕ(x)

lim

gradz V σ(z) · nx − 12 σ(x) for x ∈ Γ , (1.2.5)

z→x∈Γ,z∈Ω z→x∈Γ,z∈Ω

−

for x ∈ Γ , (1.2.3)

lim

z→x∈Γ,z∈Ω

gradz W ϕ(z) · nx

for x ∈ Γ . (1.2.6)

To be more explicit, we quote the following standard results without proof. Lemma 1.2.1. Let Γ ∈ C 2 and let σ and ϕ be continuous. Then the limits in (1.2.3)–(1.2.5) exist uniformly with respect to all x ∈ Γ and all σ and ϕ with supx∈Γ |σ(x)| ≤ 1, supx∈Γ |ϕ(x)| ≤ 1. Furthermore, these limits can be expressed by  V σ(x) = E(x, y)σ(y)dsy for x ∈ Γ, (1.2.7) y∈Γ \{x}



Kϕ(x)

= y∈Γ \{x}

K  σ(x)



= y∈Γ \{x}

∂E (x, y)ϕ(y)dsy for x ∈ Γ , ∂ny

(1.2.8)

∂E (x, y)σ(y)dsy for x ∈ Γ . ∂nx

(1.2.9)

We remark that here all of the above boundary integrals are improper with weakly singular kernels in the following sense (see [302, p. 158]): The kernel k(x, y) of an integral operator of the form  k(x, y)ϕ(y)dsy Γ

1.2 Boundary Potentials and Calder´ on’s Projector

5

is called weakly singular if there exist constants c and λ < n − 1 such that |k(x, y)| ≤ c|x − y|−λ for all x, y ∈ Γ .

(1.2.10)

For the Laplacian, for Γ ∈ C 2 and E(x, y) given by (1.1.2), one even has |E(x, y)| ≤ cλ |x − y|−λ for any λ > 0 for n = 2 and λ = 1 for n = 3 , (1.2.11) ∂E (x, y) ∂ny

=

(x − y) · ny 1 , 2(n − 1)π |x − y|n

(1.2.12)

∂E (x, y) ∂nx

=

(y − x) · nx 1 for x, y ∈ Γ . 2(n − 1)π |x − y|n

(1.2.13)

In case n = 2, both kernels in (1.2.12), (1.2.13) are continuously extendable to a C 0 -function for y → x (see Mikhlin [302] ), in case n = 3 they are weakly singular with λ = 1 (see G¨ unter [162, Sections II.3 and II.6]). For other differential 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 of Cauchy 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:   C m+α (Γ ) := {ϕ ∈ C m (Γ )  ||ϕ||C m+α (Γ ) < ∞} where the norm is defined by 

||ϕ||C m+α (Γ ) :=

sup |∂ β ϕ(x)| +

|β|≤m

x∈Γ

 |β|=m

sup x,y∈Γ

x=y

|∂ β ϕ(x) − ∂ β ϕ(y)| |x − y|α

for m ∈ IN0 and 0 < α < 1. Here, ∂ β denotes the covariant derivatives β

n−1 ∂ β := ∂1β1 · · · ∂n−1

on the (n − 1)–dimensional boundary surface Γ where β ∈ INn−1 is a multi– 0 index and |β| = β1 + . . . + βn−1 (see Millman and Parker [306]). older continuously differentiable Lemma 1.2.2. Let Γ ∈ C 2 and let ϕ be a H¨ function. Then the limit in (1.2.6) exists uniformly with respect to all x ∈ Γ and all ϕ with  ϕ C 1+α ≤ 1. Moreover, the operator D can be expressed as a composition of tangential derivatives and the simple layer potential operator V: d dϕ V ( ) (x) for n = 2 dsx ds

Dϕ(x)

=

−

Dϕ(x)

=

−(nx × ∇x ) · V (ny × ∇y ϕ)(x) for n = 3 .

(1.2.14)

and (1.2.15)

6

1. Introduction

For the classical proof see Maue [284] and G¨ unter [162, p.73ff]. Note that the differential operator (ny × ∇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 [149, 329]). In the following, we give a brief derivation for these formulae based on classical results of potential theory with H¨older continuous densid ties dϕ ds (y) and (ny × ∇y )ϕ(y), respectively. Note that dsx and (nx × ∇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 z ∈ Ω, z = y ∈ Γ ,  −nx · ∇z (ny · ∇y E(z, y)) ϕ(y)dsy Γ

 =

− 

=

 nx · ∇z ny · ∇y E(z, y) ϕ(y)dsy ,

Γ

 n x · ∇ y ∇ y E(z, y) · ny ϕ(y)ds .

Γ

Here, ⎞ dx2 ⎜ ds ⎟ ⎝ dx ⎠ , 1 − ds ⎛ nx

=

hence, with

⎞ dx1 ⎜ ds ⎟ ⊥ ⎝ dx ⎠ = nx 2 ds ⎛

tx

=

 2 is defined as counterclockwise rotation by where a⊥ := −a a1 An elementary computation shows that

π 2.

  n x Any = −tx A ty + (trace A)tx · ty

for any 2 × 2–matrix A. Hence,   −nx · ∇z W ϕ(z) = nx · ∇y ∇ y E(z, y) · ny ϕ(y)dsy Γ

 {tx · Δy E(z, y)ty − (tx · ∇y )(ty · ∇y E(z, y))}ϕ(y)dsy .

= Γ

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

1.2 Boundary Potentials and Calder´ on’s Projector

7

 −nx · ∇z W ϕ(z)

(tx · ∇z )(ty · ∇y E(z, y))ϕ(y)dsy

= Γ



(tx · ∇z )

=

(

d E(z, y))ϕ(y)dsy dsy

Γ

and, after integration by parts,

 −tx · ∇z

=

E(z, y)

dϕ (y)dsy dsy

Γ

−tx · ∇z {V (

=

dϕ ) (z)} . ds

First note that ∇z V ( dϕ older continuous function for z ∈ Ω which ds )(z) is a H¨ admits a H¨older continuous extension up to Γ (G¨ unter [162, p.68]). The definition of derivatives at the boundary gives us ∇x V (

dϕ dϕ )(x) = lim ∇z V ( )(z) z→x ds ds

which yields dϕ dϕ dϕ d V ( )(x) = tx · ∇x V ( )(x) = lim tx · ∇z V ( )(z) , z→x dsx ds ds ds i.e. (1.2.14). Similarly, for n = 3, we see that  nx · (∇y ∇Ty E(z, y)) · ny ϕ(y)dsy −nx · ∇z W ϕ(z) = Γ

and with the formulae of vector analysis  {(ny · ∇y )(nx · ∇y )E(z, y)}ϕ(y)dsy = Γ

=



−

{(ny × ∇y ) · (nx × ∇y )E(z, y)}ϕ(y)dsy Γ

 {(nx · ny )Δy E(z, y)}ϕ(y)dsy ,

+ Γ

where the last term vanishes since z ∈ Γ . Now, with elementary vector analysis,

8

1. Introduction

 −nx · ∇z W ϕ(z)

=−

{ny · (∇y × (nx × ∇y ))E(z, y)}ϕ(y)dsy Γ 

=−

ny · {∇y × ((nx × ∇y )E(z, y)ϕ(y))}dsy Γ 

−

ny · {(nx × ∇y )E(z, y) × ∇y ϕ(y)}dsy , Γ

where ϕ(y) denotes any C 1+α –extension from Γ into IR3 . The first term on the right–hand side vanishes due to the Stokes theorem, whereas the second term gives  −nx · ∇z W ϕ(z) = − ny · {∇y ϕ(y) × (nx × ∇z )E(z, y)}dsy Γ



=−

(nx × ∇z E(z, y)) · (ny × ∇y ϕ(y))dsy Γ



= −(nx × ∇z ) ·

E(z, y)(ny × ∇y )ϕ(y)dsy . Γ

Since (ny × ∇y )ϕ(y) defines tangential derivatives of ϕ, ∇z · V ((ny × ∇y ) ϕ)(z) defines a H¨older continuous function for z ∈ Ω which admits a H¨older continuous limit for z → x ∈ Γ due to G¨ unter [162, 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 singular operator. This, in fact, is a regularization of the hypersingular distribution, which will also be useful for the variational formulation and related computational procedures. A more elementary, but different regularization can be obtained as follows (see Giroire and Nedelec [149]). From the definition (1.2.6), we see that   ∂E Dϕ(x) = lim − n x · ∇z (z, y) (ϕ(y) − ϕ(x)) dsy Ω z→x∈Γ ∂ny Γ   ∂E − n x · ∇z (z, y)ϕ(x)dsy . ∂ny Γ

If we apply the representation formula (1.1.7) to u ≡ 1, then we obtain Gauss’ well known formula  ∂E (z, y)dsy = −1 for all z ∈ Ω . ∂ny Γ

1.2 Boundary Potentials and Calder´ on’s Projector

This yields

 ∇z

9

∂E (z, y)ϕ(x)dsy = 0 for all z ∈ Ω , ∂ny

Γ

hence, we find the simple regularization  ∂E (z, y) (ϕ(y) − ϕ(x)) dsy . Dϕ(x) = lim −nx · ∇z Ω z→x∈Γ ∂ny

(1.2.16)

Γ

In fact, the limit in (1.2.16) can be expressed in terms of a Cauchy principal value integral,    ∂ ∂ Dϕ(x) = − p.v. E(x, y) (ϕ(y) − ϕ(x)) dsy ∂nx ∂ny Γ   ∂  ∂ := − lim E(x, y) (ϕ(y) − ϕ(x)) dsy ε→0+ ∂nx ∂ny y∈Γ ∧|y−x|≥ε



= lim

ε→0+ y∈Γ ∧|y−x|≥ε

n · n 1 (y − x) · ny (x − y) · nx  x y + n 2(n − 1)π rn r2+n  × ϕ(y) − ϕ(x) dsy .

(1.2.17) The derivation of (1.2.17) from (1.2.16), however, requires detailed analysis (see G¨ unter [162, Section II,10]). 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), this leads to the following relations between the Cauchy data:

and

∂u 1 (x) u(x) = ( I − K)u(x) + V 2 ∂n

(1.2.18)

∂u 1 ∂u (x) = Du(x) + ( I + K  ) (x) . ∂n 2 ∂n

(1.2.19)

∂u  ) on Γ Consequently, for any solution of (1.1.6), the Cauchy data (u, ∂n are reproduced by the operators on the right–hand side of (1.2.18), (1.2.19), namely by  1 I −K , V 2 CΩ := (1.2.20) D , 12 I + K 

This operator is called the Calder´on projector (with respect to Ω) (Calder´on [59]). The operators in CΩ have mapping properties in the classical H¨older function spaces as follows:

10

1. Introduction

Theorem 1.2.3. Let Γ ∈ C 2 and 0 < α < 1, a fixed constant . Then the boundary potentials V, K, K  , D define continuous mappings in the following spaces, → C 1+α (Γ ) , (1.2.21) α 1+α 1+α → C (Γ ), C (Γ ) → C (Γ ) , (1.2.22)

V

:

C α (Γ )

K, K  D

:

C α (Γ )

:

C 1+α (Γ ) → C α (Γ ) .

(1.2.23)

For the proofs see Mikhlin and Pr¨ossdorf [305, Sections IX, 4 and 7]. Remark 1.2.1: The double layer potential operator K and its adjoint K  for 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 [305, Sections IX, 4 and 7] and Colton and Kress [75, 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  correspond to more general elliptic partial differential equations than (1.1.6). This 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 CΩ indeed is a projection. More precisely, there holds: Lemma 1.2.4. Let Γ ∈ C 2 . Then CΩ maps C 1+α (Γ ) × C α (Γ ) into itself continuously. Moreover, 2 CΩ = CΩ . (1.2.24) Consequently, we have the following identities: VD DV KV DK

= = = =

1 4I 1 4I

− K2 , 2

−K , V K , 

K D.

(1.2.25) (1.2.26) (1.2.27) (1.2.28)

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 a direct 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 equations. As is well known, for regular elliptic boundary value problems, only

1.3 Boundary Integral Equations

11

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 equations which may be used for determining the complete Cauchy data. In general, any combination of (1.2.18) and (1.2.19) can serve as a boundary integral equation for the missing part of the Cauchy data. Hence, the boundary integral equations associated with one particular boundary condition are by no means uniquely determined. The ‘direct’ approach for formulating boundary integral equations becomes particularly simple if one considers the Dirichlet problem or the Neumann problem. In what follows, we will always prefer the direct formulation. 1.3.1 The Dirichlet Problem In the Dirichlet problem for (1.1.6), the boundary values u|Γ = ϕ on Γ

(1.3.1)

are given. Hence, ∂u |Γ (1.3.2) ∂n 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 equation (1.2.18) of the Calder´on projection then σ is to be determined by the boundary integral equation σ=

V σ(x) = 12 ϕ(x) + Kϕ(x) , x ∈ Γ .

(1.3.3)

Explicitly, we have  E(x, y)σ(y)dsy = f (x) , x ∈ Γ

(1.3.4)

y∈Γ

where f is given by the right–hand side of (1.3.3) and E is given by (1.1.2), a weakly singular kernel. Hence, (1.3.4) is a Fredholm integral equation of the 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 [349]). However, the modified equation  E(x, y)σ(y)dsy − ω = f (x) , x ∈ Γ (1.3.5) y∈Γ

together with the normalizing condition  σds = Σ y∈Γ

(1.3.6)

12

1. Introduction

is always solvable for σ and the constant ω for given f and given constant Σ [198]. 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 the Calder´ on projector, we arrive at 1 2 σ(x)

− K  σ(x) = Dϕ(x) for x ∈ Γ .

In view of (1.2.9) and (1.2.11), the explicit form of (1.3.7) reads  ∂E (x, y)σ(y)dsy = g(x) for x ∈ Γ , σ(x) − 2 ∂nx

(1.3.7)

(1.3.8)

y∈Γ \{x}

∂E (x, y) is weakly singular due to (1.2.13), provided Γ is smooth. where ∂n x 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 employ different boundary integral equations. In fact, (1.3.8) is one of the celebrated integral equations of classical potential theory — the adjoint to the Neumann– Fredholm integral equation of the second kind with the double layer potential — which can be obtained by using the double layer ansatz in the indirect approach. In the classical framework, the analysis of integral equation (1.3.8) has been studied intensively for centuries, including its numerical solution. For more details and references, see, e.g., Atkinson [16], Bruhn et al. [46], Dautray and Lions [95, 97], Jeggle [214, 215], Kellogg [224], Kral et al [239, 240, 241, 242, 243], Martensen [282, 283], Maz‘ya [289], Neumann [335, 336, 337], Radon [360] and [432]. In recent years, increasing efforts 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 fundamental formulation theoretically as well as computationally. It will be seen that this equation is particularly suitable for the variational analysis.

1.3.2 The Neumann Problem In the Neumann problem for (1.1.6), the boundary condition reads as ∂u |Γ = ψ on Γ ∂n

(1.3.9)

with given ψ. For the interior problem (1.1.6) in Ω, the normal derivative ψ needs to satisfy the necessary compatibility condition  ψds = 0 (1.3.10) Γ

1.4 Exterior Problems

13

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 the Calder´on projector, then u|Γ is determined by the solution of the boundary integral equation 1 (1.3.11) 2 u(x) + Ku(x) = V ψ(x), x ∈ Γ . This is a classical Fredholm integral equation of the second kind on Γ , namely   ∂E u(x) + 2 (x, y)u(y)dsy = 2 E(x, y)ψ(y)dsy =: f (x) . ∂ny y∈Γ \{x}

y∈Γ \{x}

(1.3.12) For the Laplacian we have (1.2.12), which shows that the kernel of the integral 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) holds and that the null–space of (1.3.12) is one–dimensional, see e.g. Mikhlin [303, Chap.17, 11]. Alternatively, if we take the second equation (1.2.19) of the Calder´on projector, we arrive at the equation Du(x) = 12 ψ(x) − K  ψ(x) for x ∈ Γ .

(1.3.13)

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, we will see that, nevertheless, it has advantages for the variational formulation and corresponding numerical treatment.

1.4 Exterior Problems In many applications such as electrostatics and potential flow, one often deals with exterior problems which we will now consider for our simple model equation. 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 −Δu = 0 in Ω c ,

(1.4.1)

u|Γ = ϕ on Γ

(1.4.2)

the boundary condition and the additional growth condition u(x)

=

1 Σ log |x| + ω + o(1) as |x| → ∞ for n = 2 2π

u(x)

=

−

or

(1.4.3)

1 1 Σ + ω + O(|x|−2 ) as |x| → ∞ for n = 3 . (1.4.4) 4π |x|

The Green representation formula now reads   ∂u (x) + ω for x ∈ Ω c , u(x) = W u(x) − V ∂n

(1.4.5)

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) on Γ to the equations     1 u u I +K , −V 2 = on Γ . (1.4.6) 1  ∂u ∂u I − K −D , 2 ∂n ∂n 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 V σ(x)

=

Kϕ(x)

=

K  σ(x)

=

Dϕ(x)

=

lim V σ(z) x ∈ Γ ,

z→x

lim

W ϕ(z) − 12 ϕ(x) , x ∈ Γ ,

(1.4.8)

lim

gradz V (z) · nx + 12 σ(x) , x ∈ Γ ,

(1.4.9)

z→x,z∈Ω c z→x,z∈Ω c

−

(1.4.7)

lim

z→x,z∈Ω c

gradz W ϕ(z) · nx , x ∈ Γ.

(1.4.10)

Note that in (1.4.8) and (1.4.9) the signs at 12 ϕ(x) and 12 σ(x) are different 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 Calder´ on projector CΩ c for the Laplacian with respect to the exterior domain Ω c . Clearly, (1.4.11) CΩ c = I − C Ω , where I denotes the identity matrix operator.

1.4 Exterior Problems

15

For the solution of (1.4.1), (1.4.2) and (1.4.3), we obtain from (1.4.5) a modified boundary integral equation, V σ(x) − ω = − 12 ϕ(x) + Kϕ(x) for x ∈ Γ .

(1.4.12)

∂u This, again, is a first kind integral equation for σ = ∂n |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  σds = Σ . (1.4.13) Γ

This is the same modified system as (1.3.5), (1.3.6), which is always uniquely solvable for (σ, ω). 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 2 σ(x)

+ K  σ(x) = −Dϕ(x) for x ∈ Γ .

(1.4.14)

This is the classical integral equation associated with the exterior Dirichlet problem which has a one–dimensional space of eigensolutions. Here, the special 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 ∂u |Γ = ψ on Γ ∂n

(1.4.15)

where ψ is given. Moreover, we again need a condition at infinity. We choose the growth condition (1.4.3) or (1.4.4), respectively, where the constant Σ is given by  ψds

Σ= Γ

from (1.4.15), where ω is now an additional parameter, which can be prescribed arbitrarily according to the special situation. The representation formula (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 boundary integral equation − 12 u(x) + Ku(x) = V ψ(x) − ω for x ∈ Γ .

(1.4.16)

For any given ψ and ω, this is the classical Fredholm integral equation of the second kind which has been studied intensively (G¨ unter [162]). (See also

16

1. Introduction

Atkinson [16], Dieudonn´e [100], Kral [240, 241], Maz‘ya [289], Mikhlin [301, 303]) (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, Du(x) = − 12 ψ(x) − K  ψ(x) for x ∈ Γ .

(1.4.17)

This equation has the constants as an one–dimensional eigenspace. The special right–hand side in (1.4.17) satisfies an orthogonality condition in the classical Fredholm alternative, which is also valid for (1.4.17), e.g., in the space 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 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 the two–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 ) :

lim

∂q2 ∂q1 − = 0 in Ω c , ∂x1 ∂x2 ∂q2 ∂q1 + = 0 in Ω c , ∂x1 ∂x2

(∇ × q)3

=

divq

=

q · n|Γ

=

0 on Γ ,

|q(x)|

=

|q||T E exists at the trailing edge T E , (1.4.21)

q − q∞

=

o(1) as |x| → ∞ .

Ω c ∈x→T E

(1.4.18) (1.4.19) (1.4.20)

(1.4.22)

Here, the airfoil’s profile Γ is given by a simply closed curve with one corner point at the trailing edge T E. Moreover, Γ has a C ∞ – parametrization x(s) 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 and equal pressure at the trailing edge. (See also Ciavaldini et al [70]). 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.

1.4 Exterior Problems

17

x2 x3

Θ TE O

x1

ϑ ρ

Γ Ω

x

q∞

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 reformulation of (1.4.18)–(1.4.22) as   ω 1 −x2 + ∇u (1.4.23) q = q∞ + 2π |x|2 x1 where u is the solution of the exterior Neumann boundary value problem −Δu = 0 in Ω c ,   ∂u ω0 1 −x2 = −q ∞ · n|Γ − ·n ∂n |Γ 2π |x|2 x1 u(x) = o(1) as |x| → ∞ .

(1.4.24) on Γ ,

(1.4.25) (1.4.26)

In this formulation, u ∈ C 2 (Ω c ) ∩ C 0 (Ω c ) is the unknown disturbance potential, ω0 is 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 [226], see also Goldstein [153]. 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     −y2 ω0 · n (x) for x ∈ Ω c . (1.4.28) u(x) = W u(x) + V q ∞ · n + 2π|y|2 y1

18

1. Introduction



Since

 q ∞ · n(y)dsy =

Γ

(divq ∞ )dx = 0 , Ω

and, by Green’s theorem for     1 −y2 · n(y)ds = y |y|2 y1

|y|=R

Γ

  1 −y2 · n(y)dsy = 0 , |y|2 y1

it follows that every solution u represented by (1.4.28) satisfies (1.4.26) for any choice of ω0 . We now set u = u0 + ω 0 u 1 ,

(1.4.29)

where the potentials u0 and u1 are solutions of the exterior Neumann problems −Δui = 0 in Ω c with i = 0, 1, and   ∂u0 ∂u1 1 −2 −x2 |Γ = −q ∞ · n|Γ , |Γ = − |x| · n|Γ , ∂n ∂n 2π x1 respectively, and ui = o(1) as |x| → ∞, i = 0, 1. Hence, these functions admit the representations u0 (x)

=

u1 (x)

=

W u0 (x) + V (q ∞ · n)(x) ,     1 1 −y2 W u1 (x) + V · n (x) 2π |y|2 y1

(1.4.30) (1.4.31)

for x ∈ Ω c , whose boundary traces are the unique solutions of the boundary integral equations 1 u0 (x) − Ku0 (x) 2 1 u1 (x) − Ku1 (x) 2

= =

V (q ∞ · n)(x) , x ∈ Γ, (1.4.32)     1 1 −y2 · n (x) , x ∈ Γ . (1.4.33) V 2π |y|2 y1

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 [62] and Radon [360], there exist unique solutions u0 and u1 in the class of continuous functions. A more detailed analysis shows that the derivatives of these solutions possess singularities at the trailing edge TE. More precisely, one finds (e.g. in the book by Grisvard [156, Theorem 5.1.1.4 p.255]) that the solutions admit local singular expansions of the form  2π  π  π ui (x) = αi  Θ cos ϑ + O  Θ −ε , i = 0, 1 , (1.4.34) Θ

1.5 Remarks

19

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   2π π π ∇ui (x) = αi  Θ −1 eϑ + O  Θ −1−ε , i = 0, 1 , (1.4.35) Θ π where eϑ is a unit vector with angle (1 − 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 , (1.4.36) α 0 + ω0 α 1 = 0 .

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 [261]). Consequently, ω0 is uniquely determined from (1.4.36). We remark that this choice of ω0 shows that   2π ∇u = O  Θ −1−ε due to (1.4.35) and, hence, the singularity vanishes for Θ < 2π at the trailing edge T E; which indeed is then a stagnation point for the disturbance velocity ∇u. This approach can be generalized to two–dimensional transonic flow problems (Coclici et al [74]).

1.5 Remarks 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 of problems, the approach in the previous sections needs to be modified accordingly. To be more specific, we first consider Lyapounov boundaries. Following Mikhlin [303, Chap.18], a Lyapounov curve in IR2 or Lyapunov surface Γ in unter [162]) satisfies the following two conditions: IR3 (G¨ 1. There exists a normal nx at any point x on Γ . 2. There exist positive constants a and κ ≤ 1 such that for any two points x and ξ on Γ with corresponding vectors nx and nξ the angle ϑ between them satisfies |ϑ| ≤ arκ where r = |x − ξ| .

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 [303, Chap.18]. For a Lyapounov boundary, all results in Sections 1.1–1.4 remain valid if C 2 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 [162], Mikhlin [301, 303, 302] and Smirnov [392]. In applications, one often has to deal with boundary curves with corners, or with boundary surfaces with corners and edges. The simplest generalization of the previous approach can be obtained for piecewise Lyapounov curves in IR2 with finitely many corners where Γ = ∪N j=1 Γ j and each Γj being an open arc of a particular closed Lyapounov curve. The intersections Γ j ∩ Γ j+1 are the corner points where ΓN +1 := Γ1 . In this case it easily follows that there exists a constant C such that  ∂E | (x, y)|ds ≤ C for all x ∈ IR2 . (1.5.1) ∂ny Γ \{x}

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 C 0 (Γ ). However, (1.2.8) needs to be modified and becomes  ∂E ϕ(y) (x, y)dsy Kϕ(x) := ∂ny Γ \{x}  (1.5.2)  1 ∂E + − (x, y)dsy ϕ(x) , 2 ∂ny Γ \{x}

where the last expression takes care of the corner points and vanishes if x is not a corner point. Here K is not compact anymore as in the case of a Lyapounov boundary, however, it can be shown that K can be decomposed into 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 [360]. For the most general two–dimensional case we refer to Kral [240]. For the Neumann problem, one needs a generalization of the normal derivative in terms of the so–called boundary flow, which originally was introduced by Plemelj [349] and has been generalized by Kral [241]. It should be mentioned that in this case the adjoint operator K  to K is no longer a bounded operator on the space of continuous functions (Netuka [334]). 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 valid

1.5 Remarks

21

even for general piecewise Lyapounov surfaces with finitely many corners and edges; see, e.g., Angell et al [13], Burago et al [55, 56], Kral et al [242, 243], Maz‘ya [289] and [432]. On the other hand, as we will see, in the variational formulation of the boundary integral equations, many of these difficulties can be circumvented for even more general boundaries such as Lipschitz boundaries (see Section 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 corresponding traces of u. The functions ϕ+ and ϕ− are given with the additional requirement that ϕ+ − ϕ− = 0 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 the regular 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   ∂u u(x) = WΓ [u](x) − VΓ (x) + ω for x ∈ Ω c , (1.5.4) ∂n where WΓ , VΓ are the corresponding boundary potentials with integration over Γ only,    ∂ WΓ ϕ(x) := E(x, y) ϕ(y)dsy , x ∈ Γ ; (1.5.5) ∂ny y∈Γ  VΓ σ(x) := E(x, y)σ(y)dsy , x ∈ Γ . (1.5.6) y∈Γ

 [u] = u+ − u− , σ :=

 ∂u ∂u+ ∂u− = − on Γ ∂n ∂n ∂n

(1.5.7)

with n the normal to Γ pointing in the direction of the side Γ+ . If we substitute 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  σds = Σ . (1.5.8) Γ

22

1. Introduction

By taking x to Γ+ (or Γ− ) we obtain (in both cases) the boundary integral equation of the first kind for σ on Γ ,  VΓ σ(x) − ω = − 12 ϕ+ (x) + ϕ− (x) + KΓ (ϕ+ − ϕ− )(x) =: f (x) (1.5.9) where KΓ is defined by  KΓ ϕ(x) = y∈Γ \{x}

 ∂  E(x, y) ϕ(y)dsy for x ∈ Γ . ∂ny

(1.5.10)

As for the previous case of a closed curve or surface Γ , respectively, the system (1.5.8), (1.5.9) admits a unique solution pair (σ, ω) for any given ϕ+ , ϕ− and Σ. Here, however, σ will have singularities at the endpoints of Γ or the boundary edge of Γ for n = 2 or 3, respectively, and our classical approach, presented here, requires a more careful justification in terms of appropriate function spaces and variational setting. Γ

˜ \ Γ Γ

Figure 1.5.1: Configuration of an arc Γ in IR2

In a similar manner, one can consider the Neumann problem: Find u satisfying (1.4.1) in Ω c subject to the boundary conditions ∂u+ ∂u− = ψ+ on Γ+ and = ψ− on Γ− ∂n ∂n

(1.5.11)

where ψ+ and ψ− are given smooth functions. By applying the normal derivatives ∂/∂nx 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, DΓ ϕ = − 12 (ψ+ + ψ− ) − KΓ [ψ] on Γ ,

(1.5.12)

1.5 Remarks

23

where the operators DΓ and KΓ again are given by (1.2.6) and (1.2.5) with W and V replaced by WΓ and VΓ , respectively. As we will see later on 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 [210], Stephan et al [403, 406], Costabel et al [79, 85].

2. Boundary Integral Equations

In Chapter 1 we presented basic ideas for the reduction of boundary value problems of the Laplacian to various forms of boundary integral equations based on the direct approach. This reduction can be easily extended to more general partial differential equations. Here we will consider, in particular, the Helmholtz equation, the Lam´e system, the Stokes equations and the biharmonic equation. For the Helmholtz equation, we also investigate the solution’s asymptotic behavior for small wave numbers and the relation to solutions of the Laplace equation by using the boundary integral equations. For the Lam´e system of elasticity, we first present the boundary integral equations of the first kind as well as of the second kind. Furthermore, we study the behavior of the solution and the boundary integral equations for incompressible materials. As will be seen, this has a close relation to the Stokes system and its boundary integral equations. In the two–dimensional case, both the Stokes and the Lam´e problems can be reduced to solutions of biharmonic boundary value problems which, again, can be solved by using boundary integral equations based on the direct approach. In this chapter we consider these problems for domains whose boundaries are smooth enough, mostly Lyapounov boundaries, and the boundary charges belonging to H¨older spaces. Later on we shall consider the boundary integral equations again on Sobolev trace spaces which is more appropriate for stability and convergence of corresponding discretization procedures.

2.1 The Helmholtz Equation A slight generalization of the Laplace equation is the well–known Helmholtz equation −(Δ + k 2 )u = 0 in Ω (or Ω c ) . (2.1.1) This equation arises in connection with the propagation of waves, in particular in acoustics (Filippi [122], Kupradze [251] and Wilcox [438]) and electromagnetics (Ammari [8], Cessenat [63], Colton and Kress [75], Jones [220], M¨ uller [317] and Neledec [331]). In acoustics, k with Im k ≥ 0 denotes the complex wave number and u corresponds to the acoustic pressure field. © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_2

26

2. Boundary Integral Equations

The reduction of boundary value problems for (2.1.1) to boundary integral equations can be carried out in the same manner as for the Laplacian in Chapter 1. For (2.1.1), in the exterior domain, one requires at infinity the Sommerfeld radiation conditions,   ∂u u(x) = O(|x|−(n−1)/2 ) and (2.1.2) (x) − iku(x) = o |x|−(n−1)/2 ∂|x| where i is the imaginary unit. (See the book by Sommerfeld [395] and the further references therein; see also Neittaanm¨ aki and Roach [333] and Wilcox [438]). These conditions select the outgoing waves; they are needed for uniqueness of the exterior Dirichlet problem as well as for the Neumann problem. The pointwise condition (2.1.2) can be replaced by a more appropriate and weaker version of the radiation condition given by Rellich [363, 364],  ∂u | (x) − iku(x)|2 ds = 0 . (2.1.3) lim R→∞ ∂n |x|=R

This form is to be used in the variational formulation of exterior boundary value problems. The fundamental solution E(x, y) to (2.1.1), subject to the radiation condition (2.1.2) for fixed y ∈ Rn is given by ⎧ i (1) ⎪ H (kr) in IR2 , ⎪ ⎪ ⎨4 0 with r = |x − y| (2.1.4) Ek (x, y) = ⎪ ikr ⎪ e ⎪ 3 ⎩ in IR , 4πr (1)

where H0 denotes the modified Bessel function of the first kind. We note, that for n = 2, E(x, y) has a branch point for C  k → 0. Therefore, in the following we confine ourselves first to the case k = 0. In terms of these fundamental solutions, which obviously are symmetric, the representation of solutions to (2.1.1) in Ω or in Ω c with (2.1.2) assumes the same forms as (1.1.7) and (1.4.5), namely     ∂u ∂Ek (x, y) Ek (x, y) (y)dsy − u(y) dsy u(x) = ± ∂n ∂ny (2.1.5) y∈Γ y∈Γ∂u  = ± Vk (x) − Wk u(x) ∂n c for all x ∈ Ω or Ω respectively, where the ±sign corresponds to the interior and the exterior domain. Here, Vk is defined by  Ek (x, y)σ(y)dsy Vk σ(x) := y∈Γ

= V σ(x) + Sk σ(x) +

 ik {δn3 4π

− δn2 (log k + γ0 )}

(2.1.6) σds ,

Γ

2.1 The Helmholtz Equation

27

where V is given by (1.2.1) with (1.1.2) and Sk is a k–dependent remainder defined by ⎧  i  1 (1) ⎪ H0 (k|x − y|) + log |x − y| σ(y)dsy , ⎪ ⎪ 4 2π ⎪ ⎨ y∈Γ  Sk σ(x) = (2.1.7) ∞   1 (m − 1) 1 ⎪ m ⎪ ⎪ (ik|x − y|) σ(y)ds − y ⎪ ⎩ 4π m! |x − y| m=2 y∈Γ

for n = 2 or 3, respectively. The potential Wk is defined by  ∂Ek (x, y) ϕ(y)dsy = W ϕ(x) + Rk ϕ(x) Wk ϕ(x) := ∂ny

(2.1.8)

y∈Γ

where W is given by (1.2.2) with (1.1.2) and Rk is a k–dependent remainder defined by  ⎧ ∂ i (1) 1 ⎪ ⎪ log |x − y|}ϕ(y)dsy , { H0 (k|x − y|) + ⎪ ⎪ 2π ⎪ ⎨ y∈Γ \{x} ∂ny 4   Rk ϕ(x) = ∞  ∂ (m − 1) 1  1 ⎪ ⎪ ⎪ − (ik|x − y|)m ϕ(y)dsy ⎪ ⎪ m! ∂ny |x − y| ⎩ 4π m=2 y∈Γ \{x}

(2.1.9) for n = 2 or 3, respectively. Note that Sk and Rk have bounded kernels for all y ∈ Γ and x ∈ IRn and for k = 0 in the case n = 2. Hence, Sk σ(x) and Rk ϕ(x) are well defined for all x ∈ IRn . Moreover, the properties of the operators V and W , as given by Lemmata 1.2.1, 1.2.2, 1.2.4 and Theorem 1.2.3 for the Laplacian, remain valid for Vk and Wk . Since the kernels of Vk and Wk , Sk and Vk depend analytically on k ∈ C \ {0} for n = 2 and k ∈ C for n = 3, the solutions of the corresponding boundary integral equations will depend analytically on the wave number k, as well. Here again we consider the interior and exterior Dirichlet problem for (2.1.1), where u|Γ = ϕ on Γ is given . (2.1.10) In acoustics, (2.1.10) models a ”soft” boundary for the interior and a ”soft scatterer” for the exterior problem. As in Section 1.3.1, here the missing ∂u Cauchy datum on Γ is = σ. ∂n |Γ The boundary integral equation for the interior Dirichlet problem reads Vk σ(x) = where

1 ϕ(x) + Kk ϕ(x), 2

x∈Γ,

(2.1.11)

28

2. Boundary Integral Equations

 Kk ϕ(x) := y∈Γ \{x}

∂Ek (x, y) ϕ(y)dsy = Kϕ(x) + Rk ϕ(x) ∂ny

for x ∈ Γ

(2.1.12) with K given by (1.2.8) with (1.2.12) or (1.2.13), respectively. As before, (2.1.11) is a Fredholm boundary integral equation of the first kind for σ on Γ . In the classical H¨older continuous function space C α , 0 < α < 1, this integral equation has been studied by Colton and Kress in [75, Chap.3]. In particular, for k = 0 and ϕ ∈ C 1+α (Γ ), (2.1.11) is uniquely solvable with σ ∈ C α (Γ ), except for certain values of k ∈ C which are the exceptional or irregular frequencies of the boundary integral operator Vk . For any irregular frequency k0 , the operator Vk0 has a nontrivial nullspace ker Vk0 = span {σ0j }. The eigensolutions σ0j are related to the eigensolutions u 0j of the interior Dirichlet problem for the Laplacian, −Δ u0 = k02 u 0 in Ω , u 0|Γ = 0 on Γ .

(2.1.13)

That is, σ0j =

∂ u0j . ∂n|Γ

(2.1.14)

Moreover, the solutions are real–valued and dim kerVk0 = dimension of the eigenspace of (2.1.13) . As is known, see e.g. Hellwig [177, p.229], the eigenvalue problem (2.1.13) 2 . They are all real and admits denumerably infinitely many eigenvalues k0 have at most finite multiplicity. Moreover, they can be ordered according to 2 2 < · · · and have +∞ as their only limit point. For any of size 0 < k01 < k02 the corresponding eigensolutions u 0j , (2.1.14) can be obtained from (2.1.5) applied to u 0j . In this case, when k0 is an eigenvalue, the interior Dirichlet problem (2.1.1), (2.1.10) admits solutions in C α (Γ ) if and only if the given boundary values ϕ ∈ C 1+α (Γ ) satisfy the orthogonality conditions   ∂ u0 ds = 0 for all σ0 ∈ kerVk0 . ϕσ0 ds = ϕ (2.1.15) ∂n Γ

Γ

Correspondingly, for ϕ ∈ C 1+α (Γ ), the boundary integral equation (2.1.11) has solutions σ ∈ C α (Γ ) if and only if (2.1.15) is satisfied. For the exterior Dirichlet problem, i.e. (2.1.1) in Ω c with the Sommerfeld radiation conditions (2.1.2) and boundary condition (2.1.10), from (2.1.5) again we obtain a boundary integral equation of the first kind, Vk σ(x) = − 12 ϕ(x) + Kk ϕ(x) , x ∈ Γ ,

(2.1.16)

which differs from (2.1.11) only by a sign in the right–hand side. Hence, the exceptional values k0 are the same as for the interior Dirichlet problem,

2.1 The Helmholtz Equation

29

namely the eigenvalues of (2.1.13). If k = k0 , (2.1.16) is always uniquely solvable for σ ∈ C α (Γ ) if ϕ ∈ C 1+α (Γ ). For k = k0 , in contrast to the interior Dirichlet problem, the exterior Dirichlet problem remains uniquely solvable. However, (2.1.16) now has eigensolutions, and the right–hand side always satisfies the orthogonality conditions   1 − 2 ϕ(x) + Kk0 ϕ(x) σ0 (x)dsx x∈Γ

 =

  ϕ(x) − 12 σ0 (x) + Kk 0 σ0 (x) dsx = 0 for all σ0 ∈ ker Vk0 ,

x∈Γ

since σ0 is real valued and the simple layer potential Vk0 σ0 (x) vanishes identically for x ∈ Ω c . The latter implies ∂ 1 Vk0 σ0 (x) = − σ0 (x) + Kk 0 σ0 (x) = 0 for x ∈ Γ , ∂nx 2 where Kk σ(x)

 := y∈Γ \{x}

 ∂  Ek (x, y) σ(y)dsy = K  σ(x) + Rk σ(x) ∂nx

with

 ⎧  ∂  i (1) 1 ⎪ ⎪ H log |x − y| σ(y)dsy , (k|x − y|) + ⎪ 0 ⎪ 2π ⎪ ⎨ y∈Γ \{x} ∂nx 4   Rk σ(x) = ∞  ∂ (m − 1) 1  1 ⎪ ⎪ ⎪ − (ik|x − y|)m σ(y)dsy , ⎪ ⎪ m! ∂nx |x − y| ⎩ 4π m=2 y∈Γ \{x}

for n = 2 and n = 3, respectively. Accordingly, the representation formula ∂u (2.1.5) with u|Γ = ϕ and ∂n = σ will generate a unique solution for any σ |Γ solving (2.1.16). Alternatively, both, the interior and exterior Dirichlet problem can also be solved by the Fredholm integral equations of the second kind as (1.3.7) and (1.4.14) for the Laplacian. In order to avoid repetition we summarize the different direct formulations of the interior and exterior Dirichlet and Neumann problems which will be abbreviated by (IDP), (EDP), (INP) and (ENP), accordingly, in Table 2.1.1. The Neumann data in (INP) and (ENP) will be denoted by ψ. In addition to the previously defined integral operators Vk , Kk , Kk we also introduce the hypersingular integral operator, Dk for the Helmholtz equation, namely  ∂Ek (x, y) ∂ ∂ ϕ(y)dsy = Dϕ(x) + Rk ϕ(x) on Γ , Dk ϕ(x) := − ∂nx ∂ny ∂nx Γ \{x}

(2.1.17)

ENP

INP

(2)( 12 I − Kk )u = −Vk ψ

(1)Dk u = −( 12 I + Kk )ψ

(2)( 12 I + Kk )u = Vk ψ

(1)Dk u =

− ∂u ˜1 ∂n |Γ

−(Δ +

(D0 )

= 0 on Γ

= 0 in Ω

Kk )ψ

( 12 I

k12 )˜ u1

(N0 ) :

= −Dk ϕ

Kk )σ

+

u ˜0|Γ = 0 on Γ

(1)Vk σ = (− 12 I + Kk )ϕ

(2)( 12 I

−(Δ + k02 )˜ u0 = 0 in Ω,

(2)( 12 I − Kk )σ = Dk ϕ

EDP

(D0 ):

(1)Vk σ = ( 12 I + Kk )ϕ ∂u ˜0 ∂n |Γ

Dk 0 u0 =

∂u ˜0 ∂n |Γ

u1 = u ˜1|Γ on Γ

Vk 1 σ 1 = u ˜1 on Γ

σ0 =

or u0 , u1

Exceptional values k0 , k1

IDP

for BIE, σ0 , σ1

Eigensolutions

for BVP and

BIE

BVP

Eigensolutions u ˜0 or u ˜1

Γ



Γ



None

u1 ψds = 0

None

σ0 ϕds = 0

given ϕ, ψ

Conditions for

Solvability

Table 2.1.1. Summary of the boundary integral equations for the Helmholtz equation and the related eigenvalue problems

30 2. Boundary Integral Equations

2.1 The Helmholtz Equation

31

where D is the hypersingular integral operator (1.4.10) of the Laplacian and Rk is the remainder in (2.1.9). Note that the relations between the eigensolutions of the BIEs and the interior eigenvalue problems of the Laplacian are given explicitly in column three of Table 2.1.1. We also observe that for the exterior boundary value problems, the exceptional values k0 and k1 of the corresponding boundary integral operators depend on the type of boundary integral equations derived by the direct formulation. For instance, we see that for (EDP), k0 are the exceptional values for Vk whereas k1 are those for ( 12 I +Kk ). Similar relations hold for (ENP). In Table 2.1.1, the second column contains all of the boundary integral equations (BIE) obtained by the direct approach. As we mentioned earlier, for the exterior boundary value problems, the solvability conditions of the corresponding boundary integral equations at the exceptional values are always satisfied due to the special forms of the corresponding right-hand sides. For the indirect approach, this is not the case anymore; see Colton and Kress [75, Chap. 3]. There are various ways to modify the boundary integral equations so that some of the exceptional values will not belong to the spectrum of the boundary integral operator anymore. In this connection, we refer to the work by Brakhage and Werner [42] Colton and Kress [75], Jones [220], Kleinman and Kress [228] and Ursell [423], to name a few. 2.1.1 Low Frequency Behaviour Of particular interest is the case k → 0 which corresponds to the low– frequency behaviour. This case also determines the large–time behaviour of the solution to time–dependent problems if (2.1.1) is obtained from the wave equation by the Fourier–Laplace transformation (see e.g. MacCamy [274, 275] and Werner [436]). As will be seen, some of the boundary value problems will exhibit a singular behaviour for k → 0. The main results are summarized in the Table 2.1.2 below. To illustrate the singular behaviour we begin with the explicit asymptotic expansions of the boundary integral equations in Table 2.1.1. Our presentation here follows [203]. In particular, we begin with the fundamental solution for small kr having the series expansions: For n = 2: Ek (x, y) =

i (1) 1 H0 (kr) = E(x, y) − (log k + γ0 ) + Sk (x, y) , 4 2π

where γ0 = c0 − log 2 − i and

π with c0 ≈ 0.5772 , 2

Euler’s constant,

(2.1.18)

32

2. Boundary Integral Equations

Sk (x, y)

i (1) 1 H (kr) − {E(x, y) + (log(kr) + γ0 )} 4 0 2π ∞ ∞   1 − {log(kr) am (kr)2m + bm (kr)2m } , 2π m=1 m=1

= =

am

(−1)m

=

22m (m!)2

,

bm = (γ0 − 1 −

1 1 − · · · − )am . 2 m

For n = 3: Ek (x, y) = E(x, y) +

ik + Sk (x, y) , 4π

where ∞ 1 1  (ikr)m ikr (e − 1 − ikr) = . Sk (x, y) = 4πr 4πr m=2 m!

(2.1.19)

Correspondingly, for the double layer potential kernel we obtain ∂ ∂ Ek (x, y) = E(x, y) + Rk (x, y) , ∂ny ∂ny where Rk (x, y)

=

∞ 

(am (1 + 2m log(kr)) + 2mbm ) (rk)2m

m=1

for n = 2 , and Rk (x, y)

=

−

 ∂ E(x, y) ∂ny (2.1.20)

∞   ∂ (m − 1) (ikr)m E(x, y) m! ∂ny m=2

for n = 3 .

(2.1.21)

The kernel of the adjoint operator Rk can be obtained by interchanging the variables x and y. Hence, Rk has the same asymptotic behaviour as Rk for k → 0. Similarly, for the hypersingular kernel we have, for n = 2, ∞  ∂E ∂E ∂Rk (x, y) = −4π (m − 1)cm (k)(rk)2m (x, y) ∂nx ∂n x ∂ny m=2

+

∞ nx · ny 1  , cm (k)(rk)2m 2π m=1 r2

+ log r

∞  m=1

where

2mam (rk)2m

(2.1.22)

 1 n ·n ∂E ∂E  x y − 4π(m − 1) 2π r2 ∂nx ∂ny

2.1 The Helmholtz Equation

33

cm (k) = (1 + 2m log k)am + 2mbm , and, for n = 3, ∞ ∂Rk nx · ny 1  (m − 1) (ikr)m (x, y) = − ∂nx 4π m=2 m! r3 ∞ 

∂E ∂E (m − 1) r(ikr)m − 4π (3 + m) (x, y) . m! ∂n x ∂ny m=2

(2.1.23)

∂Rk (x, y) is symmetric. ∂nx As can be seen from the above expansions, the term log k appears in (2.1.18) explicitly which shows that Vk is a singular perturbation of V whereas the other operators are regular perturbations of the corresponding operators of the Laplacian as k → 0. Note that the kernel

IDP: Let us consider first the simplest case, i.e. Equation (2) for (IDP) in Table 2.1.1, 1   (2.1.24) 2 I − K − Rk σ = Dk ϕ on Γ . Since, for given ϕ ∈ C 1+α (Γ ), the equation σ = Dϕ ( 12 I − K  ) has a unique solution σ  ∈ C α (Γ ), we may rewrite (2.1.24) as σ

1 1 ( I − K  )−1 Rk σ + ( I − K  )−1 Dk ϕ , 2 2 ∂ 1 1 Rk ϕ , = σ  + ( I − K  )−1 Rk σ + ( I − K  )−1 2 2 ∂nx  O(k 2 log k) for n = 2 , = σ + 2 for n = 3 , O(k )

=

(2.1.25)

where the last expressions can be obtained from the expansions (2.1.20) and (2.1.22) in case n = 2 and from (2.1.21) and (2.1.23) in case n = 3 (see MacCamy [274]). The analysis for the integral equation of the first kind (1) for (IDP) in Table 2.1.1 is more involved, depending on n = 2 or 3. For n = 2, from (2.1.11) with the expansion (2.1.18) of (2.1.7) we have V σ + ω + Sk σ = where ω=−

1 ϕ + Kϕ + Rk ϕ 2

1 (log k + γ0 ) 2π

(2.1.26)

 σds . Γ

(2.1.27)

34

2. Boundary Integral Equations

Similar to (2.1.25), we seek the solution of (2.1.26) and (2.1.27) in the asymptotic form, σ = σ  + α1 (k) σ1 + σ R , (2.1.28) ω = ω  + α1 (k) ω1 + ω R , where σ , ω  correspond to the solution of the interior Dirichlet problem for the Laplacian (1.1.6), (1.3.1). Hence, σ , ω  satisfy (1.3.3), namely Vσ +ω =

1 ϕ + Kϕ 2

(2.1.29)

subject to the constraints  σ ds = 0 and ω  = 0. Γ

The first perturbation terms σ 1 , ω 1 are independent of k with the coefficient α1 (k) = o(1) for k → 0. The functions σR , ωR are the remainders which 1 and ω 1 , we employ equation (2.1.26) are of order o(α1 (k)). To construct σ inserting (2.1.28). For k → 0 we arrive at Vσ  +ω 1  1 σ 1 ds

=

0,

=

1,

(2.1.30)

Γ

where we appended the last normalizing condition for σ 1 , since from the  1 ds = 0 would yield the previous results in Section 1.3 we know that σ Γ

1 = 0. Inserting (2.1.28) into (2.1.27), it follows trivial solution σ 1 = 0, ω 1 ds = 1 with ω  = 0 that α1 (k) = O(σR ). Hence, without loss of from σ Γ

generality, we may set α1 (k) = 0 in (2.2.28). Now (2.1.26) and (2.1.27) with (2.1.28) imply that the remaining terms σR , ωR satisfy the equations 

, V σR + ωR + Sk σR = Rk ϕ − Sk σ σR ds + ωR 2π(log k + γ0 )−1 = 0 ;

(2.1.31)

Γ

which are regular perturbations of equations (1.3.5), (1.3.6) due to the expansions for Sk in (2.1.18). The right–hand side of (2.1.31) is of order O(k 2 log k) because of the expansions of Sk and of Rk in (2.1.20). Therefore, σR and ωR are, indeed, of order O(k 2 log k) as in (2.1.25), which was already obtained with the integral equations (2.1.24) of the second kind. For n = 3, the integral equation of the first kind takes the form  1 1 σds + Sk σ = ϕ + Kϕ + Rk ϕ , (2.1.32) V σ + ik 4π 2 Γ

2.1 The Helmholtz Equation

35

where the kernels of the integral operators Sk and Rk are given by (2.1.19) and (2.1.21), respectively. Both are of order O(k 2 ). Hence, (2.1.32) is a regular perturbation of equation (1.3.3). If we insert (2.1.28) with α1 (k) = k then the function σ  is given by the solution of Vσ  = 12 ϕ + Kϕ , which corresponds to the interior Dirichlet problem of the Laplacian. Hence,  σ ds = 0 and σ 1 ≡ 0 Γ

since it is the solution of Vσ 1 +

i 4π

 σ ds = 0 . Γ

Therefore, the solution of (2.1.32) is of the form σ = σ  + O(k 2 ). EDP: By using the indirect formulation of boundary integral equations, this case was also analyzed by Hariharan and MacCamy in [171]. In case n = 2, for the exterior Dirichlet problem (EDP), Equation (1) in Table 2.1.1 has the form (2.1.33) V σ + ω + Sk σ = − 12 ϕ + Kϕ + Rk ϕ with ω again defined by (2.1.27). Again, the solution admits the asymptotic expansion (2.1.28) with σ , ω  being the solution of  Vσ +ω  = − 12 ϕ + Kϕ , σ ds = 0 . (2.1.34) Γ

Hence, σ , ω  correspond to the exterior Dirichlet problem in Section 1.4.1. Therefore, in contrast to the (IDP), here ω  = 0, in general. Again, σ 1 , ω 1 are solutions of equations (2.1.30). In contrast to the (IDP), the coefficient α1 (k) is explicitly given by  −1 1 α1 (k) = − (log k + γ0 ) + ω ω 1 , π which is not identically equal to zero, in general. The remainders σR , ωR satisfy equations similar to (2.1.31) and are of order O(k 2 log k). For the case n = 3, the integral equation(1) for (EDP) is of the form  1 V σ + ik σds + Sk σ = − 12 ϕ + Kϕ + Rk ϕ 4π Γ

and σ is of the form

36

2. Boundary Integral Equations

σ=σ  + k σ 1 + σR . This yields Vσ  = − 12 ϕ + Kϕ for σ  corresponding to the (EDP) for the Laplacian . For the next term σ 1 we have  i Vσ 1 = − σ ds . 4π Γ

Therefore, σ 1 is proportional to the natural charge q which is the unique eigensolution of   1 q + K q = 0 on Γ with qds = 1 . 2 Γ

Note that the corresponding simple layer potential satisfies V q(x) = c0 = const. for all x ∈ Ω .

(2.1.35)

For n = 3, a simple contradiction argument shows c0 = 0. Hence,  i   (x) = − σ ds q(x) . σ 1 4πc0 Γ

The remainder term σR is of order k 2 which follows easily from  2 i σ + k σ1 ) − k σ 1 ds − Rk ϕ , V σR + Sk σR = −Sk ( 4π Γ

which is a regular perturbation of (1.4.12). For the integral equation of the second kind (2) of the (EDP) and k = 0, the homogeneous reduced integral equation reads 1 q + K q = 0 2

(2.1.36)

with the natural charge q as an eigensolution. We therefore modify boundary integral equation (2) by using a method by Wielandt [437] (see also Werner [435]). Using Equation (1), we obtain a normalization condition for σ,  σds = lk (ϕ) +  lk (σ) , Γ

where the linear functionals lk and lk are given by

2.1 The Helmholtz Equation

 ⎧ −1    1 ⎪ ϕqds + (Rk ϕ)qds ⎨ 2π (log k + γ0 ) − c0 Γ  lk (ϕ) = Γ i −1 ⎪ ϕqds + (Rk ϕ)qds ⎩(c0 + k 4π ) Γ

37

for n = 2 , for n = 3 ;

Γ

and

⎧ −1  1 ⎪ (Sk σ)qds ⎨ 2π (log k + γ0 ) − c0 Γ  lk (σ) =   −1 i ⎪ (Sk σ)qds ⎩ c0 + k 4π

for n = 2 , for n = 3 .

Γ

The modified boundary integral equation of the second kind reads   1 lk (σ) = −Dk ϕ + qlk (ϕ) , (2.1.37) ( 2 I + K )σ + q σds + Rk σ − q Γ

which is a regular perturbation of the equation   1 σ+q σ ds = −Dϕ + ql0 (ϕ) . ( 2 I + K ) Γ

The latter is uniquely solvable. The next term σ 1 satisfies the equation    1 σ1 + q σ 1 ds = ϕqds . ( 2 I + K ) Γ

Γ

It is not difficult to see that both boundary integral equations (2.1.33) and (2.1.37) provide the same asymptotic solutions  σ  + α1 (k) σ1 + O(k 2 log k) for n = 2 , σ= 2 for n = 3 . σ  + k σ1 + O(k ) INP: Since the boundary integral equation of the second kind (see (2) in Table 2.1.1) for the (INP) and k = 0 has the constant functions as eigensolutions, we need an appropriate modification of (2). This modification can be derived from the Helmholtz equation (2.1.1) in Ω together with the Neumann boundary condition ∂u |Γ = ψ on Γ , (2.1.38) ∂n namely from the solvability condition (Green’s formula for the Laplacian )   1 u(x)dx = − 2 ψds . (2.1.39) k Ω

Γ

This condition can be rewritten in terms of the representation formula (2.1.5),

38

2. Boundary Integral Equations



 Wk u(x)dx =

Ω

1 Vk ψ(x)dx + 2 k

Ω

 ψds ,

(2.1.40)

Γ

the left–hand side of which actually depends on the boundary values u|Γ . It is not difficult to replace the domain integration on the left–hand side by appropriate boundary integrals. With the help of (2.1.40), the boundary integral equation of the second kind (see (2) in Table 2.1.1) for (INP) can be modified,   1 1 1 u + Ku − W u(x)dx + Rk u − Rk u(x)dx 2 |Ω| |Ω| Ω

Ω



 1 ψds + V ψ − V ψ(x)dx |Ω| Γ Ω  1 + Sk ψ − Sk ψ(x)dx on Γ |Ω|

1 =− |Ω|k 2

(2.1.41)

Ω

where |Ω| = meas(Ω). Note, that the constant term of order log k in Vk for n = 2 and of order k for n = 3, respectively, has been canceled in (2.1.41) due to (2.1.40). The boundary integral operator on the left–hand side in (2.1.41) is a regular perturbation of the reduced modified operator  1 1 A := 2 I + K − W • dx on Γ |Ω| Ω

due to (2.1.20) and (2.1.21). Moreover, we will see that the reduced homogeneous equation Av0 = 0 on Γ admits only the trivial solution v0 = 0, since for the equivalent equation,  1 1 W v0 dx = κ = const , (2.1.42) 2 v0 + Kv0 = |Ω| Ω

the orthogonality condition for the original reduced operator on the left–hand side reads  qκds = κ = 0 . Γ

This implies from (2.1.42) that v0 = const as the eigensolution discussed in Section 1.3.2. Hence, with (1.1.7), we have  1 W v0 dx = −v0 . 0=κ= |Ω| Ω

2.1 The Helmholtz Equation

39

Consequently, A−1 exists in the classical function space C α (Γ ) due to the Fredholm alternative. As for the asymptotic behaviour of u, it is suggested from (2.1.40) or (2.1.41) that u admits the form u=

1 α+u  + uR . k2

(2.1.43)

The first term α satisfies the equation   1 1 W αdx = − ψds Aα = 12 α + Kα − |Ω| |Ω| Ω

Γ

which has the unique solution α=−

1 |Ω|

 ψds = const .

(2.1.44)

Γ

The second term u  is the unique solution of the equation    1 1 1 V ψdx − α lim 2 Rk 1 − Rk 1dx . A u=Vψ− k→0 k |Ω| |Ω| Ω

(2.1.45)

Ω

Now we investigate the last term in (2.1.45). For n = 2, we have  |Ω| k2 Rk 1 = (log k + γ0 )k 2 + log |x − y|dy + O(k 4 log k). 2π 2π Ω

This implies   1 1 R 1 − R 1dx , k k k2 |Ω| Ω     1  1 = log |x − y|dy − log |x − y|dydx + O(k 2 log k) . 2π |Ω| Ω

Ω Ω

Hence, the limit on the right–hand side in (2.1.45) exists. For n = 3, the limit is a well defined function in C α (Γ ) due to (2.1.21). The remainder uR satisfies the equation  1 Rk uR (x)dx = fR on Γ (2.1.46) AuR + Rk uR − |Ω| Ω

where

40

2. Boundary Integral Equations

 fR =

O(k 2 log k) O(k 2 )

for n = 2 , for n = 3 .

Hence, uR is uniquely determined by (2.1.46) and is of the same order as fR . For the first kind hypersingular equation (1) in (INP) we use the same normalization condition (2.1.40) by subtracting it from the equation and obtain the modified hypersingular equation   ∂ Rk u − Rk udz Du − W u(z)dz − ∂nx Ω

( 12 − K  )ψ − Rk ψ −

=



Ω

Ω



+ |Ω| δn2



V ψdz −

Sk ψdz Γ

1 ik  (log k + γ0 ) − δn3 2π 4π

 ψds − Γ

(2.1.47) 1 k2

 ψds . Γ

Again, the operator on the left-hand side is a regular perturbation of the reduced operator  Bu := Du − W u(z)dz, Ω

which can be shown to be invertible, in the same manner as for A. From the previous analysis, we write the solution in the form (2.1.43), namely u=

α +u  + uR . k2

We note that B1 = |Ω|, and therefore (2.1.47) for k → 0 again yields (2.1.44) for α. We further note that for n = 2, it can be verified from (2.1.20) that  α 1 |Ω|2 α log k + O(1) for k → 0 , Rk dz = k2 2π Ω

which shows that the choice of α from (2.1.44) cancels the term 1 |Ω| log k ψds on the right–hand side of (2.1.47). 2π Γ Consequently in both cases n = 2 and n = 3, u  can be obtained as the unique solution of the equation  1 B u = ψ − K  ψ − V ψdz + αχn , 2 Ω

where

2.1 The Helmholtz Equation

⎧ 1   |Ω|2 γ0 ⎪ 2 ⎪ |Ω| log k , + lim R 1dz − − ⎪ k ⎪ ⎨ 2π k→0 k 2 2π Ω χn = 1 ∂ ⎪ ⎪ lim { R 1 + Rk 1dz} , ⎪ k ⎪ ⎩k→0 k 2 ∂nx

41

n=2 , n=3 .

Ω

Similarly, again we can show that uR is the unique solution of the equation  ∂ Rk uR − Rk uR (z)dz = fR BuR − ∂nx Ω

with fR

=

−Rk ψ −

 Sk ψdx + Ω

α ∂ Rk 1 + + 2 k ∂nx



∂ Rk u + ∂nx

Rk u dx Ω



Rk 1dx − δn2 Ω

1 ∂ Rk 1 + −α lim 2 k→0 k ∂nx

|Ω|2 2 |Ω|2 3  k log k + δn3 ik 2π 4π

 Rk 1dx − δn2

 |Ω|2 2 k log k . 2π

Ω

For n = 2, the above relations show that fR = O(k 2 log k), which implies that uR = O(k 2 log k), as well. In case n = 3, by using the Gaussian theorem for the first two terms from (2.1.21) we obtain explicitly   1 i 3 1 Rk 1 = Rk (x, y)dsy = − k 2 dy − k |Ω| + O(k 4 ) . 4π |x − y| 4π Γ

Ω

Hence, ∂ Rk 1 ∂nx  Rk 1dx Ω

= =



∂ 1 dy + O(k 4 ) , ∂nx |x − y| Ω   1 i 3 2 1 2 dydx − k |Ω| + O(k 4 ) . − k 4π |x − y| 4π

−

1 2 k 4π

Ω Ω

Together with (2.1.44) this yields fR = O(k 2 ) which implies that uR is of the same order as k → 0. ENP: Finally, let us consider the boundary integral equations of (ENP) and begin with the simplest case, i.e. the integral equation of the second kind (see (2) in Table 2.1.1),

42

2. Boundary Integral Equations

    1 1 ik u − Ku − Rk u = −V ψ + δn2 log k + γ0 − δn3 ψds − Sk ψ . 2 2π 4π Γ

(2.1.48) The operator on the left–hand side is a regular perturbation of the reduced invertible operator 12 I − K (see Section 1.4.2). For n = 2, Sk ψ is of order O(k 2 log k) in view of (2.1.19). Hence, for u|Γ we use the expansion 1 u= (log k + γ0 )α + u  + uR . 2π The highest order term  ψds

α= Γ

is the unique solution of the reduced equation  1 ψds ; 2 α − Kα = Γ

the second term u  is the unique solution of the reduced equation 1 − 2u

Ku  = −V ψ .

(2.1.49)

For the remainder uR we obtain the equation 1 2 uR

− KuR − Rk uR =

α (log k + γ0 )Rk 1 + Rk u  − Sk ψ . 2π

The dominating term on the right–hand side is defined by Rk 1 =

1 |Ω|k 2 log k + O(k 2 ) , 2π

therefore uR is of order O((k log k)2 ). For n = 3, Sk ψ in (2.1.48) is of order O(k 2 ) in view of (2.1.19). Hence, now u is of the form ik + uR u=u −α 4π  is the unique solution of the reduced equation (2.1.49) and α = where u ψds. The remainder uR satisfies the equation Γ 1 2 uR

− KuR − Rk uR = Rk u  − Sk ψ −

ik Rk α , 4π

and, therefore, is of the order O(k 2 ). The hypersingular boundary integral equation (1) of the first kind in Table 2.1.2 reads

2.1 The Helmholtz Equation

Du −

 1 ∂ Rk u = − I + K  ψ − Rk ψ , ∂nx 2

43

(2.1.50)

where D has the constant functions as eigensolutions. Multiplying (2.1.48) by the natural charge q, integrating over Γ and using (2.1.36) and Vq = c0 , we obtain the relation   uqds − q(Rk u)ds Γ



Γ

1 ik = − c0 − δn2 (log k + γ0 ) + δn3 2π 4π



 ψds −

Γ

(2.1.51) q(Sk ψ)ds

Γ

with c0 given by (2.1.35). Now (2.1.51) can be used as a normalizing condition by combining (2.1.50) and (2.1.51); we arrive at the modified equation   ∂ Rk u Du + uqds − (Rk u)qds − ∂nx Γ Γ   1 ik = −c0 + δn2 (log k + γ0 ) − δn3 ψds (2.1.52) 2π π Γ   − 12 I + K  ψ − Rk ψ − q(Sk ψ)ds . Γ

Again, we assume u in the form u = δn2

1 ik (log k + γ0 − 2πc0 )α − δn3 α + u  + uR , 2π 4π

where k → 0 yields

 ψds .

α= Γ

The term u  is the unique solution of   qds = −δn3 c0 ψds − ( 12 I + K  )ψ . D u+ u Γ

Γ

The remainder uR satisfies the equation   ∂ DuR + uR qds − q(Rk uR )ds − R k u R = fR , ∂nx Γ

where

Γ

44

2. Boundary Integral Equations

  ik  ∂ α δn2 (log k + γ0 − 2πc0 ) − δn3 Rk 1 + q(Rk 1)ds 4π ∂nx Γ   ∂ + Rk u  + q(Rk u )ds − Rk ψ − q(Sk ψ)ds ∂nx 

fR

=

Γ

Γ

 is of order O (k log k)2 for n = 2 and of order O(k 2 ) for n = 3, which is in agreement with the result from the previous analysis of the boundary integral equation of the second kind. With the solutions of the BIEs available, we now summarize the asymptotic behaviour of the solutions of the BVPs for small k by substituting the boundary densities into the representation formula (2.1.5). In all the cases we arrive at the following asymptotic expression u(x) = ±[V σ (x) − W u (x)] + C(k; x) + R(k; x)

(2.1.53)

where the ± sign corresponds to the interior and exterior domain and x ∈ Ω or Ω c as in (2.1.5). For the Dirichlet problems, u |Γ = ϕ, and for the Neumann problems, σ |Γ = ψ on Γ , are the given boundary data, respectively, whereas the missing densities are the solutions of the corresponding BIEs presented above. In Formula (2.1.53), C(k; x) denotes the highest order terms of the perturbations in Ω or Ω c ; whereas R(k; x) denotes the remaining boundary potentials. The behaviour of C and R for k → 0 is summarized in Table 2.1.2 below. Table 2.1.2. Low frequency characteristics BVP

C(k; x)

IDP

0

−k{V σ 1 (x) + −{ k12 −

INP

−{ k12

ENP

Γ

σ ds}

 + γ0 ) ψds Γ  ik − 4π ψds Γ

1

i 4π

Γ

1 (log k 2π

n=2

O((k log k)2 ) 



Ω

O((k log k)2 ) O(k )

 1 log |x − y|dy} |Ω| ψds Ω Γ   1 1 1 + 4π dy} |Ω| ψds |x−y| 1 2π

n

2

− ω EDP

R(k; x)

n=3 1

2

n=2

O(k )

n=3

O(k2 log k)

n=2

O(k2 )

n=3

O((k log k)2 )

n=2

O(k2 )

n=3

For this case a sharper result with O(k2 ) is given by MacCamy [274]

2.2 The Lam´e System

45

The remainders R(x; k) are of the orders as given in the table, uniformly in x ∈ Ω for the interior problems and in compact subsets of Ω c only, for the exterior problems.

2.2 The Lam´ e System In linear elasticity for isotropic materials, the governing equations are −Δ∗ v := −μΔv − (μ + λ) graddiv v = f in Ω ( or Ω c ) ,

(2.2.1)

where v is the desired displacement field and f is a given body force. The parameters μ and λ are the Lam´e constants which characterize the elastic material. (See e.g. Ciarlet [68], Fichera [119], Gurtin [165], Kupradze et al [253] and Leis [263]). For n = 3, one also has the relation λ = 2μν/(1 − 2ν) with 0 ≤ ν < 12 , the Poisson ratio. The latter relation is also valid for the plane strain problem in two dimensions, where (2.2.1) is considered with n = 2. In the special case of so–called generalized plane stress problems one still has (2.2.1) with n = 2 for the first two displacement components (v1 , v2 ) but with a modified λ = 2μν/(1 − 2ν) and a modified Poisson ratio ν = ν/(1 + ν). In this case and in what follows, we will keep the same notation for λ. For the Lam´e system, the fundamental solution is given by   λ+3μ λ+μ 1  γn (x, y)I + λ+3μ , (2.2.2) E(x, y) = 4π(n−1)μ(λ+2μ) r n (x − y)(x − y) a matrix–valued function, where I is the identity matrix, r = |x − y| and  − log r for n = 2 , γn (x, y) = 1 (2.2.3) for n = 3 . r The boundary integral equations for the so–called fundamental boundary value problems are based on the Green representation formula, which in elasticity also is termed the Betti–Somigliana representation formula. For interior problems, we have the representation     v(x) = E(x, y)T v(y)dsy − (Ty E(x, y)) v(y)dsy + E(x, y)f (y)dy Γ

Γ

Ω

(2.2.4) for x ∈ Ω. Here the traction on Γ is defined by   ∂v  T v |Γ = λ(div v)n + 2μ + μn × curlv  ∂n Γ

(2.2.5)

for n = 3 which reduces to the case n = 2 by setting u3 = 0 and the third component of the normal n3 = 0. The subscript y in Ty E(x, y) again denotes differentiations in (2.2.4) with respect to the variable y.

46

2. Boundary Integral Equations

The last term in the representation (2.2.4) is the volume potential (or Newton potential) due to the body force f defining a particular solution v p of (2.2.1). As in Section 2.1, we decompose the solution in the form v = vp + u where u now satisfies the homogeneous Equation (2.2.1) with f = 0 and has a representation (2.2.4) with f = 0, i.e. u(x) = V σ(x) − W ϕ(x) .

(2.2.6)

Here V and W are the simple and double layer potentials, now defined by  E(x, y)σ(y)dsy , (2.2.7) V σ(x) := Γ



W ϕ(x)

:=



(Ty (x, y)E(x, y)) ϕ(y)dsy ;

(2.2.8)

Γ

and where in (2.2.6) the boundary charges are the Cauchy data ϕ(x) = u(x)|Γ , σ(x) = T u(x)|Γ of the solution to −Δ∗ u = 0 in Ω .

(2.2.9)

For linear problems, because of the above decomposition, in the following, we shall consider, without loss of generality, only the case of the homogeneous equation (2.2.9), i.e., f = 0. For the exterior problems, the representation formula for v needs to be modified by taking into account growth conditions at infinity. For f = 0, the growth conditions are u(x) = −E(x, 0)Σ + ω(x) + O(|x|1−n ) as |x| → ∞ , where ω(x) is a rigid motion defined by  a + b(−x2 , x1 ) ω(x) = a+b×x

for n = 2 , for n = 3 .

(2.2.10)

(2.2.11)

Here, a, b and Σ are constant vectors; the former denote translation and rotation, respectively. The representation formula for solutions of −Δ∗ u = 0 in Ω c

(2.2.12)

with the growth condition (2.2.10) has the form u(x) = −V σ(x) + W ϕ(x) + ω(x) with the Cauchy data ϕ = u|Γ and σ = T u|Γ and with  Σ = σds . Γ

(2.2.13)

(2.2.14)

2.2 The Lam´e System

47

2.2.1 The Interior Displacement Problem The Dirichlet problem for the Lam´e equations (2.2.9) in Ω is called the displacement problem since here the boundary displacement u|Γ = ϕ on Γ

(2.2.15)

is prescribed. The missing Cauchy datum on Γ is the boundary traction σ = T u|Γ . Applying the trace and the traction operator T (2.2.5) to both sides of the representation formula (2.2.4), we obtain the overdetermined system of boundary integral equations ϕ(x) σ(x)

= =

( 12 I − K)ϕ(x) + V σ(x) , Dϕ +

( 12 I

(2.2.16)



+ K )σ(x) on Γ .

(2.2.17)

Here, the boundary integral operators are defined as in (1.2.3) –(1.2.6), however, it is understood that the fundamental solution now is given by (2.2.2) and the differentiation ∂n∂|Γ (1.2.4)–(1.2.6) is to be replaced by the traction operator T|Γ (2.2.5). Explicitly, we also have: Lemma 2.2.1. Let Γ ∈ C 2 and let ϕ ∈ C α (Γ ), σ ∈ C α (Γ ) with 0 < α < 1. Then, for the case of elasticity, the limits (1.2.3)–(1.2.5) exist uniformly with respect to all x ∈ Γ and all ϕ and σ with ||ϕ||C α ≤ 1, ||σ||C α ≤ 1. Furthermore, these limits can be expressed by  E(x, y)σ(y)dsy , x ∈ Γ , (2.2.18) V σ(x) = y∈Γ \{x}

Kϕ(x)

=



p.v.



(Ty E(x, y)) ϕ(y)dsy ,

x ∈ Γ , (2.2.19)

(Tx E(x, y)) σ(y)dsy ,

x ∈ Γ . (2.2.20)

y∈Γ \{x} 

K σ(x)



=

p.v. y∈Γ \{x}

These results are originally due to Giraud [145], see also Kupradze [252, 253] and the references therein. We remark that the integral in (2.2.18) is a weakly singular improper integral, whereas the integrals in (2.2.19) and (2.2.20) are to be defined as Cauchy principal value integrals, i.e.     (Ty E(x, y)) ϕ(y)dsy = lim (Ty E(x, y)) ϕ(y)dsy , p.v. y∈Γ \{x}

ε→0 |y−x|≥ε>0∧y∈Γ

since the operators K and K  have Cauchy singular kernels:

48

2. Boundary Integral Equations

(Ty E(x, y)) =



  n(λ + μ) μ  ∂γn I+ (x − y)(x − y) (x, y) 2(n − 1)π(λ + 2μ) μ|x − y|2 ∂ny   1   + (x − y)n − n (x − y) , (2.2.21) y y |x − y|n

(Tx E(x, y)) =

  n(λ + μ) μ  ∂γn I+ (x − y)(x − y) (x, y) 2(n − 1)π(λ + 2μ) μ|x − y|2 ∂nx   1   (x − y)n . (2.2.22) − n (x − y) − x x |x − y|n

We note that in case n = 2, the last term in (2.2.21) can also be written in the form    1 0 , 1 d   (x − y)n = − n (x − y) log |x − y| . (2.2.23) y y −1 , 0 dsy |x − y|2 The last terms in the kernels (2.2.21) and (2.2.22) are of the order O(r−n ) as r = |x − y| → 0 and, in addition, satisfy the Mikhlin condition (see Mikhlin [302, Chap.5] or [305, Chap.IX]) which is necessary and sufficient for the existence of the above Cauchy principal value integrals for any x ∈ Γ . We shall return to this condition later on. The proof of Lemma 2.2.1 can be found in [302, Chap.45 p.210 ff]. It turns out that there exists a close relation between the single and double layer operators of the Laplacian and the Lam´e system based on the following lemma and the G¨ unter derivatives   ∂ ∂ − nj (y) = −mkj ∂y , n(y) mjk ∂y , n(y) := nk (y) ∂yj ∂yk

(2.2.24)

(see Kupradze et al [253, (4.7), p. 99]) which are particular tangential derivatives; in matrix form    M ∂y , n(y) := mjk ∂y , n(y) 3×3 . (2.2.25) Lemma 2.2.2. For the G¨ unter derivatives there hold the identities  mj ∂y , n(y) γn n 

=

nj (y − x ) − n (yj − xj ) , |x − y|n

(2.2.26)

 (y − x )(yk − xk ) mj ∂y , n(y) |x − y|n =1  n(yj − xj )(yk − xk )  ∂ = − δjk − γn .(2.2.27) |x − y|2 ∂ny

2.2 The Lam´e System

49

The proof is straight forward by direct computation. Inserting the identities (2.2.26) and (2.2.27) into (2.2.21) we arrive at  ∂     1 Ty E(x, y) = γn (x, y) + M ∂y , (n(y) γn (x, y) 2(n − 1)π ∂ny    (2.2.28) +2μ M ∂y , n(y) E(x, y) . In the same manner we find  ∂    1 Tx E(x, y) = γn (x, y) − M ∂x , (n(x) γn (x, y) 2(n − 1)π ∂nx   (2.2.29) +2μ M ∂x , n(x) E(x, y) . As a consequence, we may write the double layer potential operator in (2.2.19) after integration by parts in the form of the Stokes theorem as   ∂  1 γn (x, y) ϕ(y)dsy Kϕ(x) = 2(n − 1)π ∂ny Γ    − γn (x, y)M ∂y , n(y) ϕ(y)dsy (2.2.30) Γ



+ 2μ

 E(x, y)M ∂y , n(y) ϕ(y)dsy ,

Γ

see Kupradze et al [253, Chap. V Theorem 6.1]. Finally, the hypersingular operator D in (2.2.17) is defined by Dϕ(x)

=

−Tx W ϕ(x)

:=

−

  λ divz W ϕ(z)) + 2μ(gradz W ϕ(z) · nx z→x∈Γ,z∈Γ  + μnx × curlz W ϕ(z) . (2.2.31) lim

Lemma 2.2.3. Let Γ ∈ C 2 and let ϕ be a H¨ older continuously differentiable function. Then in the case of elasticity, the limits (1.2.6) exist uniformly with respect to all x ∈ Γ and all ϕ with ϕC 1+α ≤ 1. Moreover, the operator D can be expressed as a composition of tangential differential operators and the simple layer operators of the Laplacian and the Lam´e system: For n = 2:   d  dϕ Dϕ(x) = − (x) (2.2.32) V dsx ds where μ(λ + μ) V χ(x) := π(λ + 2μ)

  − log |x − y| + Γ

(x − y)(x − y) |x − y|2

 χ(y)dsy . (2.2.33)

50

2. Boundary Integral Equations

For n = 3:

 1 μ Dϕ(x) = − (nx × ∇x ) · (ny × ∇y )ϕ(y)dsy 4π |x − y| Γ      2 1 μ 4μ E(x, y) − I M ∂y , n(y) ϕ(y)dsy − M ∂x , n(x) 2π |x − y| Γ

 3    1  μ  mkj ∂y , n(y) ϕ (y)dsy + mk ∂x , n(x) . 4π |x − y| j=1,2,3 ,k=1

Γ

(2.2.34) Proof: The proof for n = 2 was given by Bonnemay [36] and Nedelec [330]; here we follow the proof given by Houde Han [169, 170] for n = 3. The proof is based on a different representation of the traction operator T by employing G¨ unter’s derivatives M; more precisely we have:   ∂u  + Mu . T ∂x , n(x) u(x) = (λ + μ)(divu)n(x) + μ ∂n

(2.2.35)

We note

∂u − (divu)n(x) + n(x) × curl u . (2.2.36) ∂n Now we apply T in the form (2.2.35) to the three individual terms in the double layer potential K in (2.2.30) successively, and begin with Mu =

∂  γn (z, y)I T ∂z , n(x) ∂ny  ∂ ∂ γn (z, y)I + μM ∂z , n(x) γn (z, y)I = μ(n(x) · ∇z ) ∂ny ∂ny   ∂ + (λ + μ)n(x) gradz γn (z, y) . ∂ny Next we apply T to the remaining two terms by using (2.2.29) to obtain  1 γn (z, y)I 2(n − 1)π   2  2μ γn (x, y)I = M ∂z , n(x) 4μ E(x, y) − 2(n − 1)π  1 2μ n(x) · ∇z γn (z, y)I − T ∂z , n(x) γn (z, y)I . + 2(n − 1)π 4π

  T ∂z , n(x) 2μE(z, y) −

Now we apply (2.2.35) to the last term in this expression and find

2.2 The Lam´e System

51

 T ∂z , n(x)

1 γn (z, y)I 2(n − 1)π   μ 1 (λ + μ)n(x) ∇z γn (z, y) + n(x) · ∇z γn (z, y)I = 2(n − 1)π 2(n − 1)π  μ M ∂z , n(x) γn (z, y)I . + 2(n − 1)π Hence,   T ∂z , n(x) 2μE(z, y) −

 1 γn (z, y)I 2(n − 1)π      2 3μ λ+μ γn I − n(x) ∇z γn (x, z) = M z, n(x) 4μ E(z, y) − 2(n − 1)π 2(n − 1)π μ n(x) · ∇z (z, y)I . + 2(n − 1)π

Collecting these results we obtain  T ∂z , n(x) (Kϕ)(z)  ∂ μ n(x) · ∇z γn (z, y)ϕ(y)dsy = 2(n − 1)π ∂ny Γ      2 3μ 4μ E(z, y) − γn (z, y) M ∂y , n(y) ϕ(y)dsy + M ∂z , n(x) 2(n − 1)π Γ    ∂ λ+μ  n(x) + γn (z, y) · ϕ(y)dsy ∇z 2(n − 1)π ∂ny Γ      − n(x) ∇z γn (z, y) · M ∂y , n(y) ϕ(y) dsy   μ + M ∂z , n(x) 2(n − 1)π

Γ



Γ



+ n(x) · ∇z

∂ γn (z, y)ϕ(y)dsy ∂ny   γn (z, y)M ∂y , n(y) ϕ(y)dsy

Γ

   ∂ μ = n(x) · ∇z γn (z, y) ϕ(y)dsy 2(n − 1)π ∂ny Γ      2 3μ 4μ E(z, y) − γn (z, y) M ∂y , n(y) ϕ(y)dsy + M ∂z , n(x) 2(n − 1)π Γ

(λ + μ) μ J1 ϕ(x) + J2 ϕ(x) + 2(n − 1)π 2(n − 1)π

52

2. Boundary Integral Equations

where J1 ϕ(x) :=

and J2 ϕ(x) :=



  ∂ n(x) ∇z γn (z, y) · ϕ(y) ∂ny Γ     − n(x) ∇z γn (z, y) · M ∂y , n(y) ϕ(y) dsy 

  Γ

∂  γn (z, y) M ∂z , n(x) ∂ny

   + n(x) · ∇z γn (z, y) M ∂y , n(y) ϕ(y)dsy .

The product rule gives   M ∂y , n(y) ∇z γn (z, y) · ϕ(y) n     ∂ = mk ∂y , n(y) γn (z, y) ϕk (y) ∂z k,=1 n  

  ∂  γn (z, y)ϕk (y) mk ∂y , n(y) ∂z k,=1   ∂  γn (z, y) − mk ∂y , n(y) ϕk (y) ∂z n     ∂  = γn (z, y)ϕk (y) − mk ∂y , n(y) ∂z k,=1    ∂  γn (z, y) mk ∂y , n(y) ϕk (y) . + ∂z

=

(2.2.37)

The symmetry of γn (z, y) and Δy γn (z, y) = 0 for z = y implies n  k=1

 ∂2 ∂γn (z, y)  ∂2 = nk (y) γn (z, y) − nj (y) γn (z, y) ∂zk ∂yj ∂zk ∂yk ∂zk n

mjk (y)

=

k=1 n  k=1

= i.e., ∇z

n

k=1

∂2 nk (y) γn (z, y) + nj (y)Δy γn (z, y) ∂yk ∂zj

∂ ∂γn (z, y) , ∂zj ∂ny

 ∂γn (z, y) = M ∂y (y) ∇z γn (z, y) . ∂ny

(2.2.38)

Using (2.2.37) and (2.2.38) for the first term of the integrand of J1 , we find

2.2 The Lam´e System

 J1 ϕ(x) = n(x)



53

 M ∂y , n(y) ∇z γn (z, y) · ϕ(y)

Γ

   − ∇z γn (z, y) · M ∂y (y) ϕ(y) dsy   n    ∂ = −n(x) mk ∂y , n(y) γn (z, y)ϕk (y) dsy = 0 ∂zk ,k=1

Γ

for x ∈ Γ due to the Stokes theorem. For the integral J2 we use ∂    γn (z, y) − M ∂y , n(y) ∇z · n(x) γn (z, y)I M ∂z , n(x) ∂ny       = M ∂y , n(y) M ∂z , n(x) − M ∂z , n(x) M ∂y , n(y) γn (z, y)I to obtain J2 ϕ(x) =

 

   M ∂z , n(x) γn (z, y)IM ∂y , n(y) ϕ(y)dsy

Γ

 + M ∂z , n(x)



 γn (z, y)M ∂y , n(y) ϕ(y)dsy .

Γ

The final result is then with (2.2.36):   K(z, y)ϕ(y)dsy −T ∂z , n(x) Γ

   μ n(x) × ∇z · γn (z, y) n(y) × ∇y ϕ(y)dsy =− 2(n − 1)π Γ      2 2μ 4μ E(z, y) − γn (z, y)I M ∂y , n(y) ϕdsy − M ∂z , n(x) 2(n − 1)π Γ      μ + ϕ(y)dsy , M ∂z , n(x) γn (z, y)M ∂y , n(y) 2(n − 1)π Γ

where z ∈ Γ . As can be seen, this is a combination of applications of tangential derivatives to weakly singular operators operating on tangential derivatives of ϕ. Therefore, the limits z → x ∈ Γ with z ∈ Ω or z ∈ Ω c exist, which leads to the desired result involving Cauchy  singular integrals (see Hellwig [177,  p.197]). Note that for n = 2 we have n(x) × ∇x |Γ = dsdx . The singular behaviour of the hypersingular operator now can be regularized as above. This facilitates the computational algorithm for the Galerkin method (Of et al [340]).

54

2. Boundary Integral Equations

As for the hypersingular integral operator associated with the Laplacian in Section 1.2, here we can apply a more elementary, but different regularization. Based on (2.2.31), we get  Dϕ(x)

=

−

lim

Ω z→x∈Γ

 + Tz

{Tz

(Ty E(x, y) (ϕ(y) − ϕ(x))dsy

Γ

(Ty E(x, y)) ϕ(x)dsy } .

Γ

Here, in the neighborhood of Γ , the operator Tz is defined by (2.2.22) where we identify nz = nx for z ∈ Γ . If we apply the representation formula (2.2.4) to any constant vector field a, representing a rigid displacement, then we obtain  a = − (Ty E(z, y)) adsy for z ∈ Ω, Γ

which yields for z ∈ Ω in the neighborhood of Γ  Tz (Ty E(z, y)) dsy ϕ(x) = 0 . Γ

Hence,  Dϕ(x) = −

lim

Ω z→x∈Γ



(Ty E(z, y)) (ϕ(y) − ϕ(x))dsy ,

Tz Γ

from which it can finally be shown that   Dϕ(x) = −p.v. Tx (Ty E(x, y)) (ϕ(y) − ϕ(x)) dsy ,

(2.2.39)

Γ

(see Kupradze et al [253, p.294], Schwab et al [380]). If the boundary potential operators in (1.2.18) and (1.2.19) are replaced by the corresponding elastic potential operators, then the Calder´on projector CΩ for solutions u of (2.2.9) in Ω again is given in the form of (1.2.20) with the corresponding elastic potential operators. Also, Theorem 1.2.3. and Lemma 1.2.4. remain valid for the corresponding elastic potentials V, K, K  and D. For the solution of the interior displacement problem we now may solve the Fredholm boundary integral equation of the first kind V σ = 12 ϕ + Kϕ on Γ , or the Cauchy singular integral equation of the second kind

(2.2.40)

2.2 The Lam´e System 1 2σ

− K  ϕ = Dϕ on Γ ,

55

(2.2.41)

which both are equations for σ. The first kind integral equation (2.2.40) may have eigensolutions for special Γ similar to (1.3.4) for the Laplacian (see Steinbach [400]). Again we can modify (2.2.40) by including rigid motions (2.2.11). More precisely, we consider the system V σ − ω 0 = 12 ϕ + Kϕ on Γ ,  σds = 0

(2.2.42)

Γ

together with  (−σ1 x2 + σ2 x1 )ds Γ

=

0

for n = 2 or

=

0

for n = 3 ,

 (σ × x)ds

(2.2.43)

Γ

where σ and the unknown constant vector ω 0 are to be determined. As was shown in [205], the rotation b in (2.2.11) can be prescribed as b = 0; in this case the side conditions (2.2.43) will not be needed. For n = 3, many more choices in ω 0 can be made (see [205] and Mikhlin et al [304]). The modified system (2.2.42) and (2.2.43) is always uniquely solvable in the H¨older space, σ ∈ C α (Γ ) for given ϕ ∈ C 1+α (Γ ). For the special Cauchy singular integral equation of the second kind (2.2.41), Mikhlin showed in [300] that the Fredholm alternative, originally designed for compact operators K, remains valid here. Therefore, (2.2.41) admits a unique classical solution σ ∈ C α (Γ ) provided ϕ ∈ C 1+α (Γ ), 0 < α < 1. (See Kupradze et al [253, Chapt. VI], Mikhlin et al [305, p.382 ff]). 2.2.2 The Interior Traction Problem The Neumann problem for the Lam´e system (2.2.9) in Ω is called the traction problem, since here the boundary traction T u|Γ = ψ on Γ

(2.2.44)

is given, whereas the missing Cauchy datum u|Γ needs to be determined. Corresponding to (2.2.16) and (2.2.17), we have the overdetermined system 1 (2.2.45) 2I + K u = V ψ , 1  Du = (2.2.46) 2 I − K ψ on Γ for the unknown boundary displacemant u|Γ .

56

2. Boundary Integral Equations

As for the Neumann Problem for the Laplacian, here ψ needs to satisfy certain equilibrium conditions for a solution to exist. These can be obtained from the Betti formula, which for u and any rigid motion ω(x) reads  0 = (ω · Δ∗ u) − (u · Δ∗ ω)dx Ω



 ω · T uds −

= Γ

u · T ωds . Γ

This implies, with T ω = 0, the necessary compatibility conditions for the given traction ψ, namely  ω · ψds = 0 for all rigid motions ω (2.2.47) Γ

given by (2.2.11). This condition turns out to be also sufficient for the existence of u in the classical H¨older function spaces. If ψ ∈ C α (Γ ) with 0 < α < 1 is given satisfying (2.2.47), then the right–hand side, V ψ in (2.2.45), automatically satisfies the orthogonality conditions from Fredholm’s alternative; and the Cauchy singular integral equation (2.2.45) admits a solutionu ∈ C 1+α (Γ ). The solution, however, is unique only up to all rigid motions ω, which are eigensolutions. For further details see [205]. The hypersingular integral equation of the first kind (2.2.46) also has eigensolutions which again are given by all rigid motions (2.2.11). As will be seen, the classical Fredholm alternative even holds for (2.2.46), and the right– hand side 12 ψ − K  ψ satisfies the corresponding orthogonality conditions, provided, ψ ∈ C α (Γ ) satisfies the equilibrium conditions (2.2.47). In both cases, the integral equations , together with appropriate side conditions, can be modified so that the resulting equations are uniquely solvable (see [204]). 2.2.3 Some Exterior Fundamental Problems In this section we shall summarize the approach from [205]. For the exterior displacement problem for u satisfying the Lam´e system (2.2.12) and the Dirichlet condition (2.2.15), u|Γ = ϕ on Γ , we require at infinity appropriate conditions according to (2.2.10); namely that there exist Σ and some rigid motion ω such that u(x) + E(x, 0)Σ − ω = O(|x|1−n ) as |x| → ∞ . In general, the constants in Σ and ω are related to each other. However, some of them can still be specified.

2.2 The Lam´e System

57

For n = 2, we consider the following two cases. In the first case, b in ω = a + b(−x2 , x1 )T is given . In addition, the total forces Σ in (2.2.14) are also given, often as Σ = 0 due to equilibrium. Then a in ω is an additional unknown vector. The representation formula (2.2.13) for the Dirichlet problem (2.2.12) with (2.2.14) yields the modified boundary integral equation of the first kind Vσ−a  σds

=

− 12 ϕ(x) + Kϕ(x) + b(−x2 , +x1 ) on Γ ,

(2.2.48)

=

Σ.

(2.2.49)

Γ

Here, ϕ, Σ and b are given and σ, a are the unknowns. As we will see, these equations are always uniquely solvable. In particular, for any given ϕ ∈ C 1+α (Γ ), Σ and b, one finds in the classical H¨older–spaces σ ∈ C α (Γ ). In the second case, in addition to the total force Σ, the total momentum  (−x2 σ1 + x1 σ2 )dsx = Σ3 Γ

is also given, whereas b is now an additional unknown. Now the modified boundary integral equation of the first kind reads 1 V σ(x) − a − b(−x2 , +x1 ) = ϕ(x) + Kϕ(x) on Γ ,   2 σds = Σ, (−x2 σ1 + x1 σ2 )dsx = Σ3 , Γ

(2.2.50) (2.2.51)

Γ

where ϕ ∈ C 1+α (Γ ), Σ, Σ3 are given and σ, a and b are to be determined. The system (2.2.50), (2.2.51) always has a unique solution σ ∈ C α (Γ ), a, b. Both these problems can also be reduced to Cauchy singular integral equations by applying the traction operator to (2.2.13). This yields the singular integral equation 1 2 σ(x)

+ K  σ(x) = −Dϕ(x) for x ∈ Γ ,

(2.2.52)

with the additional equation (2.2.49) in the first case or the additional equations (2.2.51) in the second case, respectively. The operator 12 I +K  is adjoint to 12 I + K in (2.2.17). Due to Mikhlin [300], for these special operators, Fredholm’s classical alternative is still valid in the space C α (Γ ). Since 1 2ω

+ Kω = 0 on Γ

for all rigid motions ω, the adjoint equation (2.2.52) has an 3(n − 1)– dimensional eigenspace, as well. Moreover, Dω = 0 for all rigid motions;

58

2. Boundary Integral Equations

hence, the right–hand side of (2.2.52) always satisfies the orthogonality conditions for any given ϕ ∈ C 1+α (Γ ). This implies that equation (2.2.52) always admits a solution σ ∈ C α (Γ ) which is not unique. If, for n = 2, the total force and total momentum in addition are prescribed by (2.2.51), i.e. in the second case, then these equations determine σ(x) uniquely. For finding a and b we first determine three vector–valued functions λj (x), j = 1, 2, 3, satisfying on Γ the equations   a · λj ds = aj and (−x2 , x1 ) · λj (x)ds = 0 for j = 1, 2 , Γ 

Γ 

a · λ3 ds

=

0

(−x2 , x1 ) · λ3 ds

and

Γ

= 1.

Γ

(2.2.53) Since σ(x) on Γ is already known from solving (2.2.52), equation (2.2.50) can now be used to find a and b; namely     1 for j = 1, 2 : aj = V σ(x) · λj (x)dx − ϕ · λj ds − (Kϕ) · λj ds . for j = 3 :b 2 Γ

Γ

Γ

(2.2.54) If, as in the first case, b and Σ are given, then the additional equations  σds = Σ and Γ



λ3 (x) · V σ(x)dsx

=

b−

1 2

Γ



 λ3 · ϕds +

Γ

λ3 · Kϕds Γ

determine σ uniquely; and a1 , a2 can be found from (2.2.54), afterwards. In the case n = 3, there are many more possible choices of additional conditions. To this end, we write the rigid motions (2.2.11) in the form ω(x) =

3 

aj mj (x) +

j=1

where mj (x) is the j–th ⎛ 1, ⎝0 , 0,

6 

bj−3 mj (x) =:

j=4

6 

ωj mj (x)

j=1

column vector of the matrix 0, 1, 0,

0, 0, 1,

0, −x3 , x2 ,

x3 , 0, −x1 ,

⎞ −x2 x1 ⎠ . 0.

(2.2.55)

Let J ⊂ F := {1, 2, 3, 4, 5, 6} denote any fixed set of indices in F. Then we may prescribe aj−3 , bj for j ∈ J , i.e., some of the parameters in ω subject to the behavior of (2.2.10) at infinity. If J is a proper subset of F then we must include additional normalization conditions,

2.2 The Lam´e System

59

 mk (y) · σ(y)dsy = Σk for k ∈ F \ J .

(2.2.56)

Γ

By taking the representation formula (2.2.13) on Γ , we obtain from the direct formulation the boundary integral equation of the first kind on Γ , V σ(x) −

 k∈F \J

 1 ωk mk (x) = − ϕ(x) + Kϕ(x) + ωj mj (x) , 2

(2.2.57)

j∈J

together with the additional equations,  mk (y) · σ(y)dsy = Σk for k ∈ F \ J .

(2.2.58)

Γ

In the right–hand side of (2.2.57), the ωj are given, whereas the ωk in the left–hand side are unknown. For given ϕ ∈ C α (Γ ), 0 < α < 1, and given constants ωj , j ∈ J , the unknowns are σ ∈ C α (Γ ) together with ωk for k ∈ F \J. Again, we may take the traction of (2.2.13) on Γ to obtain a Cauchy singular boundary integral equation instead of (2.2.57) and (2.2.58), namely (2.2.52) together with (2.2.58). Since (2.2.52) is always solvable for any given ϕ ∈ C 1+α (Γ ) due to the special form of the right–hand side, and since the eigenspace of 12 I + K  is the linear space of all rigid motions, the linear equations (2.2.58) need to be completed by including (card J ) additional equations which resembles the required behavior of u(x) at infinity for those ωj given already with j ∈ J . For these constraints, we again choose the vector–valued functions, λ on Γ ,  ∈ F, (e.g. linear combinations of m |Γ ) which are orthonormalized to mj |Γ , i.e.,  mj (x) · λ (x)dsx = δj, , j,  ∈ F . (2.2.59) Γ

Now, the complete system of equations for the modified Dirichlet problem can be formulated as  1 σ(x) + K  σ(x) − α m (x) = −Dϕ(x) for x ∈ Γ 2 ∈F  (2.2.60) mk · σ k (y)dsy = Σk , k ∈ F \ J , Γ

together with     1 ϕ(x) + K  ϕ(x) ds + ωj for j ∈ J . λj (x) · V σds = λj · 2 Γ

Γ

60

2. Boundary Integral Equations

The desired displacement u(x)|Γ then is obtained from (2.2.13) with σ(x) determined from (2.2.60) and ωk given by     1 ωk = λk · V σds + λk · ϕ − Kϕ ds for k ∈ F \ J . 2 Γ

Γ

Note that we have included the extra unknown term α m (x) in (2.2.60) so that the number of unknowns and equations coincide. One can show that in fact α = 0 for  ∈ F. The last set of equations has been obtained from (2.2.50) with (2.2.59). For the exterior traction problem, the Neumann datum is given by T u|Γ = ψ on Γ , and the total forces and momenta by  ψ · mk ds = Σk ,

k ∈ F.

Γ

Here, the standard direct approach with x → Γ in (2.2.13) yields the Cauchy singular boundary integral equation for u|Γ , 1 2 u(x)

− Ku(x) = V ψ(x) + ω(x) for x ∈ Γ .

(2.2.61)

As is well known, (2.2.61) is always uniquely solvable for any given ψ ∈ C α (Γ ) and given ω with u ∈ C 1+α (Γ ); we refer for the details to Kupradze [252, p.118]. If we apply T to (2.2.13) then we obtain the hypersingular boundary integral equation Du(x) = − 12 ψ(x) − K  ψ(x) for x ∈ Γ .

(2.2.62)

It is easily seen that the rigid motions ω(y) on Γ are eigensolutions of (2.2.62). Therefore, in order to guarantee unique solvability of the boundary integral equation , we modify (2.2.62) by including restrictions and adding unknowns, e.g.,  1 α m (x) for x ∈ Γ and Du0 (x) = − ψ(x) − K  ψ(x) + 2 =1  mk (y) · u0 (y)dsy = 0 , k = 1, · · · , 3(n − 1) . 3(n−1)

(2.2.63)

Γ

Also here, the unknowns α are introduced for obtaining a quadratic system. For the Neumann problem, they all vanish because of the special form of the right–hand side. As we will see, the system (2.2.63) is always uniquely

2.2 The Lam´e System

61

solvable; for any given ψ ∈ C α (Γ ) we find exactly one u0 ∈ C 1+α (Γ ). In (2.2.63), the additional compatibility conditions can also be incorporated into the first equation of (2.2.63) which yields the stabilized uniquely solvable version  0 (x) := Du0 (x) + Du

3(n−1) 



mk (y)u0 (y)dsy mk (x)

k=1 Γ 

(2.2.64)

= − 12 ψ(x) − K ψ(x) for x ∈ Γ . Once u0 is known, the actual displacement field u(x) is then given by u(x) = −V ψ(x) + W u0 (x) for x ∈ Ω c .

(2.2.65)

Note that the actual boundary values of u|Γ may differ from u0 by a rigid motion. u|Γ can be expressed via (2.2.65) in the form u(x)|Γ =

1 u0 (x) + Ku0 (x) − V ψ(x) . 2

(2.2.66)

In concluding this section we remark that the boundary integral equations on H¨ older continuous charges are considered in most of the more classical works on this topic with applications in elasticity (e.g., Ahner et al [7], Balaˇs et al [23], Bonnemay [37], Kupradze [251, 252], Muskhelishvili [319] and Natroshvili [322]). However, as mentioned before, we shall come back to these equations in Chapters 5–11. Now we extend our approach to incompressible materials. 2.2.4 The Incompressible Material If the elastic material becomes incompressible, the Poisson ratio ν := λ/2(λ + μ) tends to 1/2 or λ = 2μν/(1 − 2ν) → ∞ for n = 3 and for the plane strain case where n = 2. However, for the plane stress case we have ν → 1/3 and λ → 2μ if the material is incompressible; and our previous analysis remains valid without any restriction. In order to analyze the incompressible case, we now rewrite the Lam´e equation (2.2.1) in the form of a system −Δu + ∇p = 0 , div u = −cp where p = −

λ+μ divu μ

(2.2.67)

μ → 0+ (see Duffin and Noll [108]). This system correand c = 1 − 2ν = λ+μ sponds to the Stokes system. In terms of the parameter c, the fundamental solution (2.2.2) now takes the form

62

2. Boundary Integral Equations

E(x, y) =

 1 + 2c γn (x, y)I + 4π(n − 1)μ(1 + c)

1 (1+2c)r n (x

− y)(x − y)



(2.2.68) which is well defined for c = 0, as well. Consequently, the Betti–Somigliana representation formula (2.2.4) remains valid, where the double layer potential kernel now reads as  n 1 ∂γ (cI + 2 (x − y)(x − y) ) (Ty E(x, y)) = 2π(n − 1)(1 + c) r ∂ny  c   + n (x − y)n , (2.2.69) y − ny (x − y) r which again is well defined for the limiting case c = 0. However, the limiting case of the differential equations (2.2.63) leads to the more complicated Stokes system involving a mixed variational formulation (see Brezzi and Fortin [45]) whereas the associated boundary integral equations remain valid. In fact, from (2.2.65) and (2.2.66) it seems that the case c = 0 does not play any exceptional role. However, for small c > 0, a more detailed analysis is required. We shall return to this point after discussing the Stokes problem in Section 2.3.

2.3 The Stokes Equations The linearized and stationary equations of the incompressible viscous fluid are modeled by the Stokes system consisting of the equations in the form −μΔu + ∇p = f , div u = 0 in Ω (or Ω c ) .

(2.3.1)

Here u and p are the velocity and pressure of the fluid flow, respectively, which are the unknowns; f corresponds to a given forcing term, while μ is the given dynamic viscosity of the fluid. We have already seen this system previously in (2.2.67) for the elastic material when it becomes incompressible, although for viscous flow with given fluid density ρ, one introduces μ ν :=  1 ρ which is usually referred to as the kinematic viscosity of the fluid, not the Poisson ratio as in the case of elasticity. The fundamental solution of the Stokes system (2.3.1) is defined by the pair of distributions v k , and q k satisfying −{μΔx v k (x, y) − ∇x q k (x, y)} = δ(x, y)ek , (2.3.2) divx v k = 0 , where ek denotes the unit vector along the xk -axis, k = 1, ., n with n = 2 or 3 (see Ladyˇzenskaya [256]). By using the Fourier transform, we may obtain

2.3 The Stokes Equations

63

the fundamental solution explicitly: For n = 2, v k (x, y) =

2  (xk − yk )(xj − yj )ej  1 1  , log ek + 4πμ |x − y| |x − y|2 j=1

(2.3.3)

1 ∂ 1 log }; {− q (x, y) = ∂xk 2π |x − y| k

and for n = 3, v k (x, y) =

3  (xk − yk )(xj − yj )ej  1  1 , ek + 8πμ |x − y| |x − y|3 j=1

(2.3.4)

1 ∂ 1 }. {− q (x, y) = ∂xk 4π |x − y| k

We note that from their explicit forms, v k (x, y) and q k (x, y) also satisfy the adjoint system in the y–variables, namely −{μΔy v(x, y) + ∇y q k (x, y)} = δ(x, y)ek , −divy v k = 0 .

(2.3.5)

This means that we may use the same fundamental solution for the Stokes system and for its adjoint depending on which variables are differentiated. As will be seen, we do not need to switch the variables x and y in the representation of the solution of (2.3.1) from Green’s formula (see [256]). For the flow (u, p), we define the stress operators as in elasticity, T (u) := −pn + μ(∇u + ∇u )n

(2.3.6)

= σ(u, p)n , 

T (u) := pn + μ(∇u + ∇u )n = σ(u, −p)n , where

σ := −pI + μ(∇u + ∇u )

denotes the stress tensor in the viscous flow. We remark that it is understood that the stress operator T is always defined for the pair (u, p). For smooth (u, p) and (v, q), we have the second Green formula   u · {−μΔv − ∇q)} dy − {−μΔu + ∇p} · vdy Ω

 = Γ

Ω

[T (u) · v − u · T  (v)] ds

(2.3.7)

64

2. Boundary Integral Equations

provided div u = div v = 0 . k

Now replacing (v, q) by (v , q k ), and by following standard arguments, the velocity component of the nonhomogeneous Stokes system (2.3.1) has the representation   ! " uk (x) = Ty (u) · v k (x, y) − u · Ty (v k )(x, y) dsy + f · v k dy for x ∈ Ω . Γ

Ω

(2.3.8) To obtain the representation of the pressure, we may simply substitute the relation ∂p = μΔuk + fk ∂xk into (2.3.8). To simplify the representation, we now introduce the fundamental velocity tensor E(x, y) and its associated pressure vector Q(x, y), respectively, as E(x, y) = [v 1 , ·, v n ],

Q(x, y) = [q 1 , ·, q n ] ,

and

which satisfy −μΔx E + ∇x Q = δ(x, y)I ,

(2.3.9)

divx E = 0 . As a result of (2.3.4) and (2.3.5), we have explicitly  1 (x − y)(x − y)  E(x, y) = γn I + , 4(n − 1)πμ |x − y|n 1 1 (−∇x γn ) = (∇y γn ) Q(x, y) = 2(n − 1)π 2(n − 1)π 

with γn (x, y) =

− log |x − y| 1 |x−y|

(2.3.10)

for n = 2 , for n = 3

as in (2.2.2). In terms of E(x, y) and Q(x, y), we finally have the representation for solutions in the form:      Ty E(x, y) u(y)dsy E(x, y)T (u)(y)dsy − u(x) = Γ



E(x, y)f (y)dy for x ∈ Ω ,

+  p(x)

Γ

Ω



Q(x, y) · T (u)(y)dsy − 2μ

= Γ



(

∂Q (x, y)) · u(y)dsy ∂ny

Γ

Q(x, y) · f (y)dy for x ∈ Ω .

+ Ω

(2.3.11)

(2.3.12)

2.3 The Stokes Equations

65

It is understood that the representation of p is unique only up to an additive constant. Also, as was explained before,  Ty E(x, y) := σ E(x, y), −Q(x, y) n(y) . 2.3.1 Hydrodynamic Potentials The last terms in the representation (2.3.11) and (2.3.12) corresponding to the body force f define a particular solution (U , P ) of the nonhomogeneous Stokes system (2.3.1). As in elasticity, if we decompose the solution in the form u = uc + U , p = pc + P, then the pair (uc , pc ) will satisfy the corresponding homogeneous system of (2.3.1). Hence, in the following, without loss of generality, we shall confine ourselves only to the homogeneous Stokes system. The solution of the homogeneous system now has the representation from (2.3.11) and (2.3.12) with f = 0, i.e., u(x) = V τ (x) − W ϕ(x) , p(x) = Φτ (x) − Πϕ(x).

(2.3.13) (2.3.14)

(The subscript c has been suppressed.) Here the pair (V, Φ) and (W, Π) are the respective simple– and double layer hydrodynamic potentials defined by  V τ (x) := E(x, y)τ (y)dsy , Γ



(2.3.15) Q(x, y) · τ (y)dsy ;

Φτ (x) := Γ

 W ϕ(x) :=



Ty (E(x, y))



ϕ(y)dsy ,

Γ

   ∂ Πϕ(x) := 2μ Q(x, y) · ϕ(y)dsy for x ∈ Γ . ∂ny

(2.3.16)

Γ

In (2.3.13) and (2.3.14) the boundary charges are the Cauchy data ϕ = u(x)|Γ and τ (x) = T u(x)|Γ of the solution to the Stokes equations −μΔu + ∇p = 0 , div u = 0 in Ω .

(2.3.17)

For the exterior problems, the representation formula for u needs to be modified by taking into account the growth conditions at infinity. Here proper growth conditions are

66

2. Boundary Integral Equations

 u(x)

=

p(x)

=

Σ log |x| + O(1) O(|x|−1 )

O(|x|1−n )

as

for n = 2 , for n = 3 ;

|x| → ∞.

(2.3.18) (2.3.19)

In the two–dimensional case Σ is a given constant vector. The representation formula for solutions of the Stokes equations (2.3.17) in Ω c with the growth conditions (2.3.18) and (2.3.19) has the form u(x)

=

−V τ (x) + W ϕ(x) + ω ,

(2.3.20)

p(x)

=

−Φτ (x) + Πϕ(x)

(2.3.21)

with the Cauchy data ϕ = u|Γ and τ = T (u)|Γ satisfying  Σ = τ ds ;

(2.3.22)

Γ

and ω is an unknown constant vector which vanishes when n = 3. 2.3.2 The Stokes Boundary Value Problems We consider two boundary value problems for the Stokes system (2.3.17) in Ω as well as in Ω c . In the first problem (the Dirichlet problem), the boundary trace of the velocity on Γ (2.3.23) u|Γ = ϕ is specified, and in the second problem (the Neumann problem), the hydrodynamic boundary traction T (u)|Γ = τ

on Γ

(2.3.24)

is given. As consequences of the incompressible flow equations and the Green formula for the interior problem, the prescribed Cauchy data need to satisfy, respectively, the compatibility conditions  ϕ · n ds = 0 , 

Γ

(2.3.25)

τ · (a + b × x) ds = 0 for all a ∈ IRn and b ∈ IR1+2(n−2) , Γ

with b × x := b(x2 , −x1 ) for n = 2. For the exterior problem we require the decay conditions (2.3.18) and (2.3.19). We again solve these problems by using the direct method of boundary integral equations. Since the pressure p will be completely determined once the Cauchy data for the velocity are known, in the following, it suffices to consider only the

2.3 The Stokes Equations

67

boundary integral equations for the velocity u. We need, of course, the representation formula for p implicitly when we deal with the stress operator. In analogy to elasticity, we begin with the representation formula (2.3.13) for the velocity u in Ω and (2.3.20) and (2.3.21) in Ω c . Applying the trace operator and the stress operator T to both sides of the representation formula, we obtain the overdetermined system of boundary integral equations (the Calder´ on projection) for the interior problem ϕ(x)

=

( 12 I − K)ϕ(x) + V τ (x) ,

(2.3.26)

τ (x)

=

Dϕ + ( 12 I + K  )τ (x) on Γ .

(2.3.27)

Hence, the Calder´on projector for Ω can also be written in operator matrix form as  1 I −K V 2 . CΩ = 1  D 2I + K Here V, K, K  and D are the four corresponding basic boundary integral operators of the Stokes flow. Hence, the Calder´on projector CΩ for the interior domain has the same form as in (1.2.20) with the corresponding hydrodynamic potential operators. For the exterior problem, the Calder´on projector on solutions having the decay properties (2.3.18) and (2.3.19) with Σ given by (2.3.22) is also given by (1.4.11), i.e.,  1 I +K −V 2 . (2.3.28) CΩ c = I − C Ω = 1  −D 2I − K As always, the solutions of both Dirichlet problems as well as both Neumann problems in Ω and Ω c can be solved by using the boundary integral equations of the first as well as of the second kind by employing the relations between the Cauchy data given by the Calder´ on projectors. The four basic operators appearing in the Calder´on projectors for the Stokes problem are defined in the same manner as in elasticity (see Lemmata 2.3.1 and 2.2.3) but with appropriate modifications involving the pressure terms. More specifically, the double layer operator is defined as     1 Kϕ(x) := ϕ(x) + lim Ty E(z, y) ϕ(y)dsy (2.3.29) Ω z→x∈Γ 2 Γ     = Ty E(x, y) ϕ(y)dsy  = Γ \{x}

Γ \{x}



n 

Qk (x, y)δij + μ

i,j,k,=1

=

n 2(n − 1)π

 Γ \{x}



 ∂E (x, y) ∂E (x, y)  ik jk + nj (y)ϕi (y)dsy ∂yj ∂yi

 (x − y) · n(y) (x − y) · ϕ(y) (x − y) dsy |x − y|n+2

68

2. Boundary Integral Equations

having a weakly singular kernel for Γ ⊂ C 2 , and, hence, defines a continuous mapping K : C α (Γ ) → C 1+α (Γ ) (see Ladyˇzenskaya [256, p.35] where the fundamental solution and the potentials carry the opposite sign). The hypersingular operator D is now defined by Dϕ

=

−Tx W ϕ(x)

:=

−

=

 Tz (∂z , x) W ϕ(z) (2.3.30) Ω z→x∈Γ      Πϕ(z) n(x) − μ ∇z W ϕ(z) + ∇z W ϕ(z) n(x) . lim lim

Ω z→x∈Γ

With the standard regularization this reads       Dϕ(x) = −p.v. Tx Ty E(x, y) ϕ(y) − ϕ(x) dsy Γ

=

−μ p.v. 2(n − 1)π

  2 Γ

(2.3.31)

 1 n(y) · ϕ(y) − ϕ(x) n |x − y|

 !  " n n(y) · n(x) (x − y) · ϕ(y) − ϕ(x) (x − y) dsy |x − y|n+2   " 2n(n + 2) ! μ (x − y) · n(x) + n+4 2(n − 1)π |x − y| Γ ! "!  " × (x − y) · n(y) (x − y) · ϕ(y) − ϕ(x) (x − y) ! "!  " n − (x − y) · n(x) (x − y) · ϕ(y) − ϕ(x) n(y) |x − y|n+2  ! " + n(x) · ϕ(y) − ϕ(x) (y − x) · n(y) (x − y) ! "! "  + (x − y) · n(x) (x − y) · n(y) ϕ(y) − ϕ(x) dsy . +

Again, as in Lemma 2.2.3, the hypersingular operator can be reformulated. Lemma 2.3.1. Kohr et al [234] Let Γ ∈ C 2 and let ϕ be a H¨ older continuously differentiable function. Then the operator D in (2.3.30) can be expressed as a composition of tangential differential operators and simple layer potenλ+μ = 1 in (2.2.33) tials as in (2.2.32)–(2.2.34) where in the case n = 2 set λ+2μ and in the case n = 3 take E(x, y) from (2.3.10) in (2.2.34). Now let us assume that the boundary Γ =

L #

Γ consists of L separate,

=1

mutually non intersecting compact boundary components Γ1 , . . . , ΓL . Before we exemplify the details of solvability of the boundary integral equations, we first summarize some basic properties of their eigenspaces. Theorem 2.3.2. (See also Kohr and Pop [233].) Let n = 3. Then we have the following relations.

2.3 The Stokes Equations

69

i) The normal vector fields n ∈ C α (Γ ) where n |Γj = 0 for  = j generate L # Γ the L–dimensional eigenspace or kernel of the exterior to Ω on Γ = =1

simple layer operator V as well as of ( 12 I − K  ). Then the operator ( 12 I − K) also has an L–dimensional eigenspace generated by ϕ0 ∈ C 1+α (Γ ) with ϕ0 |Γj = 0 for  = j satisfying the equations n = Dϕ0 for  = 1, L .

(2.3.32)

L

Any eigenfunction

γj nj generates a solution j=1

0≡

L 

γj V nj

and p0 = γ1 in Ω

j=1

(see Kohr and Pop [233], Reidinger and Steinbach [361]). ii) On each component Γ of the boundary, the boundary integral operators D|Γ as well as ( 12 I + K)|Γ have the 6–dimensional eigenspace v  = (a + b × x)|Γ for all a ∈ IR3 with b ∈ IR3 . If v j, with j = 1, . . . , 6 and  = 1, . . . , L denotes a basis of this eigenspace then there exist 6L linearly independent eigenvectors τj, ∈ C α (Γ ) of the adjoint operator ( 12 I + K  )|Γ ; and there holds the relation v j, = V |Γ τj,

(2.3.33)

between these two eigenspaces. Any of the eigenfunctions vj, on Γ generates a solution   0 for x ∈ Ω if  = 2, L, u0j, (x) = − K(x, y)v j, (x)dsx = for x ∈ Ω if  = 1 vj,1 (x) Γ

⎧ ⎨0 p0j, (x) =

μ ⎩divx π(n−1)

  Γ1

∂ ∂ny γn (x, y)



for x ∈ Ω if  = 2, L, v j,1 (y)dsy for x ∈ Ω if  = 1 .

In the case n = 2, the operator V needs to be replaced by  V τ := V τ + α( τ ds) with α > 0 Γ

an appropriately large chosen scaling constant α and a + b × x replaced by a + b(x2 , −x1 ) and 6 by 3 in ii). Proof: Let n = 3 and, for brevity, L = 1. i) It is shown by Ladyˇzenskaya in [256, p.61] that n is the only eigensolution of ( 12 I − K  ). Therefore, due to the classical Fredholm alternative, the adjoint

70

2. Boundary Integral Equations

operator ( 12 I − K) has only one eigensolution ϕ0 , as well. It remains to show that n is also the only linear independent eigensolution to V and satisfies (2.3.32). As we will show later on, for V , the Fredholm theorems are also valid and V : C α (Γ ) → C α+1 (Γ ) has the Fredholm index zero. Let τ0 be any solution of V τ0 = 0. Then the single layer potential u0 = V τ0 with p0 = Φτ0 is a solution of the Stokes system in Ω as well as in Ω c with u0 |Γ = 0. Then u0 ≡ 0 in IR3 and the associated pressure is zero in Ω c and p0 = β = constant in Ω. As a consequence we have from the jump relations ! " Tx (u0− , p0 ) − Tx (u0+ , 0) = σ(u0 , p0 )n |Γ = −βn , therefore τ0 = −βn. On the other hand, V n|Γ = 0 follows from the fact that u := V n and p := Πn is the solution of the exterior homogeneous Neumann problem of the Stokes system since T (V u)|Γ = (− 12 I + K  )n = 0 . and therefore vanishes identically (see [256, Theorem 1 p.60]). In order to show (2.3.32), we consider the solution of the exterior Dirichlet Stokes problem with u+ |Γ = ϕ0 = 0 which admits the representation u(x) = W ϕ0 − V τ . Then it follows that the corresponding simple layer term has vanishing boundary values, −V τ |Γ = ϕ0 − ( 12 I + K)ϕ0 = ( 12 I − K)ϕ0 = 0 . Hence, τ = βn with some constant β ∈ IR. Application of Tx |Γ gives τ = βn = −Dϕ0 + ( 12 I − K  )βn = −Dϕ0 . The case β = 0 would imply τ = 0 and then u(x) solved the homogeneous Neumann problem which has only the trivial solution [256, p.60] implying ϕ0 = 0 which is excluded. So, β = 0 and scaling of ϕ0 implies (2.3.32). ii) For the operator ( 12 I +K) having the eigenspace (a+b×x)|Γ of dimension 6 we refer to [256, p.62]. Hence, the adjoint operator ( 12 I + K  ) also has a 6–dimensional eigenspace due to the classical Fredholm theory since K is a compact operator. For the operator D let us consider the potential u(x) = W v(x) in Ω c with v = a+b×x|Γ . Then u is a solution of the Stokes problem and on the boundary we find u+ |Γ = ( 12 I + K)v = 0 .

2.3 The Stokes Equations

71

Therefore u(x) = 0 for all x ∈ Ω c and, hence, T W v = −Dv = 0 and v ∈ kerD . Conversely, if Dv = 0 then let u be the solution of the interior Dirichlet problem with u− |Γ = v which has the representation u(x) = V τ − W τ

for x ∈ Ω

with an appropriate τ . Then applying T we find τ = ( 12 I + K  )τ + Dv = ( 12 I + K  )τ . Therefore τ satisfies ( 12 I −K  )τ = 0 which implies τ = βn with some β ∈ IR. Hence, u(x) = βV n(x) − W v(x) = −W v(x) and its trace yields ( 12 I + K)v = 0 on Γ . Therefore v = a + b × x with some a, b ∈ IR3 . Now let τ0 ∈ ker( 12 I + K  ) , τ0 = 0. Then u(x) := V τ0 (x) in Ω is a solution of the homogeneous Neumann problem in Ω since T u|Γ = ( 12 I + K  )τ0 = 0. Therefore u = a + b × x with some a, b ∈ IR3 and V τ0 ∈ ker( 12 I + K) . The mapping V : ker( 12 I + K  ) → ker( 12 I + K) is also injective since for τ0 = 0, V τ0 = 0 would imply τ0 = βn and, hence, ( 12 I + K  )τ0 = 0 = β( 12 I + K  )n = βn − ( 12 I − K  )βn = βn . So, β = 0, which is a contradiction. The case L > 1 we leave to the reader (see [207]). For n = 2 the proof follows in the same manner and we omit the details.  In the Table 2.3.3 below we summarize the boundary integral equations of the first and second kind for solving the four fundamental boundary value problems together with the corresponding eigenspaces as well as the compatibility conditions. We emphasize that, as a consequence of Theorem 2.3.2, the orthogonality conditions for the right–hand side given data in the boundary integral equations will be automatically satisfied provided the given Cauchy data satisfy the compatibility conditions if required because of the direct approach. In the case of n = 2 in Table 2.3.3, replace V by V and b × x by (bx2 , −x1 ) , b ∈ IR , a ∈ IR2 .

ENP

EDP

INP

IDP

BVP

(2)(− 12 I + K)u = V ψ

(1)Du = −( 12 + K  )ψ

Du0 = n ∈ kerV

ker( 12 I − K) = span {u0 },

kerD = span {v j, }

V τj, = v j, ∈ ker( 12 I + K)

ker( 12 I + K  ) = span {τj, },

kerV = span {n }

(1)V τ = (− 12 I + K)ϕ

(2)( 12 I + K  )τ = −Dϕ

v j, basis of {v|Γ = a + b × x|Γ , v|Γk = 0,  = k}

kerD = ker( 12 I + K) = span {v j, }

ψ · (a + b × x)ds = 0

ϕ · nds = 0

None

None

for all a, b ∈ IR3 .

Γ



Γ



for given ϕ, ψ

, k = 1, L ; j = 1, 3(n − 1)

kerV = ker( 12 − K  ) = span {n }

Compability conditions

Eigenspaces of BIO

(2)( 12 I + K)u = V ψ

(1)Du = ( 12 I − K  )ψ

(2)( 12 I − K  )τ = Dϕ

(1)V τ = ( 12 I + K)ϕ

BIE

Table 2.3.3. Boundary Integral Equations for the 3–D Stokes Problem

72 2. Boundary Integral Equations

=1

L 

j=1

=1

ωj, τj, = −Dϕ and

=1

ω n = −V ψ and Γ



u · V τj, ds =

u · v j, ds = 0

Γ



u · n ds = 0

u · v j, ds = 0,

τ · v j, ds = 0

u · Du0 ds =

Γ



Γ



 Γ

u · v j, ds = 0

τ · n ds = 0

ωj, v j, = −( 12 I + K  )ψ and

L 

3(n−1) 

j=1

=1

L 

(2)( 12 I − K)u +

(1)Du +

3(n−1) 

L 

Γ



Γ



Γ



τ · n ds = 0

ωj, v j, = V ψ and

ω n = (− 12 I + K)ϕ and

j=1

=1

(2)( 12 I − K  )τ +

(1)V τ +

Γ



τ · n ds = 0

ωj, v j, = ( 12 I − K  )ψ and

3(n−1) 

j=1

=1

 Γ

ω n = Dϕ and

L 

3(n−1) 

L 

(2)( 12 I + K)u +

(1)Du +

=1

 

ω n = ( 12 I + K)ϕ and

(2)( 12 I − K  )τ +

=1

L 

Table 2.3.4. Modified Integral Equations for the 3–D Stokes Problems

ENP

EDP

INP

IDP

(1)V τ +

Modified Equations I ,  = 1, L , j = 1, 3(n − 1)

=1 Γ

L  

=1

L 

=1 Γ

L  

=1

L 

Γ

u · v j, dsv j = V ψ

Γ

Γ



τ · v j, dsτj, = −Dϕ

=1 Γ

u · n dsn = −V ψ

u · v j, dsv j, = −( 12 I + K  )ψ

j=1

3(n−1) 

L  

j=1



=1

L 

3(n−1) 

( 12 I − K)u +

Du +

j=1

τ · n dsn = (− 12 I + K)ϕ ( 12 I − K  )τ +

Vτ +

=1



u · v j, dsv j, = ( 12 I − K  )τ

τ · n dsn = Dϕ

3(n−1) 

Γ



L 

j=1

3(n−1) 

( 12 I + K)u +

Du +

=1 Γ

L  

τ · n dsn = ( 12 I + K)ϕ

( 12 I − K  )τ +

Vτ +

Modified Equations II

2.3 The Stokes Equations 73

74

2. Boundary Integral Equations

Since each of the integral equations in Table 2.3.3 has a nonempty kernel, we now modify these equations in the same manner as in elasticity by incorporating eigenspaces to obtain uniquely solvable boundary integral equations. Again, in order not to be repetitious, we summarize the modified equations in Table 2.3.4. A few comments are in order. In the two–dimensional case, it should be understood that V should  x2 be }|Γ replaced by V and that kerD = span {v j, } with v j, a basis of {a+b −x 1 with a ∈ IR2 , b ∈ IR. Moreover, as in elasticity in Section 2.2, one has to incorporate σds appropriately, in order to take into account the decay Γ

conditions (2.3.18). For exterior problems, special attention has to be paid to the behavior at infinity. In particular, u has the representation (2.3.20), i.e., u = W ϕ − V τ + ω in Ω c . Then the Dirichlet condition leads on Γ to the system  V τ − ω = −( 12 I − K)ϕ and τ ds = Σ ,

(2.3.34)

Γ

where in the last equation Σ is a given constant vector determining the logarithmic behavior of u at infinity (see (2.3.18)). For uniqueness, this system is modified by adding the additional conditions  τ · n ds = 0  = 1, L . Γ

Then the system (2.3.34) is equivalent to the uniquely solvable system Vτ −ω+

L 

ω3 n = −( 12 I − K)ϕ ,

=1

 Γ

Vτ −ω+

L   =1 Γ

τ · n ds = 0 ,



σds = Σ ,  = 1, L ,

or

(2.3.35)

Γ

τ · n dsn = −( 12 I − K)ϕ ,



σds = Σ .

(2.3.36)

Γ

These last two versions (2.3.35) and (2.3.36) correspond to mixed formulations and have been analyzed in detail in Fischer et al [124] and [196, 197, 199, 202]. In the same manner, appropriate modifications are to be considered for other boundary co nditions and the time harmonic unsteady problems and corresponding boundary integral equations as well [199, 202], Kohr et al [233, 232] and Varnhorn [424]. There, as in this section, the boundary integral equations are considered for charges in H¨ older spaces. We shall come back to these problems in a more general setting in Chapter 5.

2.3 The Stokes Equations

75

Note that for the interior Neumann problem, the modified integral equations will provide specific uniquely determined solutions of the integral equations, whereas the solution of the original Stokes Neumann problem still has the nullspace a + b × x for n = 3 and {a + b(x2 , −x1 ) } for n = 2. Finally, the second versions of the modified integral equations (II) in Table 2.3.4 are often referred to as stabilized versions in scientific computing. Clearly, the two versions are always equivalent Fischer et al [123]. 2.3.3 The Incompressible Material — Revisited With the analysis of the Stokes problems available, we now return to the interior elasticity problems in Section 2.2.4 for almost incompressible materials, i.e., for small c ≥ 0, but restrict ourselves to the case that Γ is one connected compact manifold (see also [207], and Steinbach [398]). The case L # of Γ = Γ as in Theorem 2.3.2 is considered in [207]. =1

For the interior displacement problem, the unknown boundary traction σ satisfies the boundary integral equation (2.2.40) of the first kind, Ve σ = ( 12 I + Ke )ϕ on Γ .

(2.3.37)

where the index e indicates that these are the operators in elasticity where the kernel Ee (x, y) can be expressed via (2.2.68). Then with the simple layer potential operator Vst of the Stokes equation and its kernel given in (2.3.10) we have the relation Ve =

1 2c 1 Vst + VΔ I 1+c 1+c μ

(2.3.38)

where VΔ denotes the simple layer potential operator (1.2.1) of the Laplacian. Inserting (2.3.38) into (2.3.37) yields the equation 2 Vst σ = (1 + c)( 12 I + Ke )ϕ − c VΔ σ , μ

(2.3.39)

which corresponds to the equation (1) of the interior Stokes problem in Table 2.3.3. As was shown in Theorem 2.3.2, the solution of (2.3.39) can be decomposed in the form  σ = σ 0 + αn with σ 0 · nds = 0 and α ∈ IR . (2.3.40) Γ

Hence, 2 Vst σ 0 = (1 + c)( 12 I + Ke )ϕ − c VΔ (σ 0 + αn) , μ  σ 0 · nds = 0 . Γ

(2.3.41)

76

2. Boundary Integral Equations

A necessary and sufficient condition for the solvability of this system is the orthogonality condition    2 2 (1 + c)( 12 I + Ke )ϕ − c VΔ σ 0 − αc VΔ n · nds = 0 . μ μ Γ

Now we combine (2.2.68) with (2.3.29) and obtain the relation (1 + c)Ke ϕ = Kst ϕ + c(KΔ ϕ + L1 ϕ)

(2.3.42)

between the double layer potential operators of the Lam´e and the Stokes systems where KΔ is the double layer potential operator (1.2.8) of the Laplacian and L is the linear Cauchy singular integral operator defined by   n(y) · ϕ(y)(x − y) − (x − y) · ϕ(y)n(y)  1 dsy . p.v. L1 ϕ = 2π(n − 1) |x − y|n Γ \{x}

(2.3.43) Therefore the orthogonality condition becomes $  1 {( 2 I + Kst )ϕ} · nds + c {( 12 I + KΔ + L1 )ϕ} · nds Γ

Γ

2 % 2 (VΔ σ 0 ) · nds − α βΔ = 0 μΓ μ   := (VΔ n) · nds. Since nds = 0, it can be shown that βΔ > 0 −

where βΔ

Γ

Γ

(see [201], [204, Theorem 3.7]). In the first integral, however, we interchange orders of integration and obtain     1  ( 2 I + Kst )ϕ · nds = ϕ · ( 12 I + Kst )n ds = ϕ · nds Γ

Γ

Γ

from Theorem 2.3.2. Hence, the orthogonality condition implies that α must be chosen as μ  1 1  1 μ  ϕ · nds + {( 2 I + KΔ + L1 )ϕ} · nds − (VΔ σ 0 ) · nds . c 2βΔ Γ 2βΔ Γ βΔ Γ (2.3.44) Replacing α from (2.3.44) in (2.3.41) we finally obtain the corresponding stabilized equation,  Vst σ 0 + σ 0 · ndsn + cBσ 0 = f (2.3.45)

α=

Γ

where the linear operator B is defined by  2 1  Bσ 0 := VΔ σ 0 − (VΔ σ 0 ) · ndsVΔ n μ βΔ Γ

2.3 The Stokes Equations

77

and the right–hand side f is given by f = ( 12 I + Kst )ϕ −

1  ϕ · ndsVΔ n βΔ Γ

% $ 1  1 {( 2 I + KΔ + L1 )ϕ} · ndsVΔ n . +c ( 12 I + KΔ + L1 )ϕ − βΔ Γ Since for c = 0 the equation (2.3.45) is uniquely solvable, the regularly perturbed equation (2.3.45) for small c but c = 0 is still uniquely solvable. With σ 0 available, α can be found from (2.3.44) and, finally, the boundary traction σ is given by (2.3.40). Then the representation formula (2.2.6) provides us with the elastic displacement field u and the solution’s behavior for the elastic, but almost incompressible materials, which one may expand with respect to small c ≥ 0, as well. In particular we see that, for the almost incompressible material  ue = ust + β1Δ ϕ · ndsVΔ n + O(c) as c → 0 Γ

where ust is the unique solution of the Stokes problem with  ust |Γ = ϕ − β1Δ ϕ · ndsVΔ n + O(c) . Γ

We also have the relation    ϕ · nds = divudx = −c pdx . Γ

Ω

This shows that only if the given datum

Ω



ϕ · nds = O(c) then we have

Γ

ue = ust + O(c) . Next, we consider the interior traction problem for the almost incompressible material. For simplicity, we now employ Equation (2.2.46),  )ψ on Γ De u = ( 12 I − Ke

(2.3.46)

where now ψ, the boundary stress, is given on Γ satisfying the compatibility conditions (2.2.47), and the boundary displacement u is the unknown. With (2.3.38), i.e., Ee (x, y) =

1 1 c Est (x, y) + γn (x, y)I 1+c 1 + c 2(n − 1)πμ

(2.3.47)

and with Lemma 2.3.1 we obtain for the hypersingular operators De ϕ = Dst ϕ + cL2 ϕ

(2.3.48)

78

2. Boundary Integral Equations

where for n = 3



L2 ϕ(x) = Mx

4μ2

Γ \{x}

 1  1 Est (x, y)− γn (x, y)I My ϕ(y)dsy , 1+c 2(n − 1)μπ

and for n = 2 the differential operators can be replaced as Mx = My = dsdy . Hence, (2.3.46) can be written as  )ψ − cL2 u . Dst u = ( 12 I − Ke

(2.3.49) d dsx and

(2.3.50)

In view of Theorem 2.3.2, one may decompose the solution u in the form u(x) = u0 (x) +

M 

αj mj (x)

(2.3.51)

j=1

where 

u0 · mj ds = 0 for j = 1, . . . , M with M := 12 n(n + 1) ,

Γ

and mj (x) are the traces of the rigid motions given in (2.2.55). These vector valued functions form a basis of the kernel to De as well as to Dst which implies also that L2 mj = 0 for j = 1, . . . , M and c ∈ IR .

(2.3.52)

Substituting (2.3.51) into (2.3.50) yields the uniquely solvable system of equations  )ψ , Dst u0 + cL2 u0 = ( 12 I − Ke  (2.3.53) u · m ds = 0 for j = 1, . . . , M ; 0

j

Γ

or, in stabilized form Dst u0 +

M  

 u0 · mj dsmj + cL2 u0 = ( 12 I − Ke )ψ .

(2.3.54)

j=1 Γ

The right–hand side in (2.3.53) satisfies the orthogonality conditions  1  ( 2 I − Ke )ψ · mj ds = 0 for j = 1, . . . , M Γ

since the given ψ satisfies the compatibility conditions  ψ · mj ds = 0 for j = 1, . . . , M Γ

and the vector valued function mj satisfies

2.4 The Biharmonic Equation

79

( 12 I + Ke )mj = 0 on Γ . The equations (2.3.53) or (2.3.54) are uniquely solvable for every c ∈ [0, ∞) and so, the general elastic solution ue for almost incompressible material has the form ue (x) = Ve ψ(x) − We u0 (x) +

M 

αj mj (x)

j=1 M   1  2c = ust + VΔ ψ − c(WΔ u0 + L1 u0 (x) + αj mj (x) 1+c μ j=1

for x ∈ Ω with arbitrary αj ∈ IR and where ust is the solution of the Stokes problem with given boundary tractions ψ, and L1 is defined in (2.3.43). For c → 0 we see that for the elastic Neumann problem ue = ust + O(c) up to rigid motions, i.e., a regular perturbation with respect to the Stokes solution.

2.4 The Biharmonic Equation In both problems, plane elasticity and plane Stokes flow, the systems of partial differential equations can be reduced to a single scalar 4th–order equation, Δ2 u = 0 in

Ω (or Ω c ) ⊂ IR2 ,

(2.4.1)

kwown as the biharmonic equation. In the elasticity case, u is the Airy stress function, whereas in the Stokes flow u is the stream function of the flow. The Airy function W (x) is defined in terms of the stress tensor σij (u) for the displacement field u as σ11 (u) =

∂2W , ∂x22

σ12 (u) = −

∂2W , ∂x1 ∂x2

σ22 (u) =

∂2W , ∂x21

which satisfies the equilibrium equation divσ(u) = 0 automatically for any smooth function W . Then from the stress-strain relation in the form of Hooke’s law, it follows that ΔW = σ11 (u) + σ22 (u) = (λ + 2μ)divu ; and thus, W satisfies (2.4.1) since Δ(div u) = 0 from the Lam´e system. On the other hand, the stream function u is defined in terms of the velocity u in the form

80

2. Boundary Integral Equations

u = (∇u)⊥ . Here ⊥ indicates the operation of rotating a vector counter–clockwise by a right angle. From the definition, the continuity equation for the velocity is satisfied for any choice of a smooth stream function u. One can verify directly that u satisfies (2.4.1) by taking the curl of the balance of momentum equation in the Stokes system. We note that in terms of the stream function, the vorticity is equal to ωk = ∇ × u = Δuk, where k is the unit vector perpendicular to the (x1 , x2 )− plane of the flow. For the homogeneous Stokes system, the vorticity is a harmonic function, and as a consequence, u satisfies the biharmonic equation (2.4.1). To discuss boundary value problems for the biharmonic equation (2.4.1), it is best to begin with Green’s formula for the equation in Ω. As is well known, for fourth-order differential equations, the Green formula generally varies and depends on the choice of boundary operators, i.e., how to apply the integration by parts formulae. In order to include boundary conditions arising for the thin plate, we rewrite Δ2 u in terms of the Poisson ratio ν in the form ∂2  ∂2u ∂2u  ∂2u  ∂2  ∂2u  ∂2  ∂2u Δ2 u = +ν +ν +2(1−ν) + . ∂x21 ∂x21 ∂x22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x21 Now integration by parts leads to the first Green formula in the form     ∂v M u + vN u ds , (2.4.2) (Δ2 u) vdx = a(u, v) − ∂n Ω

Γ

where the bilinear form a(u, v) is defined by   a(u, v) := Ω

νΔu Δv + (1 − ν)

2   ∂2u ∂ 2 v  dx ; ∂xi ∂xj ∂xi ∂xj i,j=1

and M and N are differential operators defined by  M u := νΔu + (1 − ν) (n(z) · ∇x )2 u)|z=x ,  ∂  d Δu + (1 − ν) (n(z) · ∇x )(t(z) · ∇x )u(x) N u := − ∂n ds |z=x

(2.4.3)

(2.4.4) (2.4.5)

where t = n⊥ is the unit tangent vector, i.e. t1 = −n2 , t2 = n1 . Then $ ∂2u ∂2u ∂2u 2% 2 n + 2 n n + n , M u = νΔu + (1 − ν) 1 2 ∂x21 1 ∂x1 ∂x2 ∂x22 2  ∂ d  ∂ 2 u ∂ 2 u  ∂2u 2 2 N u = − Δu + (1 − ν) n − n − (n − n ) . 1 2 2 ∂n ds ∂x21 ∂x22 ∂x1 ∂x2 1 For the interior boundary value problems for (2.4.1), the starting point is the representation formula

2.4 The Biharmonic Equation

 u(x)

{E(x, y)N u(y) +

= Γ



−

 ∂E ∂ny

 (x, y) M u(y)}dsy

81

(2.4.6)

 ∂u  { My E(x, y) + Ny E(x, y) u(y)}dsy for x ∈ Ω , ∂ny

Γ

where E(x, y) is the fundamental solution for the biharmonic equation given by 1 E(x, y) = |x − y|2 log |x − y| (2.4.7) 8π which satisfies Δ2x E(x, y) = δ(x − y) in IR2 . As in case of the Laplacian, we may rewrite u in the form  ∂u  u(x) = V (M u, N u) − W u, ∂n where

 {E(x, y)N u(y) +

V (M u, N u) := 

∂u  W u, := ∂n

Γ





My E(x, y)

 ∂E ∂ny

 (x, y) M u(y)}dsy ,

 ∂u  (y) + Ny E(x, y) u(y) dsy ∂n

(2.4.8)

(2.4.9) (2.4.10)

Γ ∂u are the simple and double layer potentials, respectively, and u|Γ , ∂n |Γ , M u|Γ and N u|Γ are the (modified) Cauchy data. This representation formula (2.4.6) suggests two basic types of boundary conditions: ∂u |Γ are prescribed The Dirichlet boundary condition, where u|Γ and ∂n on Γ , and the Neumann boundary condition, where M u|Γ and N u|Γ are prescribed on Γ . In thin plate theory, where u stands for the deflection of the middle surface of the plate, the Dirichlet condition specifies the displacement and the angle of rotation of the plate at the boundary, whereas the Neumann condition provides the bending moment and shear force at the boundary. Clearly, various linear combinations will lead to other mixed boundary conditions, which will not be discussed here. From the bilinear form (2.4.3), we see that

v ∈ R := {v = c1 x1 + c2 x2 + c3 | for all c1 , c2 , c3 ∈ IR} . (2.4.11) This implies that the Neumann data need to satisfy the compatibility condition    ∂v M u + vN u ds = 0 for all v ∈ R . (2.4.12) ∂n a(u, v) = 0

for

Γ

82

2. Boundary Integral Equations

We remark that looking at (2.4.2), one might think of choosing Δu and ∂ Δu as the Neumann boundary conditions which correspond to the Pois− ∂n son ratio ν = 1. This means that the compatibility condition (2.4.12) requires that it should hold for all harmonic functions v. However, the space of harmonic functions in Ω has infinite dimension, and this does not lead to a regular boundary value problem in the sense of Agmon [3, p. 151]. As for the exterior boundary value problems, in order to ensure the uniqueness of the solution of (2.4.1) in Ω c , we need to augment (2.4.1) with an appropriate radiation condition (see (2.3.18)). We require that  A1 · x  r log r + O(r) u(x) = A0 r + |x|

as

r = |x| → ∞

(2.4.13)

for given constant A0 and constant vector A1 . Under the condition (2.4.9), we then have the representation formula for the solution of (2.4.1) in Ω c ,  ∂u  + p(x), u(x) = −V (M u, N u) + W u, ∂n

(2.4.14)

where p ∈ R is a polynomial of degree less than or equal to one. Before we formulate the boundary integral equations we first summarize some classical basic results. Theorem 2.4.1. (Gakhov [135], Mikhlin [298, 299, 301] and Muskhelishvili [319]). Let Γ ∈ C 2,α , 0 < α < 1. i) Let  ∂u  |Γ ∈ C 3+α (Γ ) × Γ 2+α (Γ ) ϕ = (ϕ1 , ϕ2 ) = u|Γ , ∂n be given. Then there exists a unique solution u ∈ C 3+α (Ω) ∩ C 4 (Ω) of the interior Dirichlet problem satisfying the Dirichlet conditions u|Γ = ϕ1 and

∂u |Γ = ϕ2 . ∂n

(2.4.15)

For given A0 ∈ IR and A1 ∈ IR2 there also exists a unique solution u ∈ C 3+α (Ω c ∪ Γ ) ∩ C 4 (Ω c ) of the exterior Dirichlet problem which behaves at infinity as in (2.4.13). ii) For given ψ = (ψ1 , ψ2 ) ∈ C 1+α (Γ ) × C α (Γ ) satisfying the compatibility conditions (2.4.12), i.e.,    ∂v + ψ2 v ds = 0 for all v ∈ R , ψ1 ∂n

(2.4.16)

Γ

the interior Neumann problem consisting of (2.4.1) and the Neumann conditions

2.4 The Biharmonic Equation

M u|Γ = ψ1 and N u|Γ = ψ2

83

(2.4.17)

(Ω) ∩ C (Ω) which is unique up to a linear function has a solution u ∈ C p ∈ R. If, for the exterior Neumann problem in Ω c , in addition to ψ the linear function p ∈ R is given, then it has a unique solution u ∈ C 3+α (Ω c ∪ Γ ) ∩ C 4 (Ω c ) with the behaviour (2.4.13), (2.4.14) where   A0 = − ψ2 ds and A1 = (ψ1 n + ψ2 x)ds . (2.4.18) 3+α

4

Γ

Γ

As a consequence of Theorem 2.4.1 type, ⎧ for  ∂p  ⎨ p 1 p for −W p, = ⎩ 2 ∂n 0 for

one has the useful identity of Gaussian x∈Ω, x∈Γ, x ∈ Ωc ,

for any p ∈ R .

(2.4.19)

2.4.1 Calder´ on’s Projector (See also [208].) In order to obtain the boundary integral operators as x approaches Γ , from the simple– and double–layer potentials in the representation formulae (2.4.8) and (2.4.14), we need explicit information concerning the kernels of the potentials. A straightforward calculation gives     ∂E V (M u, N u)(x) = (x, y) M u(y)dsy E(x, y)N u(y)dsy + ∂ny Γ Γ  1 = |x − y|2 log |x − y|N u(y)dsy (2.4.20) 8π Γ  1 + n(y) · (y − x)(2 log |x − y| + 1)M u(y)dsy 8π Γ

∂u  (x) W u, ∂n 



  ∂u(y) = dsy + My E(x, y) Ny E(x, y) u(y)dsy ∂n Γ Γ   1 = (2 log |x − y| + 1) + ν(2 log |x − y| + 3) 8π 

Γ

((y − x) · n(y))2  ∂u(y) dsy (2.4.21) |x − y|2 ∂n   ∂ 1 1 ) log( + 2π ∂ny |x − y| +2(1 − ν)

Γ

d  (x − y) · t(y)(x − y) · n(y)  1 u(y)dsy . − (1 − ν) 2 dsy |x − y|2 This leads to the following 16 boundary integral operators.

  ϕ = σ

−W (ϕ1 , 0)(z) − 12 ϕ1 (z)

⎛

⎞

⎞ V14 V24 ⎟ ⎟ V34 ⎠ 1 2 I + K44

    ∂u ϕ = (ϕ1 , ϕ2 , σ1 , σ2 ) = u, , M u, N u |Γ . σ ∂n

⎟ V24 ⎟ ⎟ V34 ⎟ ⎠ K44

V14

Mz V (σ1 , 0)(z) − 12 σ1 (x)

−Mz W (0, ϕ2 )(z) Nz V (σ1 , 0)(z)

nx · ∇z V (σ1 , 0)(z)

−nx · ∇z W (0, ϕ2 )(z) − 12 ϕ2 (x) −Nz W (0, ϕ2 )(z) ⎛ −K11 V12 V13 ⎜ ⎜ D21 K22 V23 =⎜ ⎜D ⎝ 31 D32 −K33 D41 D42 D43

V (σ1 , 0)(z)

−W (0, ϕ2 )(z)

Then the Calderon projector associated with the bi–Laplacian is defined by ⎛1 V12 V13 2 I − K11 1 ⎜ D21 1 I + K V23 22 2 CΩ := I + K = ⎜ 1 ⎝ D31 D32 I 2 2 − K33 D41 D42 D43

where we write

⎜ ⎜ −nx · ∇z W (ϕ1 , 0)(z) lim ⎜ Ω z→x∈Γ ⎜ ⎝ −Mz W (ϕ1 , 0)(z) −Nz W (ϕ1 , 0)(z)

K

⎟ ⎟ ⎟ ⎟ ⎠

⎞

(2.4.23)

(2.4.22)

Nz V (0, σ2 )(z) − 12 σ2 (x)

Mz V (0, σ2 )(z)

nx · ∇z V (0, σ2 )(z)

V (0, σ2 )(z)

84 2. Boundary Integral Equations

2.4 The Biharmonic Equation

85

Some more explanations are needed here. In order to maintain consistency with our notations for the Laplacian, we have adopted the notations Vij , Kij and Dij for the weakly and hypersingular boundary integral operators according to our terminology. These boundary integral operators are obtained by taking limits of the operations ∇z (•) · nx , Mz , Nz , respectively on the corresponding potentials V and W as Ω  z → x ∈ Γ . As in the case of the ∂u , M u, N u)Γ on Laplacian, for any solution of (2.4.1), the Cauchy data (u, ∂n on Γ are reproduced by the matrix operators in (2.4.23), and CΩ is the Calder´ projector corresponding to the bi–Laplacian. In the classical H¨ older function spaces, we have the following lemma. &3 Lemma 2.4.2. Let Γ ∈ C 2,α , 0 < α < 1. Then CΩ maps k=0 C 3+α−k (Γ ) into itself continuously. Moreover, 2 CΩ = CΩ .

(2.4.24)

As a consequence of this lemma, one finds the following specific identities: V12 D21 + V13 D31 + V14 D41

=

2 ( 14 I − K11 ),

D21 V12 + V23 D32 + V24 D42

=

2 ( 14 I − K22 ),

D31 V13 + D32 V23 + V34 D43

=

2 ( 14 I − K33 ),

D41 V14 + D42 V24 + D43 V34

=

2 ( 14 I − K44 ).

Clearly, from (2.4.24) one finds 12 more identities between these operators. In the same manner as in the case for the Laplacian, for any solution u of (2.4.1) in Ω c with p = 0, we may introduce the Calder´on projection CΩ c for the exterior domain for the biharmonic equation. Then clearly, we have CΩ c = I − C Ω , where I denotes the identity matrix operator. This relation then provides the corresponding boundary integral equations for exterior boundary value problems. As will be seen, the boundary integral operators in CΩ are pseudodifferential operators on Γ and their orders are summarized systematically in the following: ⎞ ⎛ 0 −1 −3 −3 ⎜ +1 0 −1 −3 ⎟ ⎟ (2.4.25) Ord(CΩ ) := ⎜ ⎝ +1 +1 0 −1 ⎠ +3 +1 +1 0 The orders of these operators can be calculated from their symbols and provide the mapping properties in the Sobolev spaces to be discussed in Chapter 11.

86

2. Boundary Integral Equations

2.4.2 Boundary Value Problems and Boundary Integral Equations We begin with the boundary integral equations for the Dirichlet problems. For the integral equations of the first kind we employ the second and the first row of CΩ which leads to the following system for the interior Dirichlet problem, (' ( ' (' ( ' 1 σ1 −D21 ϕ1 V23 V24 2 I − K22 = 1 =: f i . Vσ := V13 V14 σ2 ϕ2 −V12 2 I + K11 (2.4.26) The solution of the interior Dirichlet problem has associated Cauchy data σ which satisfy the three compatibility conditions:   (σ1 n + σ2 x)dsx = 0 , − σ2 ds = 0 . (2.4.27) Γ

Γ

As we shall see in Chapter 11 , V is known as a strongly elliptic operator for which the classical Fredholm alternative holds. Hence uniqueness will imply the existence of exactly one solution σ ∈ C 1+α (Γ ) × C α (Γ ). For the exterior Dirichlet problem, by using CΩ c and the representation (2.4.13) we obtain the system with integral equations of the first kind, (' ( ' 1 ϕ1 I + K D21 22 2 =: f e , Vσ + Rω = − 1 ϕ2 V12 2 I − K11 

 (σ1 n + σ2 x) = A1 , − σ2 ds = A0

Γ

where



0 R(x) = − 1

(2.4.28)

Γ

n1 x1

n2 x2



and ω = (ω0 , ω1 , ω2 ) ∈ IR3 .

Lemma 2.4.3. The homogeneous system corresponding to (2.4.28) has only the trivial solution in C 1+α (Γ ) × Cα (Γ ) × IR3 . Proof: Let σ 0 , ω 0 be any solution of Vσ 0 + Rω 0 = 0 on Γ ,

  ∂v  , v σ 0 ds = 0 for all v ∈  Γ ∂n

(2.4.29)

and consider the solution of (2.4.1), ◦

◦

◦

u0 (x) := V σ 0 (x) + p0 (x) with p0 (x) = ω 0 + ω 1 x1 + ω 2 x2 for x ∈ Ω c . Then A0 = 0 , A1 = 0 because of (2.4.29) and (2.4.18), hence u0 = O(|x|)at infinity due to (2.4.13) which implies u0 (x) = p0 (x) for all x ∈ Ω c ∪ Γ . On the other hand, u0 (x) is also a solution of (2.4.1) in Ω and is continuous

2.4 The Biharmonic Equation

87

across Γ where u0 |Γ = 0. Hence, due to Theorem 2.4.1, u0 (x) = 0 for all x ± in Ω. Consequently, M u± 0 |Γ = 0 and N u0 |Γ = 0. Then the jump relations corresponding to CΩ c − CΩ imply σ 0 = ([M u]|Γ , [N u]|Γ ) = 0 on Γ and ◦ + 0 = u−  0 |Γ = u0 |Γ = p0 implies p0 (x) = 0 for all x, i.e., ω = 0. As a consequence, both, interior and exterior Dirichlet problems lead to the same uniquely solvable system (2.4.28) where only the right–hand sides are different and, for the interior Dirichlet problem, ω = 0. Clearly, the solution of the Dirichlet problems can also be treated by using the boundary integral equations of the second kind. To illustrate the idea we ∂u |Γ = ϕ2 . consider again the interior Dirichlet problem where u|Γ = ϕ1 and ∂n From the representation formula (2.4.6) we obtain the following system for the unknown σ = (M u, N u) on Γ : (' ( ' (' ( ' 1 σ1 D31 D32 ϕ1 −V34 2 I + K33 = =: Dϕ . (2.4.30) 1 σ2 D41 D42 ϕ2 −D43 2 I − K44 This system (2.4.30) of integral equations has a unique solution. As we shall see in Chapter 11, for 0 ≤ ν < 1 the Fredholm alternative is still valid for these integral equations and σ ∈ C 1+α (Γ ) × C α (Γ ). So, uniqueness implies existence. ◦

Lemma 2.4.4. Let σ ∈ C α (Γ ) × C 1+α (Γ ) be the solution of the homogeneous system ◦ ◦ ( 12 I + K33 )σ1 − V34 σ2 = 0 (2.4.31) ◦ ◦ −D34 σ1 + ( 12 I − K44 )σ2 = 0 on Γ . ◦

Then σ = 0. Proof: For the proof we consider the simple layer potential ◦

u0 (x) = V σ which is a solution of (2.4.1) for x ∈ Γ . Then for x ∈ Ω we obtain with (2.4.31): ◦ ◦ ◦ M u− = ( 12 I − K33 )σ1 + V34 σ2 = σ1 , 0 |Γ N u− 0 |Γ

=

◦

◦

( 12 I + K44 )σ2 + D43 σ1

◦

= σ2 .

Then the Green formula (2.4.2) implies  Γ

◦

σ1

 ∂v ◦ + σ2 v ds = 0 for all v ∈  . ∂u

(2.4.32)

+ For x ∈ Ω c , we find M u+ 0 |Γ = 0 and N u0 |Γ = 0 due to (2.4.31). Then Theorem 2.4.1 implies with (2.4.32) that

88

2. Boundary Integral Equations

u0 (x) = p(x) for x ∈ Ω c ∪ Γ

with some p ∈  .

But u0 (x) is continuously differentiable across Γ and satisfies (2.4.1) in Ω with ∂u−

∂p 0 boundary conditions u− 0 |Γ = p|Γ and ∂n |Γ = ∂n |Γ . Hence, with Theorem 2.4.1 we find ◦ V σ(x) = u0 (x) = p(x) for x ∈ IR2 .

Then

◦

◦

σ 1 = [M u0 ]|Γ = 0 and σ 2 = [N u0 ]|Γ = 0 .  We now conclude this section by summarizing the boundary integral equations associated with the two boundary value problems of the biharmonic equation considered here in the following Tables 2.4.5 and 2.4.6. However, the missing details will not be pursued here. We shall return to these equations in later chapters. We remark that in Table 2.4.5 we did not include orthogonality conditions for the right–hand sides in the equations INP (1) and (2), EDP (2) and ENP (1) since due to the direct approach it is known that the right–hand sides always lie in the range of the operators. Hence, we know that the solutions exist due to the basic results in Theorem 2.4.1, and, moreover, the classical Fredholm alternative holds for the systems in Table 2.4.5. From this table we now consider the modified systems so that the latter will always be uniquely solvable. The main idea here is to incorporate additional side conditions as well as eigensolutions. These modifications are collected in Table 2.4.6. In particular, we have augmented the systems by including additional unknowns ω ∈ IR3 in the same manner as in Section 2.2 for the Lam´e system. Note that in Table 2.4.6 the matrix valued function S is defined by ◦ ◦ ◦ S(x) := σ 1 (x), σ 2 (x), σ 3 (x) where the columns of S are the three linearly independent eigensolutions of the operator on the left–hand side of EDP (2) in Table 2.4.5. If we solve the exterior Neumann problem with the system ENP (1) in Table 2.4.6, then we obtain a particular solution with p(x) = 0 in Ω c , and for given p(x) = 0, the latter is to be added to the representation formula (2.4.14). For the interior Neumann problem, the modified boundary integral equation INP (1) and (2) provide a particular solution which presents the general solution only up to linear polynomials. Note that here we needed Γ ∈ C 2,α and even jumps of the curvature are excluded. For piecewise Γ ∈ C 2,α –boundary, Green’s formula, the representation formula as well as the boundary integral equations need to be modified appropriately by including certain functionals at the discontinuity points (Kn¨ opke [230]).



1 I 2

V13

V23



V34

σ2

σ1

1 I 2



− K33

V14

V24

− =− 



σ1

∂ p ∂n

p 

 1 I 2

+ K22

V12   ϕ1 D32

− K11

D21

D31

1 I 2





 ϕ2

ϕ1



(2)

D43

=− + K44 σ2 D41 D42 ϕ2       1 D41 D42 u D43 ψ1 I + K44 2 (1) = − 1 ∂u D31 D32 I − K V ψ 33 34 2 ∂n 2         1 u V13 V14 ψ1 p(x) I − K11 V12 2 (2) =− + ∂ ∂u 1 D21 I + K22 V23 V24 ψ2 p(x) 2 ∂n ∂n

(1)



Table 2.4.5. Boundary Integral Equations for the Biharmonic Equation

ENP

EDP

INP

IDP

Systems of BIEs       1 V23 V24 σ1 −D21 ϕ1 I − K22 2 (1) = 1 V13 V14 σ2 I + K11 −V12 ϕ2 2      1 D31 D32 ϕ1 I + K33 −V34 (2) 2 = 1 −D43 I − K D D ϕ 44 41 42 2 2       1 D41 D42 u −D43 ψ1 I − K44 2 (1) = 1 ∂u D31 D32 I + K33 −V34 ψ2 ∂n 2       1 u V13 V14 ψ1 I + K11 −V12 (2) 2 = ∂u 1 −D21 I − K V V ψ 22 23 24 2 2 ∂n

none, p(x) given

A0 , A1 given

Γ

 − σ2 ds = A0  Γ (σ1 n + σ2 x)ds = A1 .

for all v ∈ 

Γ



 ∂v + ψ2 v ds = 0 ψ1 ∂n

none

for all v ∈ 

Γ

Side conditions    ∂v σ1 ∂n + σ2 v ds = 0 ∗

∂p ∂n

p



∂p ∂n

p



◦

none



◦

,p∈

◦

σ1 , σ2 , σ3

◦

,p∈

none for σ satisfying *



none

none for σ satisfying *

◦

Eigenspaces

2.4 The Biharmonic Equation 89

90

2. Boundary Integral Equations

BVP  (1) 

V23

V24

V13

V14



1 I 2

(2)

(1)

D31

D32



1 I 2

(2)



V23

V24

u



1 I 2

1 I 2

=

σ2





+Rω =  ∂u



ϕ1

D41

D42

ϕ2

1 I 2

− K44





ψ1



+ K33 −V34 ψ2   ∂u ds = 0 , ux2 + n2 ∂n ds = 0 1 I 2

Γ



− K22  ∂u

σ1

− K33

u



 +Sω =

∂u ∂n







V13

V14

V23

V24  ∂u



ψ1



ψ2

ux2 + n2 ∂n ds = 0

Γ

D21

1 I 2

+ K22



V12

ϕ1



ϕ2



V34

σ1

 

1

x1

x2

1 I 2

n2



 ω =−

D31

D32

D41

D42



Γ

D31

D32



up ds = 0 ,

1 I 2



Γ

D42

Γ

ϕ1 ϕ2

D32

−D43

ux1 + n1 ∂n ds = 0 ,

Γ

D41





D31

+ + K44 σ2 0 n1   − σ2 ds = A0 , (σ1 n + σ2 x)ds = A1

(1)

(2)







D43



− K22

−V12

+ K11





−V12

Γ

ENP

1 I 2

+Rω = − 1 V13 V14 σ2 I − K11 2   − σ2 ds = A0 , (σ1 n + σ2 x)ds = A1 

(2)

1 I 2

σ1

ux1 + n1 ∂n

Γ

Γ

EDP



− K44

∂u ∂n

+ K11

uds = 0 ,

Γ

−D21

(σ1 n + σ2 x)dsx = 0



−D21 

 +Rω =

σ2

1 I 2

uds = 0 ,



−V34

D42

Γ

σ1

+ K33

D41



(1)

Γ

−D43 

INP



σ2 ds = 0 ,

Γ

IDP



∂up ∂n

 Γ

− K11 D21

up



 +Rω = −

up n 1 +

V12 1 I 2

+ K22

∂up x ∂n 1



u ∂u ∂n



D43

1 I 2

+ K44



ψ1



− K33 V34 ψ2

 ∂u ds = 0 , up n2 + ∂np x2 ds = 0 1 I 2

Γ



 =−

V13

V14

V23

V24



Table 2.4.6. Modified Systems for the Biharmonic Equation

ψ1 ψ2



 +

p ∂ p ∂n



ϕ1 ϕ2



2.5 Remarks

91

2.5 Remarks Very often in applications, on different parts of the boundary, different boundary conditions are required or, as in classical crack mechanics (Cruse [93]), the boundaries are given as transmission conditions on some bounded manifold, the crack surface, in the interior of the domain. A similar situation can be found for screen problems (see also Costabel and Dauge [85] and Stephan [402]). As an example of mixed boundary conditions let us consider the Lam´e system with given Dirichlet data on ΓD ⊂ Γ and given Neumann data on ΓN ⊂ Γ where Γ = ΓD ∪ ΓN ∪ γ with the set of collision points γ of the two boundary conditions (which might also be empty if ΓD and ΓN are separated components of Γ ) (see e.g. Fichera [120], [209], Kohr et al [234], Maz‘ya [289], Stephan [404]) where meas (ΓD ) > 0. For n = 2, where Γ is a closed curve, we assume that either γ = ∅ or consists of finitely many points; for n = 3, the set γ is either empty or a closed curve and as smooth as Γ . For the Lam´e system (2.2.1) the classical mixed boundary value problem reads: Find u ∈ C 2 (Ω) ∩ C α (Ω) , 0 < α < 1, satisfying in Ω with −Δ∗ u = f γ0 u = ϕD on ΓD and T u = ψ N

on ΓN .

(2.5.1)

For reformulating this problem with boundary integral equations we first extend ϕD from ΓD and ψ N from ΓN onto the complete boundary Γ such that ϕD = ϕ|ΓD and ψ N = ψ|ΓN (2.5.2) with ϕ ∈ C α (Γ ) , 0 < α < 1 and appropriate ψ. Then   , Tu = ψ + ψ γ0 u = ϕ + ϕ

(2.5.3)

 ∈ C0α (ΓN ) = {ϕ ∈ C α (Γ ) | supp ϕ ⊂ ΓN } ϕ

(2.5.4)

where now  with supp ψ ⊂ Γ D are the yet unknown Cauchy data to be determined. and ψ With (2.5.3), the representation formula (2.2.4) reads     v(x) = E(x, y)ψ(y)dsy − Ty E(x, y) ϕ(y)dsy Γ



+ Γ



Γ

 E(x, y)ψ(y)ds y −





Ty E(x, y)

Γ

E(x, y)f (y)dy for x ∈ Ω .

+ Ω



 (y)dsy ϕ

(2.5.5)

92

2. Boundary Integral Equations

 will have singularities at γ which As is well known, even if ψ ∈ C α (Γ ) then ψ 1  ∈ C α (ΓD )  with ψ need to be taken into account either by {dist (x, γ)}− 2 ψ 1 1 or by adding singular functions at γ. Taking the trace and the traction of (2.5.5) on Γ leads with (2.5.3) to the system of boundary integral equations   (x) = 12 ϕ(x) + Kϕ(x) − V ψ(x) − N f (x) V ψ(x) − Kϕ for x ∈ ΓD ,   (x) = 12 ψ(x) − K  ψ(x) − Dϕ(x) − Tx N f (x) for x ∈ ΓN . K  ψ(x) + Dϕ (2.5.6) As will be seen in Chapter 5, the system (2.5.6) is uniquely solvable for 1 ϕ  ∈ C0α (ΓN ) and ψ either in the space with the weight {dist (x, γ)}− 2 or in an augmented space according to the asymptotic behaviour of the solution and involving the stress intensity factors (Stephan et al [406] provided meas(ΓD ) > 0. In a similar manner one might also use the system of integral equations of the second kind   (x) − V ψ(x) + Kϕ = V ψ(x) − 12 ϕ(x) − Kϕ(x) + N f (x) for x ∈ ΓN ,  1 1  (x) = − ψ(x) + K  ψ(x) + Dϕ(x) + Tx N f (x) ψ(x) − K ψ(x) − Dϕ 1  2 ϕ(x)

2

2

for x ∈ ΓD .

(2.5.7)

For the Laplacian and the Helmholtz equation and mixed boundary value problems as well as for the Stokes system one may proceed in the same manner. As will be seen in Chapter 5, the variational formulation for the mixed boundary conditions provides us with the right analytical tools for showing the well–posedness of the formulation (2.5.6) (see e.g., Kohr et al [234], Sauter and Schwab [371] and Steinbach [399]). In the engineering literature, usually the system (2.5.7) is used for discretization and then the equations corresponding to (2.5.7) are obtained by assembling the discrete given and unknown Cauchy data appropriately (see e.g., Bonnet [37], Brebbia et al [43, 44] and Gaul et al [140]). For crack and insertion problems let us again consider just the example of classical linear theory without volume forces. Let us consider a bounded open domain Ω ⊂ IRn with n = 2 or 3 enclosing a given bounded crack or insertion surface as an oriented piece of a curve Γc ∈ C α , if n = 2 or, if n = 3, as an open piece of an oriented surface Γc ∈ C α , with a simple, closed boundary curve ∂Γc = γ ∈ C α . Further the crack should not reach the boundary ∂Ω = Γ of Ω, i.e., Γ c ⊂ Ω. The annulus Ωc := Ω \ Fc is not a Lipschitz domain anymore but if we distinguish the two sides of Γc assigning with + the points near Γc on the side of the normal vector nc given due to the orientation of Γc and the points of the opposite side with −, the traces from either side are still defined. For the crack or insertion problem, an elastic field u ∈ C 2 (Ωc ) is sought which satisfies the homogeneous Lam´e system

2.5 Remarks

−Δ∗ u = 0 in Ωc ,

93

(2.5.8)

in C α (Ωc ∪ Γ ) and up to Γc from either side with possibly different traces at Γc , (2.5.9) γ0± u|Γc = ϕ± and Tc± u = ψ ± where we have the transmission properties [γ0 u]|Γc := (γ0+ u − γ0− u)|Γc = [ϕ]|Γc = (ϕ+ − ϕ− )|Γc ∈ C0α (Γc )

(2.5.10)

and [Tc u]|Γc := (Tc+ u − Tc− u)|Γc = [ψ]|Γc := (ψ + − ψ − )|Γc ∈ C1α (Γc ) (2.5.11) with C0α (Γc ) := {v ∈ C α (Γ c ) | (γ0+ v − γ0− v)|γ = 0} ,

(2.5.12)

and 1

C1α (Γc ) := {ψ = {dist (x − γ)}− 2 ψ 1 (x) | ψ 1 ∈ C α (Γ c )} .

(2.5.13)

For the classical insertion problem with Dirichlet conditions γ0 u = ϕ ∈ C α (Γ ) on Γ the functions ϕ± ∈ C0α (Γc ) are given. The unknown field u then has to satisfy the boundary conditions γ0 u|Γ = ϕ on Γ

and with (ϕ+ − ϕ− )|γ = 0 ,

γ0+ u|Γc = ϕ+ and γ0− u|Γc = ϕ− on Γc

(2.5.14)

as well as the transmission conditions (2.5.10) and (2.5.11). By extending Γc up to the boundary Γ ficticiously and applying the Green formula to the two ficticiously separated subdomains of Ω one finds the representation formula     u(x) = E(x, y)ψ(y)dsy − Ty E(x, y) ϕ(y)dsy Γ



−

Γ



y∈Γc



Tyc E(x, y)

E(x, y)[ψ]|Γc (y)dsy +



(2.5.15) [ϕ]|Γc (y)dsy

Γc

for x ∈ Ωc . By taking the traces of (2.5.15) at Γ and at Γc one obtains the following system of equations on Γ and on Γc :

94

2. Boundary Integral Equations



 E(x, y)ψ(y)dsy −

y∈Γ

E(x, y)[ψ](y)dsy y∈Γc





= 12 ϕ(x) + Kϕ(x) −

Tyc E(x, y)



[ϕ]dsy for x ∈ Γ ,

y∈Γc





−

(2.5.16)

E(x, y)ψ(y)dsy + y∈Γ

 1

−



E(x, y)[ψ](y)dsy Γc



= − 2 ϕ (x) + ϕ (x) + +



Tyc E(x, y)



[ψ]dsy for x ∈ Γc .

Γc

This is a coupled system for ψ ∈ C α (Γ ) on Γ and [ψ] ∈ C1α (Γc ) on Γc , which, in fact, is uniquely solvable for any given triple (ϕ, ϕ+ , ϕ− ) with the required properties. For the classical crack problem with Dirichlet conditions on Γ , e.g. as ψ + ∈ C α (Γ c ) and ψ − ∈ C α (Γ c ) are given with (ψ + − ψ − )|γ = 0; the desired fields u has to satisfy (2.5.8) and the boundary conditions γ0 u|Γ = ϕ on Γ

and Tc+ u|Γc = ψ + , Tc− u|Γc = ψ − on Γc

(2.5.17)

as well as the transmission conditions (2.5.10), (2.5.11). Again from the representation formula (2.5.15) we now obtain the coupled system    c  Ty E(x, y) [ϕ](y)dsy E(x, y)ψ(y)dsy + y∈Γ

y∈Γc



=

1 2 ϕ(x)

 Dc [ϕ](x) − =

1 2



E(x, y)[ψ]|Γc (y)dsy for x ∈ Γ ,

+ Kϕ(x) + y∈Γc

Txc E(x, y)ψ(y)dsy

(2.5.18)

y∈Γ

ψ (x) + ψ − (x) − K c | ([ψ]|Γc )(x)    − Txc Ty E(x.y) ϕ(y)dsy for x ∈ Γc +

y∈Γ

for the unknowns ψ ∈ C α (Γ ) and [ϕ] ∈ C0α (Γc ). As it turns out, this system always has a unique solution for any given triple (ϕ, ψ + , ψ − ) with the required properties. The desired displacement field in Ωc is in both cases given by (2.5.15).

3. Representation Formulae, Local Coordinates and Direct Boundary Integral Equations

In order to generalize the direct approach for the reduction of more general boundary value problems to boundary integral equations than those presented in Chapters 1 and 2 we consider here the 2m–th order positive elliptic systems with real C ∞ –coefficients. We begin by collecting all the necessary machinery. This includes the basic definitions and properties of classical function spaces and distributions, the Fourier transform and the definition of Hadamard’s finite part integrals which, in fact, represent the natural regularization of homogeneous distributions and of the hypersingular boundary integral operators. For the definition of boundary integral operators one needs the appropriate representation of the boundary manifold Γ involving local coordinates. Moreover, the calculus of vector fields on Γ requires some basic knowledge in classical differential geometry. For this purpose, a short excursion into differential geometry is included. Once the fundamental solution is available, the representation of solutions to the boundary value problems is based on general Green’s formulae which are formulated in terms of distributions and multilayer potentials on Γ . The latter leads us to the direct boundary integral equations of the first and second kind for interior and exterior boundary value problems as well as for transmission problems. As expected, the hypersingular integral operators are given by direct values in terms of Hadamard’s finite part integrals. The results obtained in this chapter will serve as examples of the class of pseudodifferential operators to be considered in Chapters 7–11.

3.1 Classical Function Spaces and Distributions For rigorous definitions of classical as well as generalized function spaces we first collect some standard results and notations. Multi–Index Notation Let IN0 be the set of all non–negative integers and let INn0 be the set of all ordered n–tuples α = (α1 , · · · , αn ) of non–negative integers αi ∈ IN0 . Such an n–tuple α is called a multi–index. For all α ∈ INn0 , we denote by |α| = α1 + α2 + · · · + αn the order of the multi-index α. If α, β ∈ INn0 , we define α+β = (α1 +β1 , · · · , αn +βn ). The notation α ≤ β means that αi ≤ βi for 1 ≤ i ≤ n. We set © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_3

96

3. Representation Formulae

α! = α1 ! · · · αn ! and

      β β! β1 βn = = ... . α (β − α)!α! α1 αn

We use the usual compact notation for the partial derivatives: If α = (α1 , · · · , αn ) ∈ INn0 , we denote by Dα u the partial derivatives Dα u =

∂ |α| u n . . . ∂xα n

α2 1 ∂xα 1 , ∂x2

of order |α|. In particular, if |α| = 0, then Dα u = u.

Functions with Compact Support Let u be a function defined on an open subset Ω ⊂ IRn . The support of u, denoted by supp u, is the closure in IRn of the set {x ∈ Ω : u(x) = 0}. In other words, the support of u is the smallest closed subset of Ω outside of which u vanishes. We say that u has a compact support in Ω if its support is a compact (i.e. closed and bounded) subset of Ω. By the notation K  Ω, we mean not only that K ⊂ Ω but also that K is a compact subset of Ω. Thus, if u has a compact support in Ω, we may write supp u  Ω. The Spaces C m (Ω) and C0∞ (Ω) We denote by C m (Ω), m ∈ IN0 , the space of all real– (or complex–)valued functions defined in an open subset Ω ⊂ IRn having continuous derivatives of order ≤ m. Thus, for m = 0, C 0 (Ω) is the space of all continuous functions in Ω which will be simply denoted by C(Ω). We set ) C m (Ω), C ∞ (Ω) = m∈IN0

the space of functions defined in Ω having derivatives of all orders, i.e., the space of functions which are infinitely differentiable. We define C0∞ (Ω) to be the space of all infinitely differentiable functions which, together with all of their derivatives, have compact support in Ω. We denote by C0m (Ω) the space of functions u ∈ C m (Ω) with supp u  Ω. The spaces C0m (Ω) as well as C0∞ (Ω) are linear function spaces. On the linear function space C0∞ (Ω) one can introduce the notion of convergence ϕj → ϕ in C0∞ (Ω) if to the sequence of functions {ϕj }∞ j=1 there exists a common compact subset K  Ω with supp ϕj ⊂ K for all j ∈ IN and Dα ϕj → Dα ϕ uniformly for every multi–index α. This notation of

3.1 Classical Function Spaces and Distributions

97

convergence defines a locally convex topology on C0∞ (Ω) (see Treves [419]). With this concept of convergence, C0∞ (Ω) is a locally convex topological vector space called D(Ω). Similarly, one can define C m (Ω) to be the space of functions in C m (Ω) which, together with their derivatives of order ≤ m, have continuous extensions to Ω = Ω ∪ ∂Ω. If Ω is bounded and m < ∞, then C m (Ω) is a Banach space with the norm  uC m (Ω) = sup |Dα u(x)|. |α|≤m x∈Ω

The Spaces C m,α (Ω) In Section 1.2, we have already introduced H¨older spaces, however, for completeness, we restate the definition here again. Let Ω be a subset of IRn and 0 < α ≤ 1. A function u defined on Ω is said to be H¨ older continuous with exponent α in Ω if 0 < α < 1 and if there exists a constant c ≥ 0 such that |u(x) − u(y)| ≤ c|x − y|α for all x, y, ∈ Ω. The quantity [u]α;Ω := sup x,y∈Ω

x=y

|u(x) − u(y)| |x − y|α

is called the H¨older modulus of u. For α = 1, u is called Lipschitz continuous older and [u]1;Ω is called the Lipschitz modulus. We say that u is locally H¨ or Lipschitz continuous with exponent α on Ω if it is H¨ older or Lipschitz continuous with exponent α on every compact subset of Ω, respectively. By C m,α (Ω), m ∈ IN0 , 0 < α ≤ 1, we denote the space of functions in C m (Ω) whose m–th order derivatives are locally H¨older or Lipschitz continuous with exponent α on the open subset Ω ⊂ IRn . We remark that H¨older continuity may be viewed as a fractional differentiability. Further, by C m,α (Ω) we denote the subspace of C m (Ω) consisting of functions which have m–th order H¨older or Lipschitz continuous derivatives of exponent α in Ω. If Ω is bounded, we define the H¨ older or Lipschitz norm by uC m,α (Ω) := uC m (Ω) +



sup

y∈Ω |β|=m x =y

|Dβ u(x) − Dβ u(y)| |x − y|α

(3.1.1)

The space C m,α (Ω) equipped with the norm || · ||C m,α (Ω) is a Banach space. Distributions Let v be a linear functional on D(Ω). Then we denote by v, ϕ the image of ϕ ∈ D(Ω).

98

3. Representation Formulae

Definition 3.1.1. A linear functional v on D(Ω) is called a distribution if, for every compact set K  Ω, there exist constants cK ≥ 0 and N ∈ IN0 such that   |v, ϕ| ≤ cK Dα ϕC 0 (Ω) := cK sup |Dα ϕ(x)| (3.1.2) |α|≤N

|α|≤N

α∈Ω

for all ϕ ∈ D(Ω) with supp ϕ ⊂ K. The vector space of distributions is called D (Ω). For Ω = IRn we set D := D(IRn ) and D := D (IRn ). For a distribution v ∈ D (Ω), (3.1.2) implies that v, ϕj  → v, ϕ for v ∈ D (Ω) if ϕj → ϕ in D(Ω). Correspondingly, we equip D (Ω) by the weak topology; i.e., vj  v in D (Ω) if and only if vj , ϕ → v, ϕ as j → ∞ for every ϕ ∈ D(Ω). For v, w ∈ D (Ω), we say that v = w on an open set Θ ⊂ Ω if and only if v, ϕ = w, ϕ for all ϕ ∈ C0∞ (Θ). The support of a distribution v ∈ D (Ω) is the complement of the largest open set on which v = 0. The function space C ∞ (Ω) equipped with the family of seminorms  • C m (K) for all compact subsets K  Ω is denoted by E(Ω). The space of distributions with compact supports in Ω is denoted by E  (Ω) and for Ω = IRn we set E := E(IRn ) and E  := E  (IRn ). As a consequence of this definition, one obtains the following characterization of E  (Ω). Proposition 3.1.1. A linear functional w on E is an element of E  (Ω) if and only if there exist a compact set K  Ω and constants c and N ∈ IN0 such that (3.1.3) |w, ϕ| ≤ cϕC N (K) for all ϕ ∈ E(Ω) . We remark that the standard operations, such as the multiplication by a function and differentiation can be extended to distributions in the following way. Let T be any linear continuous operator on D(Ω), i.e., T ϕj → T ϕ in D(Ω) if ϕj → ϕ in D(Ω). Then the operator T on D (Ω) is defined by T v, ϕ = v, T  ϕ for v ∈ D (Ω) and ϕ ∈ D(Ω)

(3.1.4)

where the dual operator T  is given by   (T ϕ)ψdx = ϕ(T  ψ)dx for all ϕ, ψ ∈ D(Ω) .

(3.1.5)

IRn

IRn

Therefore, the linear functional (T v) in (3.1.4) is continuous on D(Ω) provided T  is continuous on D(Ω). If f ∈ E(Ω), then for v ∈ D (Ω) we can define the product (f v) ∈ D (Ω) via the continuous linear functional given by (f v), ϕ := v, f ϕ .

(3.1.6)

For the differential operator Dα = T , which is continuous on D(Ω), (−1)|α| Dα = T  is also continuous on D(Ω), hence, for v ∈ D (Ω) we define Dα v by

3.1 Classical Function Spaces and Distributions

Dα v, ϕ := v, (−1)|α| Dα ϕ for all ϕ ∈ D(Ω) .

99

(3.1.7)

Next, we collect the basic definitions of the rapidly decreasing functions and corresponding tempered distributions IRn . Definition 3.1.2. A function ϕ ∈ C ∞ (IRn ) is called rapidly decreasing if (3.1.8) ϕα,β := sup |xα Dβ ϕ| < ∞ x∈IRn

for all pairs of multi–indices α, β. The vector space of the rapidly decreasing functions equipped with this family of seminorms is called Schwartz space S. A sequence of functions {ϕk }∞ k=1 ⊂ S converges to ϕ ∈ S if and only if ϕ − ϕj α,β → 0 as j → 0 for all pairs of multi–indices α, β. Definition 3.1.3. By S  ⊂ D we denote the space of all continuous linear functionals on S, i.e., to v ∈ S  there exist constants cv ≥ 0 and N ∈ IN0 such that  |v, ϕ| ≤ cv sup |xα Dβ ϕ| for all ϕ ∈ S . (3.1.9) |α|,|β|≤N

The distribution space S  is usually referred to as the space of tempered   distributions on IRn . A sequence {vj }∞ j=1 ⊂ S is said to converge to v ∈ S  in S if and only if vj , ϕ → v, ϕ for all ϕ ∈ S as j → ∞ .

(3.1.10)

Remark 3.1.1: For the above spaces we have the following inclusions: D ⊂ E  , S ⊂ S  , E ⊂ D , D ⊂ S ⊂ E and E  ⊂ S  ⊂ D .

(3.1.11)

As will be seen, the Fourier transform will play a crucial role in the analysis of integral– and pseudodifferential operators. It is defined as follows. Fourier Transform On the function space D of functions in C0∞ (IRn ), the Fourier transform is defined by  *(ξ) := (2π)−n/2 e−ix·ξ u(x)dx for u ∈ D . (3.1.12) Fx→ξ u(x) := u IRn

100

3. Representation Formulae

Clearly, F : D → E is a continuous linear mapping. By using Definition (3.1.4), we extend the Fourier transform to distributions v ∈ E  by Fv, ϕ := v, Fϕ for all ϕ ∈ D .

(3.1.13)

Then the linear map F : E  → D is also continuous. If u ∈ S then F is also well defined by (3.1.12) and the following theorem is valid. Theorem 3.1.2. The Fourier transform is a topological isomorphism from S onto itself and the following identity holds:  −n/2 u(x) = (2π) eix·ξ Fu(ξ)dξ for all ϕ ∈ S . (3.1.14) IRn

This is the inverse Fourier transform F ∗ v for v = Fu. In addition, we have for u, v ∈ S:    *(ξ) and Dξβ u *(ξ) = Fx→ξ (−ix)β u(x) (3.1.15) Fx→ξ Dxα u(x) = (iξ)α u u, Fv = Fu, v

and Fu, Fv = u, v .

(3.1.16)

(Formula (3.1.16) is called Parseval’s formula.) Similar to the extension of F from D to E  , we now extend F from S to  S by the formula Fu, ϕ := u, Fϕ for all ϕ ∈ S

(3.1.17)

which defines a continuous linear functional on S since F : S → S is an isomorphism. The inverse Fourier transform F ∗ v for v ∈ S  is then defined by (3.1.18) F ∗ v, ϕ = v, F ∗ ϕ for all ϕ ∈ S . With these definitions, Theorem 3.1.2 remains valid for S  . A useful characterization of E  in terms of the Fourier transformed distributions is given by the fundamental Paley–Wiener–Schwartz theorem. Theorem 3.1.3. (i) If u ∈ E  (IRn ) then its Fourier transform u *(ξ) = Fx→ξ u := (2π)−n/2



u(x)e−ix·ξ dx

IRn

is analytic in ξ ∈ Cn . Here, the integration is understood in the distributional sense. (ii) For u ∈ E  (IRn ) and supp u ⊂ {x ∈ IRn | |x| ≤ a} with a > 0, there exist constants c, N ≥ 0, such that |* u(ξ)| ≤ c(1 + |ξ|)N ea|Imξ| for ξ ∈ Cn .

3.2 Hadamard’s Finite Part Integrals

101

(iii) If u ∈ C0∞ (IRn ) and supp u ⊂ {x ∈ IRn | |x| ≤ a} then for every m ∈ IN0 exists a constant cm ≥ 0 such that |* u(ξ)| ≤ cm (1 + |ξ|)−m ea|Imξ| for all ξ ∈ Cn . (see e. g. Schwartz [382] or Friedlander [128, Theorem 10.2.1]).

3.2 Hadamard’s Finite Part Integrals We now consider a subclass of S  which contains distributions defined by functions having isolated singularities of finite order. In order to give the precise definition of such distributions, we introduce the concept of Hadamard’s finite part integrals. Since our boundary integral operators have kernels defined by functions with singularities of this type, we confine ourselves to the following classes of functions. Definition 3.2.1. A function hq (z) is a C ∞ (IRn \ {0}) pseudohomogeneous function of degree q ∈ IR if hq (tz) hq (z)

= =

tq hq (z) for every t > 0 and z = 0 fq (z) + log |z|pq (z)

if q ∈ IN0 ; if q ∈ IN0 ,

(3.2.1)

where pq (z) is a homogeneous polynomial in z of degree q and where the function fq (z) satisfies fq (tz) = tq fq (z) for every t > 0 and z = 0 . In short, we denote the class of pseudohomogeneous functions of degree q ∈ IR by Ψ hfq . We note that for q > −n, the integral  hq (z)u(z)dz

(3.2.2)

Ω

C0∞ (Ω)

is well defined as an improper integral. For q ≤ −n, however, for u ∈ hq (z) is non integrable except that u(y) and its derivatives up to the order κ := [−n − q] vanish at the origin1 . Hence, (3.2.2) defines a homogeneous distribution on C0∞ (Ω \ {0}). In order to extend this distribution to all of C0∞ (Ω), we use the Hadamard finite part concept which is the most natural extension. It is based on the idea of integration by parts. Consider first the family of regular integrals: 1

As usual, [p] := max{m ∈ Z | m ≤ p} with Z the integers, denotes the Gaussian bracket for p ∈ IR.

102

3. Representation Formulae

 Iεq (u)



:=

hq (z)u(z)dz +

hq (z)u(z)dz

(3.2.3)

Ω0 \{|z| 0 where u ∈ C0∞ (Ω) and Ω0 is any fixed star–shaped domain with respect to the origin, satisfying Ω0 ⊂ Ω. The first integrals on the right–hand side has C ∞ –integrands and exists. With the decomposition of the second integral we have    1 ∂αu  Iεq (u) = z α α (0) dz hq (z) u(z) − α! ∂x Ω0 \{|z| 1 which are defined by |x − p1 | = ε = |x − p2 |. If we use the Taylor expansion of the curve Γ about x, we have

108

3. Representation Formulae

ε2 + O(ε2+α ) and 2 ε2 p2 = x − εtx − κx nx + O(ε2+α ) 2 p1 = x + εtx − κx nx

where κx denotes the curvature of Γ at x. Hence, from (1.1.2) we find p2 d 1 tx · (x − y)ϕ(y) p2  E(x; y)ϕ(x)(y) = −  dsx 2π |x − y|2 p1 p1   1  2+α 2ε + O(ε =− ) ϕ(x) + O(ε1+α ) 2 2πε ϕ(x) =− + O(εα ) , επ which is (3.2.21). Alternatively, the same result can be achieved according to Definition 3.2.2 by chosing Ω0 = Γ ∩ {y | |y − x| < ε}. Then n = 1 , q = −2 and κ = 1; 1 d0 = − 2π is given by (3.2.7) with (3.2.8) and d1 = 0 is given by (3.2.9); and the regularized part in (3.2.6) satisfies  h−2 (x − y){. . . }dsy = O(εα ) . |x−y|≤ε

3.3 Local Coordinates In order to introduce the notion of a boundary surface Γ of a bounded domain Ω ⊂ IRn , we shall first define its regularity in the classes C k,κ , k ∈ IN0 , κ ∈ [0, 1] by following Nedelec [326] and Grisvard [156], see also Adams [1, p.67] and Wloka [439]. Given a point y = (y1 , · · · , yn ) ∈ IRn , we shall write y = (y  , yn ) ,

(3.3.1)

y  = (y1 , · · · , yn−1 ) ∈ IRn−1 .

(3.3.2)

where

Definition 3.3.1. A bounded domain Ω in IRn is said to be of class C k,κ (in short Ω ∈ C k,κ ) if the following properties are satisfied: i. There exists a finite number p of orthogonal linear transformations T(r) (i.e. n × n orthogonal matrices) and the same number of points x(r) ∈ Γ and functions a(r) (y  ), r = 1, . . . , p, defined on the closures of the (n − 1)dimensional ball, Q = {y  ∈ IRn−1 | |y  | < δ} where δ > 0 is a fixed constant. For each x ∈ Γ there is at least one r ∈ {1, . . . , p} such that x = x(r) + T(r) (y  , a(r) (y  )) .

(3.3.3)

3.3 Local Coordinates

109

ii. The functions a(r) ∈ C k,κ (Q). iii. There exists a positive number ε such that for each r ∈ {1, . . . , p} the open set B(r) := {x = x(r) + T(r) y | y = (y  , yn ), y  ∈ Q and |yn | < ε} is the union of the sets − U(r)

=

B(r) ∩ Ω = {x = x(r) + T(r) y | y = (y  , yn ), y  ∈ Q and a(r) (y  ) − ε < yn < a(r) (y  )} ,

+ U(r)

=

B(r) ∩ (IRn \Ω) = {x = x(r) + T(r) y | y = (y  , yn ), y  ∈ Q and a(r) (y  ) < yn < a(r) (y  ) + ε} ,

(3.3.4)

and Γ(r) = B(r) ∩ ∂Ω = {x = x(r) + T(r) (y  , a(r) (y  )) | y  ∈ Q} .

(3.3.5)

The boundary surface Γ = ∂Ω is said to be in the class C k,κ if Ω ∈ C k,κ ; and in short we also write Γ ∈ C k,κ . The collection of the above local parametric representations is called a finite atlas and each particular mapping (3.3.3) a chart of Γ . For the geometric interpretation see Figure 3.3.1. Remark 3.3.1: Note that we have a special open covering of Γ by the open sets B(r) , p + Γ ⊂ B(r) . r=1

Moreover, the mappings y = (y  , yn ) = Φ(r) (x) where x ∈ B(r) and       −1 −1 −1 y  := T(r) (x − x(r) ) , yn = T(r) (x − x(r) ) − a(r) (T(r) (x − x(r) )) n

are one–to–one mappings from B(r) onto Q×(−ε, ε) ⊂ IRn having the inverse mappings x = Ψ(r) (y) := x(r) + T(r) (y  , a(r) (y  ) + yn ) from Q × (−ε, ε) onto B(r) as considered by Grisvard in [156] (see also Wloka k [439]). Correspondingly, we write Γ ∈ C ∞ if Γ ∈ ∩∞ k=1 C . 0,1 In the special case, when Γ ∈ C , the boundary is called a Lipschitz boundary with a strong Lipschitz property and Ω is called a strong Lipschitz domain (see Adams [1, p.67]). That means in particular that the local parametric representation (3.3.3) is given by a Lipschitz continuous function a(r) . Such a boundary can contain conical points or edges (see e.g. in Kufner et al

110

3. Representation Formulae

[249]). Note that the domains shown in Figure 3.3.2 are not strong Lipschitz domains. It was shown by Gagliardo [132] and Chenais [66] that every strong Lipschitz domain has the uniform cone property and conversely, every domain having the uniform cone property, is a strong Lipschitz domain (Grisvard [156, Theorem 1.2.2.2]). A domain Ω ∈ C 1,κ with 0 < κ < 1 is a Lyapounov surface as introduced in Section 1.5. In fact, we have C 2,0 ⊂ C 1,1 ⊂ C 1,α ⊂ C 0,1 where 0 < α ≤ 1. yn

ε ε

Γ

+ = B(r) ∩ (Rn \ Ω) U(r)

x(r)

x = x(r) + T(r)(y  , a(r)(y  )) − U(r) = B(r) ∩ Ω

δ δ Ω y

Figure 3.3.1: The local coordinates of a C k,κ domain

Figure 3.3.2: Two non–Lipschitzian domains

If f ∈ C ∞ (IRn ) then, with our definitions, the restriction of f to the boundary Γ ∈ C ∞ will be denoted by f|Γ , and it defines by f (x(y  )) a C ∞ –

3.4 Short Excursion to Elementary Differential Geometry

111

function of y  in Q. The mapping γ0 : f → f|Γ defines the linear continuous trace operator from C ∞ (IRn ) into C ∞ (Γ ). These local coordinates are necessary for obtaining explicit representations of the corresponding boundary integral operators on Γ . For this purpose we are going to rewrite the representation formulae. According to (3.3.3), let Γ be given locally by the regular parametric representation Γ : x|Γ = y(σ  ) (3.3.6) in the vicinity of a fixed point x(r) ∈ Γ , where σ  = (σ1 , . . . , σn−1 ) are the parameters and x(r) = y(0). In the following, we require only that the tangent vectors ∂y  (σ ) for λ = 1, . . . , n − 1 are linearly independent ∂σλ and sufficiently smooth but not necessarily orthonormal. We note that in (3.3.3) a particular representation was given by y(σ  ) = x(r) + T(r) (σ  , a(r) (σ  )) . For x ∈ Γ but near to Γ , we introduce a spatial transformation x(σ  , σn ) := y(σ  ) + σn n(σ  ) ,

(3.3.7)

where n(σ  ) = n is the exterior normal to Γ at y(σ  ) and σn is an additional parameter. (The coordinates (σ1 , σn ) and (3.3.7) are often called “Hadamard coordinates”.) For Γ sufficiently smooth, it can easily be shown that (3.3.7) defines a diffeomorphism between x ∈ IRn and σ = (σ  , σn ) ∈ IRn in appropriate neighbourhoods of x = x(r) and σ = 0. However, note that for x(σ) ∈ C ,κ , the surface Γ must be given in C +1,κ . We shall come back to this point more specifically in Section 3.7.

3.4 Short Excursion to Elementary Differential Geometry For the local analysis of differential as well as boundary integral operators on the boundary manifold we need some corresponding calculus in the form of elementary differential geometry. For straightforward introductions to this area we refer to the presentations in Berger and Gostiaux [30], Bishop and Goldberg [32], Klingenberg [229], Millman and Parker [306] and Sokolnikoff [394]. We begin with the differential operator ∂x∂ q which can be transformed by n  ∂u ∂σ ∂u = (3.4.1) ∂xq ∂xq ∂σ =1

112

3. Representation Formulae

∂σ where the Jacobian ∂x is to be expressed in terms of the transformation q (3.3.7). To this end, let us define the coefficients of the associated Riemannian metric ∂x ∂x gik (σ) = gki := • for i, k, = 1, . . . , n . (3.4.2) ∂σi ∂σk

The diffeomorphism defined by (3.3.7) is said to be regular if the determinant does not vanish, g := det (gik ) = 0 .

(3.4.3)

For a regular diffeomorphism, the inverse g m of gik is well defined by n 

gjm g m =

m=1

and from

n n   ∂xk m ∂xk g = δj ; ∂σ ∂σj m m=1

(3.4.4)

k=1

n  ∂σ ∂xk = δj , ∂xk ∂σj

k=1

one gets

n  ∂σ ∂xk m = g . ∂xk ∂σ m m=1

(3.4.5)

n  ∂xq m ∂u ∂u = g . ∂xq ∂σm ∂σ

(3.4.6)

Hence, (3.4.1) becomes

,m=1

This equation can be rewritten as follows. Lemma 3.4.1. Let (3.3.7) be a regular, sufficiently smooth transformation. Then n n ∂u 1  ∂ √  ∂xq m  =√ g g u . (3.4.7) ∂xq g m=1 ∂σm ∂σ =1

Proof: In order to show (3.4.7) we need to introduce the Christoffel symbols Grpq of the diffeomorphism (3.3.7) defined by2  ∂ 2 xi ∂xi = Grpq for i, p, q = 1, . . . , n . ∂σq ∂σp ∂σr r=1 n

(3.4.8)

For the Christoffel symbols, the following relations are easily established (Sokolnikoff [394, p.80 ff.]): 2

In Millman and Parker [306, (32.6)] one finds the additional term which can be shown to be zero after some manipulations.

 i ∂xs ∂xt y st ∂σp ∂σq

3.4 Short Excursion to Elementary Differential Geometry

∂gk ∂σq n  ∂g pq

∂σq q=1   n  ∂ ∂xi pq g ∂σq ∂σp p,q=1 ∂g ∂σq

n 

= =

n 

Gm q gmk +

m=1 n 

Gm kq gm ,

m=1 n 

Gpq g q −

−

q,=1 n 

Gqq g p ,

−

=

2g

n 

(3.4.9) (3.4.10)

q,=1

∂xi r pq Grq g , ∂σ p p,q,r=1

=

113

Grrq .

(3.4.11)

(3.4.12)

r=1

The right–hand side of (3.4.6) can be written as ∂u ∂xq

=

  n 1  ∂ √ ∂xq m g g u √ g ∂σ ∂σm ,m=1   n ∂ √ ∂xq m 1  −√ u. g g g ∂σ ∂σm ,m=1

Then the second term on the right–hand side vanishes, since   n 1  ∂ √ ∂xq m g g √ g ∂σ ∂σm ,m=1   n 1 ∂g ∂xq m √ ∂ 2 xq m √ ∂xq ∂ m 1  = √ g + g g + g g √ g 2 g ∂σ ∂σm ∂σ ∂σm ∂σm ∂σ =

1 √ g −

,m=1 n  ,m=1 n 

m,,t=1

=

n n   1 ∂xq m ∂xq m Grr g + Grm g √ 2g 2 g r=1 ∂σm ∂σr r,m,=1

∂xq m t G g − ∂σm t

n  m,,t=1

∂xq  tm G g ∂σm t

0,

(3.4.13) 

which gives (3.4.7).

Our coordinates (3.3.7) are closely related to the surface Γ which corresponds to σn = 0, i.e. x = x(σ  , 0) = y(σ  ). In the following, we refer to these coordinates (3.3.7) as Hadamard’s tubular coordinates. The metric coefficients of the surface Γ are given by γνμ (σ  ) = γμν (σ  ) := As usual, we denote by

∂y ∂y · ∂σν ∂σμ

for ν, μ = 1, . . . , n − 1 .

(3.4.14)

114

3. Representation Formulae

γ = det (γνμ )

(3.4.15)

λκ

the corresponding determinant and by γ the coefficients of the inverse to γνμ . Then 1 (3.4.16) ds = γ 2 dσ  is the surface element of Γ in terms of the local coordinates. In order to express gik in terms of the surface geometry we will need the Weingarten equations (Klingenberg [229], Millman and Parker [306, p. 126]), n−1  ∂n ∂y =− Lνλ ∂σλ ∂σ ν ν=1

for λ = 1, . . . , n − 1

(3.4.17)

where the coefficients Lνλ of the second fundamental form of Γ describe the curvature of Γ and are given by Lμλ =

n−1 

Lλν γ νμ , Lλν =

ν=1

∂2y • n(σ  ) . ∂σλ ∂σν

(3.4.18)

In addition, we have 2H =

n−1 

Lλλ = −(∇ • n)

(3.4.19)

λ=1

with the mean curvature H of Γ and K = det

(Lμλ ) ,

n−1 

Lνλ Lλμ = 2HLνμ − Kγνμ

(3.4.20)

λ=1

where K is the Gaussian curvature of Γ . With (3.4.2) we obtain from (3.3.7)     ∂y ∂y ∂n ∂n gνμ = + σn + σn • ∂σν ∂σν ∂σμ ∂σμ and with (3.4.17) and (3.4.20), gνμ

=

γνμ − 2σn Lνμ + σn2

=

(1 −

σn2 K)γνμ

n−1 

Lνλ Lλμ

λ=1

(3.4.21)

− 2σn (1 − σn H)Lνμ

for ν, μ = 1, . . . , n − 1 . Furthermore, we see immediately ' ( n−1  ∂x ∂y  κ ∂y • n(σ ) = − σn Lν • n(σ  ) gνn = ∂σν ∂σν ∂σ κ κ=1

gnn

=

0

=

1.

for ν = 1, . . . , n − 1 ; and

(3.4.22)

3.4 Short Excursion to Elementary Differential Geometry

115

From (3.4.21) one obtains 'n−1 (  det gνμ γ μλ = (1 − 2σn H + σn2 K)2 μ=1

and, hence, g = γ(1 − 2σn H + σn2 K)2 .

(3.4.23)

Equations (3.4.21), (3.4.22) and (3.4.23) imply in particular that for σn = 0 one has gνμ (σ  , 0) = γνμ (σ  ) and g νμ (σ  , 0) = γ νμ (σ  ) .

(3.4.24)

Similar to (3.4.22), by computing the left–hand side in (3.4.8), we find that some of the Christoffel symbols in (3.4.8) take special values, namely: Grnn = 0 Gnλn = 0

for r = 1, . . . , n; for λ = 1, . . . , n − 1 .

(3.4.25)

We also find for our special coordinates (3.3.7) with (3.4.25) and (3.4.17), n−1  ν=1

Gνnλ

n−1  ∂x ∂y ∂2x ∂n = = =− Lνλ . ∂σν ∂σn ∂σλ ∂σλ ∂σ ν ν=1

As a consequence from (3.3.7) with (3.4.17) we have ' ( n−1 n−1 n−1    ∂y ∂y ν κ ∂y Gnλ − σn Lν Lνλ =− ∂σν ∂σκ ∂σν ν=1 κ=1 ν=1 from which we obtain the relations −Lνλ = Gνnλ − σn

n−1 

Gnλ Lν

(3.4.26)

=1

for ν, λ = 1, . . . , n − 1. Note that on Γ , for σn = 0 we have Gνnλ (σ  , 0) = −Lνλ (σ  ) . ∂σ of our special diffeomorphism (3.3.7), we find from For the Jacobian ∂x k (3.4.5) and (3.4.22) the equations n−1  ∂xk ∂σλ = g μλ + nk g nλ for k = 1, . . . , n , λ = 1, . . . , n − 1 . ∂xk ∂σ μ μ=1

Multiplying by nk we obtain

(3.4.27)

116

3. Representation Formulae

0=

n n   ∂σλ ∂xk ∂σλ = nk = 1 · g nλ . ∂xk ∂σn ∂xk

k=1

k=1

Hence, g nλ = 0 and

n−1  ∂xk ∂σλ = g μλ ∂xk ∂σ μ μ=1

(3.4.28)

∂xk = nk from (3.3.7), the for k = 1, . . . , n ; λ = 1, . . . , n − 1. Moreover, with ∂σ n relation n−1  ∂xk ∂σn = g μn + nk g nn = nk g nn ∂xk ∂σ μ μ=1

yields g nn = 1 and

∂σn = nk for k = 1, . . . , n . ∂xk

(3.4.29)

We also note that for any function u,  ∂u ∂xk  ∂u ∂u ∂u . = = nk = ∂σn ∂xk ∂σn ∂xk ∂n n

n

k=1

k=1

(3.4.30)

With (3.4.28), (3.3.7) and (3.4.30) one obtains from (3.4.7), in addition to (3.4.19), the equation 1 ∂ √ 1 ∂ √ ∇·n= √ g=√ g. g ∂σn g ∂n

(3.4.31)

Finally, by collecting (3.4.27), (3.4.30) we obtain from (3.4.7) ∂u ∂ )u , = (Dq + nq ∂xq ∂n where

n−1 

∂xq μλ ∂ g ∂σμ ∂σλ λ,μ=1 ' ( n−1 n−1   ∂ ∂yq κ ∂yq g μλ = − σn Lμ ∂σμ ∂σ ∂σ κ λ κ=1

Dq :=

(3.4.32)

(3.4.33)

λ,μ=1

is a tangential operator on the surface σn = const, parallel to Γ . We note that unter derivatives the tangential operator Dq can be written in terms of the G¨ (2.2.24) in the tubular coordinates, i.e.,  ∂ ∂ mjk ∂x , n(σ  ) := nk (σ  ) − nj (σ  ) ∂xj ∂xk as

3.4 Short Excursion to Elementary Differential Geometry

Dq = −

n 

 nq (σ  )mqk ∂x , n(σ  )

117

(3.4.34)

q=1

Now, from (3.4.32), (3.4.33) we have on Γ , i.e. for σn = 0, n−1  ∂yq ∂u ∂u ∂u |Γ , |Γ = γ μλ + nq ∂xq ∂σμ ∂σλ ∂n

(3.4.35)

μ,λ=1

or from (3.4.7),   n−1  1   ∂ ∂ √ ∂u √ ∂xq μλ ( gnq u) , =√ g g u + ∂xq g ∂σλ ∂σμ ∂n μ,λ=1

and with (3.4.23), ∂u ∂xq

=

n−1 1  ∂ √ ∂xq μλ  g g u √ g ∂σλ ∂σμ μ,λ=1 √ γ ∂u , + 2 √ (σn K − H)nq u + nq g ∂n

which becomes on Γ , i.e. for σn = 0,   n−1 ∂u ∂u 1  ∂ √ ∂yq μλ . |Γ = √ γ γ u − 2Hnq u + nq ∂xq γ ∂σλ ∂σμ ∂n

(3.4.36)

μ,λ=1

Correspondingly, the so–called surface gradient GradΓ u is defined by GradΓ u =

n−1  λ,μ=1

∂y μλ ∂u γ , ∂σμ ∂σλ

(3.4.37)

which yields from (3.4.32) and (3.4.33) with σn = 0 the relation ∇u|Γ = GradΓ u + n

∂u ∂n

on Γ .

(3.4.38)

Again, with the G¨ unter derivatives (2.2.25) we have the relation GradΓ u = −(n M|Γ ) u on Γ .

(3.4.39)

In order to rewrite a differential operator P u along Γ we further need higher order derivatives: α αq  n  , α  ∂   ∂ Dq (σ , σn ) + nq (σ ) u := u (3.4.40) D u = D + n(σ ) ∂n ∂n q=1

118

3. Representation Formulae

∂ ∂ where ∂n = ∂σ∂n due to (3.4.30). Note that only the operators np (σ  ) ∂n and  ∂ nq (σ ) ∂n commute, whereas the tangential differential operators Dp and Dq ∂ and the operators np (σ  ) ∂n mutually are noncommutative. However, due to the Meinardi–Codazzi compatibility conditions, the operators   ∂  ∂  ∂ ∂ and = Dq + n q = Dp + n p ∂xq ∂n ∂xp ∂n

do commute. Similar to the arrangement in (3.4.40) we now write |α|  k=0

∂ku Cα,k k := ∂n



∂ D+n ∂n

α u = Dxα u

(3.4.41)

with tangential differential operators 

Cα,k u =





cα,γ,k (σ , σn )

∂ ∂σ 

γ u,

(3.4.42)

Cα,k u = 0 for k ≤ −1 and for k > |α| .

(3.4.43)

|γ|≤|α|−k

for γ = (γ1 , . . . , γn−1 ) , k = 0, . . . , |α| and

For convenience in calculations we set cα,γ,k (σ  , σn ) = 0 for

any γj < 0 or for |γ| > |α| − k .

(3.4.44)

Then, in particular, cα,γ,k = 0 for k > |α| in accordance with (3.4.42). The tangential operators Cα,k can be introduced from (3.4.40) and (3.4.41) recursively as follows: Set C0,0 = 1 , hence C0,0 u = u .

(3.4.45)

Then we have for β = (α1 , . . . , α + 1, . . . , αn ) with |β| = |α| + 1 the recursion formulae



Cβ,k = D Cα,k + n

∂ Cα,k ∂σn

 + n Cα,k−1

(3.4.46)

and 

cβ,,k (σ , σn ) =

n−1  λ,μ=1



+ n

∂cα,,k ∂σn



∂cα,,k + cα,(j −δjλ ),k ∂σλ  + cα,,k−1 .

∂x μλ g ∂σμ

 (3.4.47)

3.4 Short Excursion to Elementary Differential Geometry

119

In particular, we have Cβ,|β| = cβ,0,|β| = nβ (σ  ) =

n ,

n (σ  )β .

(3.4.48)

=1

Finally, inserting (3.4.40)–(3.4.42) we can write along Γ α    ∂ α u aα (x)D u(x) = aα (x) D + n P u(x) = ∂n |α|≤2m



=

|α|≤2m

|α|

aα (x)

|α|≤2m 2m 

=:

Pk

k=0



Cα,k

k=0

∂ku ∂nk

(3.4.49)

∂ku , ∂nk

where the tangential operators Pk of orders 2m − k are given by  aα (x)Cα,k Pk = |α|≤2m



=



aα (x)cα,γ,k (σ  , σn )

|α|≤2m |γ|≤|α|−k



=





 

aα (x)cα,γ,k (σ , σn )

|γ|≤2m−k |α|≤2m

∂ ∂σ  ∂ ∂σ 

γ (3.4.50) γ

in view of (3.4.44). With (3.4.43) and (3.4.48) we find   P2m = aα (x)Cα,2m = aα (x(σ  , σn ))nα (σ  ) . |α|≤2m

(3.4.51)

|α|=2m

The operator 

P0 =





aα (y)cα,γ,0 (σ , σn )

|γ|≤2m |α|≤2m



∂ ∂σ 

γ (3.4.52)

is known as the Beltrami operator along Γ associated with P and Γ . 3.4.1 Second Order Differential Operators in Divergence Form As a special example let us consider a second order partial differential operator in divergence form,    n n  ∂ ∂u ∂u bk (x) + c(x)u(x) . (3.4.53) aik (x) + P u(x) = ∂xi ∂xk ∂xk i,k=1

k=1

120

3. Representation Formulae

Then (3.4.6) yields n  i,k=1

∂ ∂xi

 aik

∂u ∂xk



n 

=

i,k,,m=1

∂ ∂xi

 aik

∂xk m ∂u g ∂σm ∂σ

 .

Now applying formula (3.4.7) to the term {· · · }, we obtain ∂  ∂u  aik (x) ∂xi ∂xk i,k=1 n n    ∂ 1 √ g √ g ∂σp n 

=

p,=1

i,k,q,m=1

aik (x)

∂xi ∂xk m qp  ∂u  g g . ∂σq ∂σm ∂σ

By using the properties g nλ = 0 and g nn = 1 in (3.4.28) and (3.4.29), formula (3.4.54) takes the form n  i,k=1

=

∂ ∂xi

 aik (x)

n−1  κ,λ=1

+

∂u ∂xk



n−1 n 1 ∂ √   ∂xi ∂xk μλ νκ  ∂u  (3.4.54) g aik g g √ g ∂σκ ∂σν ∂σμ ∂σλ μ,ν=1 i,k=1

n−1  κ=1

n n−1  ∂u  1 ∂ √   ∂xi g aik nk g νκ √ g ∂σκ ∂σν ∂n ν=1 i,k=1

n−1 n n−1 ∂xk μλ  ∂u  1 ∂   √   +√ g aik ni g g ∂n ∂σμ ∂σλ μ=1 λ=1

i,k=1

n  ∂u  1 ∂  √  +√ . g aik ni nk g ∂n ∂n i,k=0

On the other hand, (3.4.6) gives  n n−1 ( ' n n n−1     ∂xk  ∂u ∂u ∂u μλ . bk = bk g + bk nk ∂xk ∂σ ∂σ ∂n μ λ μ=1 k=1

λ=1

k=1

(3.4.55)

k=1

Substituting (3.4.54) and (3.4.55) into (3.4.53) yields P u = P0 u + P1

∂u ∂2u + P2 2 . ∂n ∂n

(3.4.56)

For ease of reading, at the end of this section we shall show how these tangential operators Pk for k = 0, 1, 2, can be derived explicitly:

3.4 Short Excursion to Elementary Differential Geometry

P2

=

n 

aik (x(σ  , σn ))ni (σ  )nk (σ  ) .

121

(3.4.57)

i,k=1

P1

=

n−1 

n n−1  

aik

i,k=1 ν=1

λ=1

 ∂ ∂xi nk g νλ ∂σν ∂σλ

(3.4.58)

n−1 n    √γ  + 2 √ (σn K − H) − Gκκn aik ni nk g κ=1 i,k=1

−

n 

n−1 

aik

i,k=1 ,λ,ν=1

+

n n   ∂aik n i nk + ∂n

i,k=1

P0

=

n−1  κ,λ=1

+

−

i,k,=1

n−1 

 ∂aik ∂x ∂xi nk g ν + bk nk , ∂x ∂σ ∂σν ,ν=1 n

k=1

n−1 n 1 ∂ √   ∂xi ∂xk νκ μλ  ∂  (3.4.59) g aik g g √ g ∂σκ ∂σν ∂σμ ∂σλ μ,ν=1

n−1 

i,k=1

n 

 n−1   2√γ ∂xk aik ni √ (σn K − H) g ∂σ μ μ=1

λ=1

i,k=1

n−1 

n−1   ∂xk ∂yk ν ∂xk ν  μλ Lμ + Gμn g − g μν Gλμn ∂σν ∂σν ∂σν

ν=1

+

∂xi ν ∂yk λ L g ∂σ λ ∂σν

μ,λ=1

n−1 n  n  μ=1

k=1

i=1

∂aik ∂xk  μλ  ∂ n i + bk + c(x) g ∂n ∂σμ ∂σλ

+ c(x) . On Γ , where σn = 0, the operators further simplify due to (3.4.24) and (3.4.26) and the definition of H. Moreover, we have the relation n−1  ∂ μλ  g σ =0 = γ μν Lλν + γ λν Lμν = 2Lλμ . n ∂σn ν=1

The operators Pk on Γ now read explicitly as

(3.4.60)

122

3. Representation Formulae

P2

=

n 

aik (y)ni (y)nk (y) ,

(3.4.61)

i,k=1

P1

=

n−1  λ=1

−

n n−1  

aik (y)

i,k=1 ν=1

n 

n−1 

aik

i,k=1 ,ν=1

 ∂ ∂yi nk (y)γ νλ ∂σν ∂σλ

∂yi ν ∂yk L ∂σ ∂σν

n n  n−1     ∂aik ∂aik ∂y ∂yi n i nk + + nk γ ν ∂n ∂x ∂σ ∂σν ,ν=1

+

i,k=1 n 

=1

bk nk ,

(3.4.62)

k=1

P0

=

n−1  κ,λ=1

+

n−1 n 1 ∂ √   ∂yi ∂yk νκ μλ  ∂  γ aik γ γ √ γ ∂σκ ∂σν ∂σμ ∂σλ μ,ν=1 i,k=1

n−1  λ=1

+

n 

aik ni

i,k=1

n−1 n n   μ=1 k=1

i=1

n−1 

∂yk (−2Hγ μλ + Lμλ ) ∂σ μ μ=1

∂aik ∂yk  μλ  ∂ n i + bk γ ∂n ∂σμ ∂σλ

+ c(y) .

(3.4.63)

Remark 3.4.1: In case of the Laplacian we have aik = δik , bk = 0, c = 0. Then the expressions (3.4.60) and (3.4.62), (3.4.63) become P2 P1

= =

1, −

n−1 

γν L

,ν=1

P0

=

n−1  λ,κ=0

=

ν

=−

n−1 

L = −2H ,

=1

n−1  ∂  1 ∂ √  γ γμν γ νκ γ μλ √ γ ∂σκ ∂σλ ν=1

n−1 1  ∂ √ κλ ∂  γγ = ΔΓ , √ γ ∂σκ ∂σλ

(3.4.64)

λ,κ=1

which is the Laplace–Beltrami operator on Γ . (See e.g. in Leis [262, p.38]).

3.4 Short Excursion to Elementary Differential Geometry

123

Derivation of P1 and P0 : We begin with the derivation of the P1 by collecting the corresponding terms from (3.4.54) and (3.4.55) arriving at P1

=

n−1  λ=1

n n−1  

aik (x(σ  , σn ))

i,k=1 ν=1

 ∂ ∂xi nk (σ  )g νλ ∂σν ∂σλ

n  1  ∂ √  +√ g aik ni nk g ∂n i,k=1

+

n−1  κ=1

+

n 

n n−1  ∂ √   ∂xi g aik nk g νκ ∂σκ ∂σν ν=1 i,k=1

bk nκ .

k=1

The following formulae are needed when we apply the product rule to the second term in the above expression. ∂ √ g ∂n ∂ √ g ∂σκ

n−1  κ=1

= =

∂ 2 xi ∂σp ∂σq

=

∂nk ∂σκ

=

∂g νκ ∂g νn + ∂σκ ∂σn

=

As a consequence, we find

√ 2 γ(σn K − H) n √  r g Grκ n 

−

from (3.4.11) ,

r=1

Grpq

r=1 n−1 

−

from (3.4.23) ,

∂xi ∂σr

Lνκ

ν=1 n  q,=1

from (3.4.8) ,

∂yk ∂σν

Gνq g q −

from (3.4.17) , n  q,=1

Gqq g ν

from (3.4.10) .

124

3. Representation Formulae

P1

n−1 

=

⎛ ⎝

n n−1  

⎞ ∂xi ∂ nk g μλ ⎠ ∂σμ ∂σλ

aik

i,k=1 μ=1

λ=1

√ n n   γ ∂aik n i nk + 2 √ (σn K − H) aik ni nk + g ∂n i,k=1 i,k=1 ' ( n−1 n n n     ∂aik ∂x ∂xi r aik + Grκ + nk g νκ ∂n ∂σ ∂σ  κ ν ν,κ=1 r=1 i,k

n 

+

=1

n−1 

aik

∂xi nk Grκν g νκ ∂σr

aik

∂xi μ ∂yk κν L g ∂σν κ ∂σμ

aik

∂xi nk (Gνq g q + Gqq g ν ) ∂σν

i,k,r=1 κ,ν=1

−

n 

n−1 

i,k=1 μ,κ,ν=1 n 

− +

n−1 

i,k,,q=1 ν=1 n 

bk n k .

k=1

g

In addition, we see that with (3.4.28) and (3.4.29), i.e. g nκ = 0 and = 1 there holds

nn

n−1 n  ∂g np ∂g nn  ∂g nκ + =0= ∂σn ∂σκ ∂σp κ=1 p=0

and (3.4.10) gives 0

=

n−1 

−

Gnκν g νκ −

κ,ν=0

−

n 

−

Gnκn g nκ − Gnnn g nn

κ=1

Gnn g n −

=1

=

n−1 

n−1 

Gκκλ g λn −

Gnκν g νκ − Gnnn −

κ,ν=1

Gnκn g nn

κ=1

κ,λ=1

n−1 

n−1 

n 

Gnn g n −

n−1 

Gκκn .

κ=1

=1

Using (3.4.25), we obtain n−1  κ,ν=1

Gnκν g νκ = −

n−1 

Gκκn .

κ=1

A simple computation with the help of (3.4.65) and (3.4.25) yields

(3.4.65)

3.4 Short Excursion to Elementary Differential Geometry

P1

=

n−1 

n n−1  

 ∂ ∂xi nk g μλ ∂σμ ∂σλ

aik

i,k=1 μ=1

λ=1

125

√ n n   γ ∂aik + 2 √ (σn K − H) aik ni nk + ni nk g ∂σn i,k=1 i,k=1  n−1 n n−1   ∂xi  + aik nk Gκν g νκ − ni nk Gκκn ∂σ  ,ν,κ=1 κ=1 i,k=1

n−1 

n−1  ∂xi ∂xi νκ − nk Gκ g − nk Gnn ∂σ ∂σ ,ν,κ=1 =1 n−1 

∂xi ν ∂yk κ − Lκ g ∂σ ∂σν  ,κ,ν=1 n 

+

n−1 

i,k,=1

-

 ∂aik ∂x ∂xi nk g ν + bk n k . ∂x ∂σ ∂σν ν,=1 n

k=1

With (3.4.25) and rearranging terms, the result (3.4.58) follows. For P0 we collect from (3.4.54) the terms P0

=

n−1  κ,λ=1

n−1 n 1 ∂ √   ∂xi ∂xk μλ νκ  ∂  g aik g g √ g ∂σκ ∂σν ∂σμ ∂σλ μ,ν=1

1 +√ g +

i,k=1

n−1  λ=1

n n−1 ∂ √   ∂xk μλ  ∂ g aik ni g ∂σn ∂σμ ∂σλ μ=1

n−1 n n−1   λ=1

k=1 μ=1

i,k=1

bk

∂xk μλ  ∂ g + c(x) . ∂σμ ∂σλ

With the product rule and (3.4.23) we obtain

126

3. Representation Formulae

P0

=

n−1  κ,λ=1

+

n−1 n 1 ∂ √   ∂xi ∂xk μλ νκ  ∂  g aik g g √ g ∂σκ ∂σν ∂σμ ∂σλ μ,ν=1 i,k=1

n−1  λ=1

+

√ n n−1   2 γ ∂xk μλ aik ni g √ (σn K − H) g ∂σ μ μ=1 i,k=1

n n−1  

aik ni

 ∂n

k

∂σμ

i,k=1 μ=1

+

n n−1   ∂aik

∂n

i,k=1 μ=1

+

n n−1  

bk

k=1 μ=1

In the third term we insert (3.4.10), which yields ∂ μλ g ∂σn

ni

g μλ +

∂xk  ∂ μλ  g ∂σμ ∂σn

∂xk μλ g ∂σμ

∂xk λμ  ∂ g + c(x) . ∂σμ ∂σλ ∂nk ∂σμ

=

=

from (3.4.17), whereas for

− −

n  ,p=1 n 

 g μ

∂ μλ ∂σn g

we use

 ∂ gp g pλ ∂σn

(g μ Gλn + g λ Gμn )

=1

=

−

n−1 

(g μν Gλνn + g λν Gμνn )

ν=1

following from (3.4.25). On the surface Γ where σn = 0 we obtain again (3.4.60).

3.5 Distributional Derivatives and Abstract Green’s Second Formula Let Γ be sufficiently smooth, say, Γ ∈ C ∞ and let χΩ (x) be the characteristic function for Ω, namely  1 for x ∈ Ω and χΩ (x) := 0 otherwise. We begin with the distributional derivatives of χΩ , and claim ∂ χΩ (x) = −ni (x)δΓ ∂xi

(3.5.1)

3.5 Distributional Derivatives and Abstract Green’s Second Formula

127

where ni (x) is the i–th component of the exterior normal n(x) to Ω at x ∈ Γ = ∂Ω extended to a C0∞ (IRn )–function, δΓ is the Dirac distribution of Γ defined by   δΓ , ϕIRn =

δΓ (x)ϕ(x)dx := IRn

ϕ(x)dsx

(3.5.2)

Γ

for every ϕ ∈ C0∞ (IRn ). Equation (3.5.1) follows immediately from the definition of the derivative of distributions and the Gaussian divergence theorem as follows:   . ∂  / ∂ϕ ∂ϕ χΩ , ϕ n = − χΩ dx = − dx ∂xi ∂xi ∂xi IR IRn Ω  = − ϕ(x)ni (x)dsx = −δΓ , ni ϕIRn = −ni δΓ , ϕIRn Γ

from (3.1.6). (1) Now we define the distribution δΓ (x) :=

∂ ∂n δΓ

(1)

δΓ , ϕIRn = (n · ∇)δΓ , ϕIRn =

given by

. ∂ / δΓ , ϕ n ∂n IR

which, by the definition of derivatives of distributions becomes / . ∂ (1) δΓ , ϕ n = δΓ , ϕIRn = −δΓ , ∇ · (nϕ)IRn for all ϕ ∈ C0∞ (IRn ) . ∂n IR (3.5.3) Consequently, we have    . ∂ ∂ / (1) δΓ , ϕIRn = − + (∇ · n) ϕdsx = − δΓ , ϕ (3.5.4) ∂n ∂n IRn Γ

which also defines the transpose of the operator

∂ ∂n ,

  ∂   ∂ ∂  +∇·n ϕ=− − 2H ϕ on Γ , ϕ=− ∂n ∂n ∂n

(3.5.5)

where H = − 12 ∇ · n is the mean curvature of Γ (see (3.4.19)). With these operators, the following relations of corresponding distributions can be derived: ∇χΩ (x)

=

−n(x)δΓ (x) ,

(3.5.6)



ΔχΩ

=

∂ δΓ , ∂n

Δ(ϕχΩ )

=

(Δϕ)χΩ − 2

(3.5.7)  ∂   ∂ϕ δΓ + ϕ δΓ ∂n ∂n

(3.5.8)

128

3. Representation Formulae

for any ϕ ∈ C0∞ . Here, the right–hand side in (3.5.7) is the composition of ∂  the differential operator ∂n given in (3.5.5) with the distribution δΓ , and it defines again a distribution. Correspondingly, (3.5.8) yields the Green identity in the form: .  ∂   / . ∂ϕ / δΓ , ψ n δΓ , ψ n − 2 ϕχΩ , ΔψIRn − (Δϕ)χΩ , ψIRn = ϕ ∂n ∂n IR IR . ∂ψ / . ∂ϕ / ,ϕ n − δΓ , ψ n , = δΓ ∂n ∂n IR IR or, in the traditional form,    ∂ψ ∂ϕ  ϕ −ψ dsx . (ϕΔψ − ψΔϕ)dx = ∂n ∂n Ω

(3.5.9)

Γ

In order to formulate the Green identities for higher order partial differential operators, we also need a relation between the distribution’s derivatives (1) ∂ ∂xi δΓ and δΓ which is given by ∂ (1) δΓ = ni (x)δΓ . ∂xi

(3.5.10)

The derivation of this relation is not straight forward and needs the concept of local coordinates near Γ , corresponding formulae from differential geometry which are given by (3.4.29) and (3.4.30), and also the Stokes theorem on Γ . For u, a sufficiently smooth p–vector valued function, e.g. u ∈ C ∞ (IRn ) and for a general 2m–th order differential operator P , we have  aα (x)Dα (uχΩ ) P (uχΩ ) = |α|≤2m

=



|α|≤2m

=



aα (x)

0≤β≤α



P uχΩ +

aα (x)

1≤|α|≤2m

where

α! Dα−β uDβ χΩ β!(α − β)!  0 0 is an appropriately chosen constant depending on x and Ω. Then E(x, y) = χE + (1 − χ)E and χE is a well defined distribution whereas φ(y) := (1 − χ)E ∈ C ∞ (IRn ) can be used in formula (3.5.19). This yields    (3.6.6) (1 − χ)E  (P u)dy − uP  (1 − χ)E dy Ω

+

Ω

     ∂  ν−k  ν ∂  k () Qν E(x, y) u dsy = 0 k ∂ny ∂n

ν  2m−1   ν=0 k=0

Γ

since χ(y) ≡ 0 along Γ . On the other hand, by definition of distributional derivatives and by χ ∈ C0∞ (Ω), we have χE, P u = P  (χE), u , 

i.e.





uP  (χE)dy = 0 .

(χE) P udy − Ω

(3.6.7)

Ω

Since χ ≡ 1 for |x − y| ≤ δ0 we find with Py E(x, y) = 0 for y = x,      uPy (χE)dy = uPy E(x, y)dy + uP  (χE)dy |x−y|≤δ0

Ω

=



P  E, uIRn +

δ0 ≤|x−y|∧y∈Ω

uP  (χE)dy ,

δ0 ≤|x−y|∧y∈Ω

formula (3.6.7) becomes  (χE) P udy − Ω



uP  (χE)dy = u(x) .

(3.6.8)

δ0 ≤|x−y|∧y∈Ω

Now, formula (3.6.5) follows immediately by adding (3.6.6) and (3.6.8) since (1 − χ) ≡ 0 for |x − y| ≤ δ0 and Py E = 0 for x = y. ∂ ν In order to use the Cauchy data ( ∂n ) u|Γ in the representation formula  ∂ λ (3.6.5), we need to replace ( ∂n ) by using (3.5.5) which yields  u(x) = Ω

E(x, y) f (y)dy +

2m−1   λ=0 Γ

{Tλy E(x, y)}

   ∂ λ u(y) dsy ∂n (3.6.9)

3.6 The Green Representation Formula

133

where the boundary operators Tλy are defined by the equations 2m−1 

(Tλ φ)

λ=0 ν ν−k 2m−1  



∂ ∂n

λ u |Γ =

ν k

(−1)

ν=0 k=0 =0

(3.6.10)

ν−k  ∂ ν−k−  ∂  k ()   ∂  2H Qν φ u |Γ .  ∂n ∂n ∂n

This formula will be used for defining general boundary potentials and Calder´ on projectors as introduced in Chapters 1 and 2 for special differential operators. Now, let x ∈ Ω c . Then (3.6.3) yields Py E(x, y) = 0 for y ∈ Ω and from (3.5.19), in exactly the same manner as before, one obtains the equation  0 = E(x, y) P u(y)dy Ω

+

ν  2m−1  

ν k

ν=0 k=0

 =

     ∂  ν−k  ∂  k () Qνy E(x, y) u dsy ∂ny ∂n Γ

E(x, y) P u(y)dy

(3.6.11)

Ω

+

2m−1  

(Tλy E(x, y))



λ=0 Γ

 ∂ λ  u dsy for every x ∈ Ω c ∂n

and for any u ∈ C ∞ (Ω). For exterior problems one can proceed correspondingly. For u ∈ C ∞ (IRn ), one has with (3.5.17) the equation P (uχΩ c ) = P (u(1 − χΩ )) = P u − P (uχΩ ) = P uχΩ c +

2m−1 

(3.6.12)

ν=0

Hence, if R > 0 is chosen large enough such that  BR := {y ∈ IRn |y| < R} ⊃ Ω c

then (3.5.19) for Ω ∩ BR gives

(ν)

(Qν u)δΓ .

134

3. Representation Formulae







uP φ dy − c

Ω ∩BR

=

φP udy c

2m−1   ν=0 Γ

Ω ∩BR

2m−1  ∂  ν    ∂  ν (φ Qν u)dsy − (φ Qν )dsy ∂n ∂n ν=0 ∂BR

∞

for every test function φ ∈ C (IR ) and every R large enough. If φ ∈ C0∞ (IRn ) has compact support then the last integral over ∂BR vanishes for supp φ ⊂ BR . In order to obtain the Green representation formula in Ω c , we again use E(x, y) instead of φ(y) and in exactly the same manner as before we now find  E(x, y) P u(y)dy u(x) = n

c

Ω ∩BR

−

2m−1  

{Tλy E(x, y)}

λ=0 Γ

+

2m−1  

 ∂ λ  u(y) dsy ∂n

{Tλy E(x, y)}

λ=0 Γ R

(3.6.13)

 ∂ λ  u(y) dsy for x ∈ Ω c ∩ BR ∂n

c

and for every u ∈ C ∞ (Ω ) and every R large enough, ΓR = ∂BR . If the right–hand side f of the differential equation P u = f in Ω c

(3.6.14)

has compact support then the limit of the volume integral in (3.6.13) exists for R → ∞ and, in this case, for any solution of (3.6.14) the existence of the limit 2m−1  ∂ λ    M (x; u) := lim {Tλy E(x, y)} u(y) dsy (3.6.15) R→∞ ∂n λ=0 Γ R

follows from (3.6.13). Since all the operators in (3.6.13) are linear, for any fixed x ∈ Ω c , the limit in (3.6.15) defines a linear functional of u. Its value depends on the behaviour of u at ∞, i.e., on the radiation conditions imposed. For instance, from the exterior Dirichlet and Neumann problems for the Helmholtz equation subject to the Sommerfeld condition we have M (x; u) = ω(u) as was explained in Chapter 2.2, whereas for exterior problems of elasticity we have 3(n−1)  M (x; u) = ωj (u)mj (x) j=1

3.7 Green’s Representation Formulae in Local Coordinates

135

for n = 2 and 3 dimensions due to (2.2.11). An immediate consequence of (3.6.4) is Px M (x; u) = 0 for all x ∈ IRn

(3.6.16)

and for every solution of (3.6.14) with f having compact support. Collecting the above arguments, we find in this case the representation formula for exterior problems,  E(x, y) f (y)dy + M (x; u) (3.6.17) u(x) = Ωc

−

2m−1  

{Tλy E(x, y)}

λ=0 Γ

  ∂ λ u(y) dsy for x ∈ Ω c . ∂n

Finally, we find from (3.6.4) in the same manner the identity  0 = E(x, y) f (y)dy + M (x; u) Ωc

−

2m−1   λ=0 Γ

{Tλy E(x, y)}

 ∂ λ  u(y) dsy for x ∈ Ω . ∂n

(3.6.18)

3.7 Green’s Representation Formulae in Local Coordinates The differential operators Tλ in the representation formulae (3.6.9), (3.6.13) and (3.6.17) and in the identities (3.6.11) and (3.6.18) can be rewritten in terms of local coordinates in the neighbourhood of Γ by using the parametric representation of Γ . This will allow us to obtain appropriate properties of the corresponding Calder´ on projections as in the special cases considered in Chapters 1 and 2. In order to simplify the Green representation formulae introduced in Section 3.6 we introduce tensor distributions subordinate to the local coordinates σ = (σ  , σn ) or τ = (τ  , τn ) of the previous sections. For simplicity, we present here the case of Hadamard’s tubular local coordinates (σ  , σn ). Let Ω1 ⊂ IRn−1 and Ω2 ⊂ IR be open subsets. If ϕ ∈ C0∞ (Ω1 ) and ψ ∈ C0∞ (Ω2 ), the tensor product ϕ ⊗ ψ of ϕ and ψ is defined by (ϕ ⊗ ψ)(σ  , σn ) := ϕ(σ  )ψ(σn )

(3.7.1)

which is a C0∞ (Ω1 × Ω2 )–function. The corresponding space C0∞ (Ω1 ) ⊗ C0∞ (Ω2 ) is a sequentially dense linear subspace of C0∞ (Ω1 × Ω2 ), that is, for every Φ ∈ C0∞ (Ω1 × Ω2 ), there exists a sequence {Φj } in C0∞ (Ω1 ) ⊗ C0∞ (Ω2 )

136

3. Representation Formulae

such that Φj = ϕj (σ  ) ⊗ ψj (σn ) → Φ in C0∞ (Ω1 × Ω2 ) (Friedlander [128, Section 4.3] and Schwartz [382]). The tensor product can be extended to distributions; and the resulting distributional tensor product then satisfies u ⊗ v , ϕ ⊗ ψIRn = u, ϕIRn−1 v, ψIR

(3.7.2)

where u and v are distributions on Ω1 and Ω2 , respectively. By density arguments, the equation (3.7.1) extends to 0 1 (u ⊗ v), ΦIRn = 0u, v, Φ(σ  , •)IR IRn−1 1 (3.7.3) = v, u, Φ(•, σn )IRn−1 IR for all Φ ∈ C0∞ (Ω1 × Ω2 ). The basic properties of the tensor product are supp(u ⊗ v)

=

supp(u) × supp(v)

(3.7.4)

Dσα Dσβn (u ⊗ v)

=

(Dα u) ⊗ (Dβ v) .

(3.7.5)

and

In order to make use of (3.7.5) we first use the separation of the differential operator P in the neighbourhood of Γ in the form (3.4.49). We begin with the following jump formula: Lemma 3.7.1. For k ∈ IN and for u and Γ sufficiently smooth one has the jump formula  k  k−1  ∂u  ∂ u ∂k  ⊗ δ (k−−1) . χ (uχ ) = − (3.7.6) Ω Ω Γ ∂nk ∂nk ∂n Γ =0

Proof: The proof is done by induction with respect to k. For k = 1 we have from (3.5.6) ∂ (uχΩ ) ∂n

= = =

∂u χΩ + un(x) · ∇χΩ ∂n ∂u χΩ − uδΓ (x) ∂n ∂u χΩ − (u|Γ ⊗ δΓ ) . ∂n

Now, if (3.7.6) holds for k, we find  ∂  ∂k (uχ ) Ω ∂n ∂nk

= =

k−1    ∂u  ∂  ∂ku  ⊗ δ k−−1 χ − Ω Γ ∂n ∂nk ∂n Γ =0  ∂ k+1 u    ∂ku   χ − ⊗ δ  Ω Γ ∂nk+1 ∂nk Γ k−1    ∂  ∂ u  (k−−1) − ⊗ δ Γ ∂σn ∂σn Γ =0

3.7 Green’s Representation Formulae in Local Coordinates

137

and, with (3.7.5), the desired formula (3.7.6) for k + 1:  k+1   k    u ∂ k+1 ∂ ∂ u  k− (uχΩ ) = ⊗ δΓ χΩ − . ∂nk+1 ∂nk+1 ∂n Γ =0

 In terms of tensor products, Formula (3.5.8) can now be rewritten in the form of the jump relation      ∂   ∂u  −2 δΓ Δ(uχΩ ) = (Δu)χΩ + u|Γ ⊗  ⊗ δΓ . ∂n ∂n Γ Now we consider the jump relation for any differential operator P as in (3.4.49), namely   k 2m 2m  ∂ u ∂ku  ∂ P u(x(σ)) = Pk k = Pk σ;  ∂n ∂σ ∂σnk k=0

k=0

in the neighbourhood of Γ where Pk is the differential operator (3.4.50) of order 2m−k acting along the parallel surfaces σn =const. By virtue of Lemma 3.7.1 we have 2m   ∂  ∂k P (uχΩ ) = Pk σ;  (uχΩ ) ∂σ ∂σnk k=0

=

P0 (uχΩ ) +

2m  k=1

=

2m  k=0

k−1     ∂  u  ∂  ∂ k u  (k−−1) Pk σ;  − ⊗ δ χ  Ω Γ ∂σ ∂nk ∂n Γ =0

2m k−1    ∂ u ∂  ∂  u   (k−−1) Pk k χ Ω − . Pk σ;   ⊗ δΓ ∂n ∂σ ∂n Γ k

k=1 =0

By interchanging the order of summation between k and  and afterwards setting p = k −  − 1 we arrive at   2m−1  2m−−1   ∂  u  (p) Pp++1   ⊗ δΓ (3.7.7) P (uχΩ ) = (P u)χΩ − ∂n Γ p=0 =0

similar to (3.5.17), the equation (3.7.7) is again the second Green formula in distributional form, which yields    uP ϕdx − ϕP udx Ω

Ω

=−

2m−1  2m−−1   =0

p=0

Γ

∂  u  ∂  p ϕds Pp++1  ∂n ∂n



(3.7.8)

138

3. Representation Formulae

¯ This formula evidently is much simpler than (3.5.19). for any ϕ ∈ C ∞ (Ω). With (3.5.5) and (3.4.31) one also obtains   ∂  p 1 ∂ √ p ϕ= −√ g ϕ. ∂n γ ∂n

(3.7.9)

In particular, the tangential operators Pk appear in the same way as in the differential equation (3.4.49) operating only on the Cauchy data of u. As in Section 3.6, one may replace ϕ in (3.7.8) by the fundamental solution E(x, y) of the differential operator P  , satisfying equations (3.6.3) and (3.6.4). This yields the representation formulae  u(x) = E(x, y) f (y)dy (3.7.10) Ω

−

2m−1  2m−−1   =0

p=0

and



 

Pp++1

∂u  (y) dsy for x ∈ Ω ∂n

E(x, y) f (y)dy

Ω

2m−1  2m−−1   =0

∂  p E(x, y) ∂ny

Γ

0=

−



p=0



∂  p E(x, y) ∂ny

(3.7.11)  

Pp++1

∂u  (y) dsy for x ∈ Ω c . ∂n

Γ

For the representation formula of the solution of P u = f in Ω c , we again require the existence of the limit M (x; u) := − lim

R→∞

(3.7.12)

2m−1  2m−−1   =0

p=0

 ∂ ∂ny

 p

E(x, y)

 

Pp++1

∂u  (y) dsy . ∂n

Γ

If f ∈ C0∞ (IRn ), then the existence of M (x; u) follows from the representation formula (3.7.10) with Ω replaced by Ω c ∩ BR as in Section 4.2; and (3.6.16) remains the same for the present case. The exterior representation formulae (3.6.17) and (3.6.18) now read as  u(x) = E(x, y) f (y)dy + M (x; u) (3.7.13) Ωc

+

2m−1  2m−−1   =0

for x ∈ Ω and c

p=0

Γ

 ∂  p   ∂u  Pp++1  (y) dsy E(x, y) ∂ny ∂n

3.8 Multilayer Potentials

 0=

E(x, y) f (y)dy + M (x; u)

139

(3.7.14)

Ωc

+

2m−1  2m−−1   =0

p=0

 ∂  p   ∂u  Pp++1  (y) dsy E(x, y) ∂ny ∂n

Γ

for x ∈ Ω. We note that the representation formulae hold equally well for systems with p equations and p unknowns, such as in elasticity where p = n. We remark that similar formulae can be obtained when using the local coordinates τ = (τ  , τn ) as in the Appendix.

3.8 Multilayer Potentials In this section we follow closely Costabel et al [90]. From the representation formulae such as (3.6.9), (3.7.10) and (3.7.13), the boundary integrals all define boundary potentials. For simplicity, we assume Γ ∈ C ∞ and consider first (3.7.10) by defining the corresponding multiple layer potentials:   (Kp φ)(x) :=

∂  ∂ny



p E(x, y)

φ(y)dsy for x ∈ Ω or x ∈ Ω c (3.8.1)

Γ

for the density φ ∈ C ∞ (Γ ). Using the tensor product of distributions, we can write (3.8.1) as    ∂ p  E(x, y) φ ⊗ δΓ dy Kp φ(x) := ∂n (3.8.2) IRn   (p) = E(x, •) , φ ⊗ δΓ Rn Γ

for x ∈ Γ since then E(x, y) is a smooth function of y ∈ Γ and x is in its vicinity. As we will see in Chapter 9, this defines a pseudodifferential operator whose mapping properties in Sobolev spaces then follow from corresponding general results. We shall come back to this point later on. Clearly, the boundary potentials (3.8.1) are in C ∞ (IRn \ Γ ) for any given φ ∈ C ∞ (Γ ). Moreover, for x ∈ Γ we have   ∂  p  P Kp φ(x) = Px E  (x, y) φ(y)dsy = 0 (3.8.3) ∂ny Γ

since here x = y for y ∈ Γ and; therefore, integration and differentiation can be interchanged. Moreover,

140

3. Representation Formulae

 P

E(x, y) f (y)dy = f (x) for all x ∈ Ω .

(3.8.4)

Ω

In terms of the above potentials, the representation formula (3.7.10) now reads for u ∈ C ∞ (Ω) and x ∈ Ω: 



E(x, y) f (y)dy −

u(x) =

2m−1  2m−−1  =0

Ω

p=0

 ∂u  Kp Pp++1  (x) . ∂n

(3.8.5)

In addition, as we already have seen in Chapters 1 and 2, these potentials admit boundary traces as x approaches the boundary Γ from Ω or Ω c , respectively, which are not equal in general. More specifically, the following lemma is needed and will be proved in Chapter 9, Theorem 9.5.8. Lemma 3.8.1. Let E be the fundamental solution of P with C ∞ –coefficients defined in Lemma 3.6.1. Let Γ ∈ C ∞ and ϕ ∈ C ∞ (Γ ). Then the multilayer potential Kp ϕ(x) can be extended from x ∈ Ω up to the boundary x ∈ Ω in C ∞ (Ω) defining a continuous mapping Kp : C ∞ (Γ ) → C ∞ (Ω) . Similarly, Kp ϕ(x) can be extended from x ∈ Ω c to Ω c defining a continuous mapping Kp : C ∞ (Γ ) → C ∞ (Ω c ) . By using the traces from Ω of both sides in equation (3.8.5) we obtain the relation  γu(x) =γ E(x, y)f (y)dy Ω

−

2m−1  2m−−1  =0

 γKp

p=0

∂u  Pp++1  (x) for x ∈ Γ , ∂n

(3.8.6)

where the trace operator γu is defined by 

∂ 2m−1 u(z)  ∂u(z) ,..., Ω z→x∈Γ ∂n ∂n2m−1 (3.8.7) for x ∈ Γ and u ∈ C 2m (Ω). Of course, γKp needs to be analyzed carefully since it contains the jump relations (see e.g. (1.2.3) – (1.2.6)). The second term on the right–hand side of (3.8.6) defines a boundary integral operator γu(x) = (γ0 , . . . , γ2m−1 )u(x) :=

QΩ Pψ(x) :=

lim

2m−1  2m−1−  =0

p=0

u(z),

γKp (Pp++1 ψ )(x)

(3.8.8)

3.8 Multilayer Potentials

141

where both, P and QΩ themselves may be considered as matrix–valued operators with ⎞ ⎛ P1 P2 P3 . . . P2m ⎜ P2 P3 0 ⎟ ⎟ ⎜ ⎜ P3 0 ⎟ (3.8.9) P=⎜ ⎟ ⎜ .. .. ⎟ ⎠ ⎝ . . 0 ... P2m 0 and ⎞ γ0 ⎟ ⎜ .. ⎟ ⎠ = ⎝ . ⎠ (K0 , · · · K2m−1 ) . γ2m−1 . . . γ2m−1 K2m−1 (3.8.10) Again, the boundary integral operator ⎛

γ0 K0 ⎜ .. QΩ = ⎝ . γ2m−1 K0

... .. .

γ0 K2m−1

⎞

⎛

CΩ ψ := −QΩ Pψ

(3.8.11)

defines the Calder´ on projector associated with the partition of the differential operator P in (3.4.49) for the domain Ω (see Calder´ on [59] and Seeley [384]). For the Laplacian, it was given explicitly in (1.2.20). If ψ is specifically chosen to be ψ = γu as the trace of a solution to the homogeneous partial differential equation  aα (x)Dα u = 0 (3.8.12) Pu = |α|≤2m

in Ω, then (3.8.6) yields with (3.8.11) the identity γu = CΩ γu .

(3.8.13)

Consequently, because of (3.8.6), u=−

2m−1  2m−−1  =0

Kp (Pp++1 ψ )

p=0

is a special solution of (3.8.12), hence we find 2 CΩ ψ = CΩ γu = γu = CΩ ψ

(3.8.14)

for every ψ ∈ (C ∞ (Ω))2m . Hence, CΩ is a projection of general boundary functions ψ|Γ onto the Cauchy data γu of solutions to the homogeneous differential equation (3.8.12). On the other hand, the relation (3.8.13) gives a necessary compatibility condition for the Cauchy data of any solution u of the differential equation (3.8.12). For solutions of the inhomogeneous partial differential equation

142

3. Representation Formulae

Pu = f

(3.8.15)

in Ω we have the corresponding necessary compatibility condition  γu = γ E(x, y) f (x)dy + CΩ γu .

(3.8.16)

Ω

In the same manner, for the exterior domain we may consider the corresponding Calder´ on projector CΩ c ψ := QΩ c Pψ :=

2m−1  2m−1−  =0

γc Kp (Pp++1 ψ )

(3.8.17)

p=0

where γc now denotes the trace operator with respect to Ω c , i.e.,  (γc0 , . . . , γc,2m−1 )u(x) = γc u(x) :=

lim

Ω c z→x∈Γ

u(z),

∂u(z) ∂ 2m−1 u  , . . . , 2m−1 (z) ∂n ∂n

on Γ . Here, QΩ c is defined by (3.8.10) with γ replaced by γc . For solutions satisfying the radiation condition (3.7.12) we find the necessary compatibility condition for its trace:  γc u = γc E(x, y) f (y)dy + CΩ c γc u + CΩ c γc M . (3.8.18) Ωc

The following theorem insures that CΩ c is also a projection. Theorem 3.8.2. (Costabel et al [90, Lemma1.1]) For the multiple layer potential operators Kj and Kjc as well as for QΩ and QΩ c (cf. (3.8.10)) there holds QΩ c − QΩ = P −1 .

(3.8.19)

Equivalently, the Calder´ on projectors satisfy CΩ + CΩ c = I .

(3.8.20)

Proof: As was shown in (3.4.51), P2m = |α|=2m aα nα is a matrix–valued function and the assumption of uniform strong ellipticity (3.6.2) for the operator P  assures |Θ(y)P2m (y)| ≥ γ0 |n|2m = γ0 > 0 , −1 hence, the existence of the inverse matrix–valued function P2m exists along Γ . Therefore, the existence of the inverse of P in (3.8.9) follows immediately −1 from that of P2m since P has triangular form. Moreover, P −1 is a lower

3.8 Multilayer Potentials

triangular matrix of similar type as P consisting operators P (j) of orders 2m − j; ⎛ 0 ... 0 0 ⎜ ⎜ ··· ⎜ P −1 = ⎜ 0 ⎜ ⎝ 0 P (3) −1 (3) P2m . . . P P (2)

143

of tangential differential −1 ⎞ P2m .. ⎟ . ⎟ ⎟ (3) ⎟ P ⎟ P (2) ⎠

(3.8.21)

P (1) .

Therefore, (3.8.19) and (3.8.20) are equivalent in view of (3.8.11) and (3.8.17). Next we shall show (3.8.19). For this purpose we consider the potentials (k)

v(x) := Kk ϕ(x) = ϕ|Γ ⊗ δΓ , E(x, •)

(3.8.22)

for x ∈ Ω and Ω c , respectively. For ϕ ∈ C ∞ (IRn ), the multilayer potential v is in C ∞ (Ω) up to the boundary and also in C ∞ (Ω c ) up to the boundary, as well, due to Lemma 3.8.1. Then, from the formula (3.7.7) we find, after changing the orders of summation, P (vχΩ ) = (P v)χΩ −

2m−1  2m−p−1 

(p)

(Pp++1 γ v)|Γ ⊗ δΓ

p=0

=0

and, similarly, P (vχΩ c ) = (P v)χΩ c +

2m−1  2m−p−1  p=0

since

(p)

(Pp++1 γc v)|Γ ⊗ δΓ ,

=0

∂ χΩ c = ni δΓ . ∂xi

Adding these two equations yields, with (3.8.3), i.e., P v = 0 in Ω ∪ Ω c , and with any test function ψ ∈ C0∞ (IRn ), the equation P (vχΩ ) + P (vχΩ c ), ψ =

2m−1  2m−p−1 

(p)

Pp++1 (γc v − γ v) ⊗ δΓ , ψ .

p=0

=0

By the definition of derivatives of distributions, the left–hand side can be rewritten as    vP  ψdx + vP  ψdx = vP  ψdx , Ω

Ωc

IRn

where the last equality holds since v was in C ∞ (Ω) up to the boundary and in C ∞ (Ω c ) ∩ C0∞ (IRn ) up to the boundary, too. With (3.8.22), this implies

144

3. Representation Formulae (k)

P (vχΩ ) + P (vχΩ c ), ψ = P v, ψ = ϕ|Γ ⊗ δΓ , ψ

(3.8.23)

since E is the fundamental solution of P . The last step will be justified at the end of the proof. Therefore, we obtain with (3.8.10), (k)

ϕ|Γ ⊗ δΓ =

2m−1  2m−p−1  p=0

=

=0

2m−1  2m−p−1  p=0

(p)

Pp++1 (γc Kk ϕ − γ Kk ϕ) ⊗ δΓ

(p)

Pp++1 (QΩ c +1,k+1 ϕ − QΩ+1,k+1 ϕ) ⊗ δΓ .

=0

Hence, from the definitions (3.8.9) and (3.8.10) we find the proposed equation I = P(QΩ c − QΩ ) . Since we have shown in (3.8.21) that P is invertible, this yields (3.8.19). To complete the proof, we return to the justification of (3.8.23). Let Φ and ψ belong to C0∞ (IRn ). Then, since E(x, y) is a fundamental solution of P satisfying (3.6.3) and (3.6.4) , we obtain the relation   φ, E(x, •)Px ψ(x)dx = P E(•, y) φ(y)dy, ψ = φ, ψ IRn

IRn



= ψ, φ =

(3.8.24) ψ, E(x, •)Px φ(x)dx ,

IRn

where the last identity follows by symmetry. Since (3.8.24) defines for any φ ∈ C ∞ a bounded linear functional on ψ ∈ C0∞ (IRn ), we can extend (3.8.24) to any distribution φ ∈ D . In particular, we may choose (k)

φ := φ|Γ ⊗ δΓ . Then (3.8.24) implies  (k) (k) v, P  ψ = φ|Γ ⊗ δΓ , E(x, •)Px ψ(x)dx = φΓ ⊗ δΓ , ψ , IRn

i.e. (3.8.23). This completes the proof.



As a consequence of Theorem 3.8.2 we have Corollary 3.8.3. For the ”rigid motions” M (x; u) in (3.7.12) and with the exterior Calder´ on projector, we find CΩ c γc M (•; u) = 0 .

(3.8.25)

3.9 Direct Boundary Integral Equations

145

Proof: Since M (x; u) is a C ∞ –solution of P M = 0 in IRn , which follows from (3.7.12), we find CΩ γM = γM = γc M . Applying CΩ c to this equation yields (3.8.25) since CΩ c CΩ = 0 because of (3.8.20).  Remark 3.8.1: From the definition of QΩ we now define the finite part multiple layer potentials on Γ ,    ∂ j−1  ∂  k−1 E(x, y) φk (y)dsy , Djk φk := p.f. ∂nx ∂ny Γ

j, k = 1, . . . , 2m

(3.8.26)

and Dφ

:=

2m  k=1

=

D1,k φk , . . . ,

2m 

D2m,k φk



k=1

1 1 QΩ φ + P −1 φ = QΩ c φ − P −1 φ on Γ 2 2

(3.8.27)

for any given φ = (φ1 , . . . , φ2m ) ∈ C ∞ (Γ ). Correspondingly, we have 1 1 D(Pψ) = −CΩ ψ + ψ = CΩ c ψ − ψ on Γ . 2 2

(3.8.28)

As we will see, (3.8.27) or (3.8.28) are the jump relations for multilayer potentials.

3.9 Direct Boundary Integral Equations As we have seen in the examples in elasticity and the biharmonic equation in Chapter 2, the boundary conditions are more general than parts of the Cauchy γu data appearing in the Calder´ on projector directly. Correspondingly, for more general boundary conditions, the boundary potentials in applications will also be more complicated. 3.9.1 Boundary Value Problems Let us consider a more general boundary value problem for strongly elliptic equations associated with (3.6.1) in Ω (and later on also in Ω c ). As before,

146

3. Representation Formulae

Ω is assumed to be a bounded domain with C ∞ –boundary Γ = ∂Ω. The linear boundary value problem here consists of the uniformly strongly elliptic system of partial differential equations  Pu = aα (x)Dα u = f in Ω (3.9.1) |α|≤2m ∞

with real C coefficient p × p matrices aα (x) (which automatically is properly elliptic as mentioned before, see Wloka [439, Sect. 9]) together with the boundary conditions Rγu = ϕ on Γ (3.9.2) where R = (Rjk ) with j = 1, . . . , m and k = 1, . . . , 2m is a rectangular matrix of tangential differential operators along Γ having orders order (Rjk ) = μj − k + 1 where 0 ≤ μj ≤ 2m − 1 and μj < μj+1 for 1 ≤ j ≤ m − 1 . If μj − k + 1 < 0 then Rjk := 0. Hence, component–wise, the boundary conditions (3.9.2) are of the form 2m  k=1



μj +1

Rjk γk−1 u =

Rjk γk−1 u = ϕj for j = 1, . . . , m .

(3.9.3)

k=1

For p–vector–valued functions u each Rjk is a p × p matrix of operators. In the simplest case μj is constant for all components of u = (u1 , . . . , up ) , and let us consider here first this simple case. We shall return to the more general case later on. In order to have normal boundary conditions (see e.g. Lions and Magenes [270]), we require the following crucial assumption. Fundamental assumption: There exist complementary tangential boundary operators S = (Sjk ) with j = 1, . . . , m and k = 1, . . . , 2m, having the orders order (Sjk ) = νj − k + 1 , where νj = μ for all j,  = 1, . . . , m , 0 ≤ νj ≤ 2m − 1 ; νj > νj+1 for 1 ≤ j ≤ m − 1 such that the square matrix of tangential differential operators   R M := S admits an inverse N

:=

M−1 ,

(3.9.4)

(3.9.5)

which is a matrix of tangential differential operators. Since M is a rearranged triangular matrix this means that the operators on the rearranged diagonal, i.e. Rj,μj +1 and Sj,νj +1 are multiplications with non vanishing coefficients.

3.9 Direct Boundary Integral Equations

147

Definition 3.9.1. The boundary value problem is called a regular elliptic boundary value problem if P in (3.9.1) is elliptic and the boundary conditions in (3.9.2) are normal and satisfy the Lopatinski–Shapiro conditions. In order to have a regular elliptic boundary value problem (see H¨ormander [187], Wloka [439]) we require in addition to the fundamental assumption, i.e. normal boundary conditions, that the Lopatinski–Shapiro condition is fulfilled (see [439, Chap.11]). This condition reads as follows: Definition 3.9.2. The boundary conditions (3.9.3) for the differential operator in (3.9.1) are said to satisfy the Lopatinski–Shapiro conditions if the initial value problem defined by the constant coefficient ordinary differential equations 2m  d k  (2m) Pk (x, iξ  ) χ(σn ) = 0 for 0 < σn (3.9.6) dσn k=0

together with the initial conditions 

rjkλ (x)(iξ  )λ

|λ|=μj −k−1

 d k−1 χ|σn =0 = 0 for j = 1, . . . , m dσn

(3.9.7)

and the radiation condition χ(σn ) = o(1) as σn → ∞ admits only the trivial solution χ(σn ) = 0 for every fixed x ∈ Γ and every ξ  ∈ IRn−1 with |ξ  | = 1. Here, the coefficients in (3.9.6) are given by the principal part of P in local coordinates via (3.4.50) as   (2m) Pk (x, ξ  ) := aα (x)cα,γ,k (σ  , 0)(iξ  )γ ; |γ|≤2m−k |α|=2m

and the coefficients in (3.9.7), from the tangential boundary operators in (3.9.2) in local coordinates are given by 

Rjk =

rjkλ (x)

|λ|≤μj −k−1

 ∂ λ . ∂σ 

The case of more general boundary conditions in which the orders μj for the different components u of u are different, with corresponding given components ϕjt of ϕj , was treated by Grubb in [158]. Here the boundary conditions have the form p μ j +1  =1

k=1



t rjkλ (x)(iξ  )λ γk−1 u = ϕjt ,

|λ|=μj −k−1

t = 1, . . . , qj ≤ p , j = 1, . . . , m and

m  j=1

qj = m · p .

(3.9.8)

148

3. Representation Formulae

Here, for each fixed  we have μj ∈ {0, . . . , 2m − 1} which are ordered as μj < μj+1, , then we also need complementing orders 2m−1 # νj ∈ {0, . . . , 2m−1}\ {μj } with νj > νj+1, and complementing boundj=1 t ary conditions Sjk . For a system of normal boundary conditions S the boundary conditions in R together with the complementing boundary conditions 2 = ((Mjk ))2m×2m should be such that the rearranged triangular matrix M with Mjk = 0 for j < k and tangential operators t Mjk = ((Mjk ))qj ×p

now define a complete system of boundary conditions p  

t Mjk γk−1 u = ϕjt , t = 1, qj , j = 1, 2m

=1 k≤j

where the diagonal matrices Mjj = ((Mjj ))qj ×p , whose entries are functions, define surjective mappings onto the qj –vector valued functions, and where 2 qj ≤ p. Then there exists the right inverse N to the triangular matrix M and, hence, to M. If we require the stronger fundamental assumption that M−1 exists, the boundary conditions still will be a normal system. In this case, in the Shapiro– Lopatinski condition, the initial conditions (3.9.7) are now to be replaced by p μ j +1  =1

k=1



t rjkλ (x)(iξ  )λ

|λ|=μj −k−1

 d k−1 χ |σn =0 = 0 dσn

(3.9.9)

for t = 1, . . . , qj with qj ≤ p and j = 1, . . . , m . In regard to the Shapiro–Lopatinski condition for general elliptic systems in the sense of Agmon–Douglis–Nirenberg, we refer to the book by Wloka et al [440, Sections 9.3,10.1], where one can find an excellent presentation of these topics. Now, for the special cases for m = 1 we consider the following second order systems: In this case, only the following two combinations of boundary conditions are possible: i) R = (R00 , 0)

with order (R00 )

=0

S = (S00 , S01 )

with order (S00 )

= 1 and order (S01 ) = 0 .

(3.9.10)

−1 and For (3.9.5) to be satisfied, we require that the inverse matrices R00 exist along Γ . Then M and N are given by

−1 S01

3.9 Direct Boundary Integral Equations



R00 S00

M=

, ,

0 S01

'

 ; N =

−1 R00

,

0

−1 −1 S00 R00 −S01

,

−1 S01

149

( (3.9.11)

ii) R = (R00 , R01 )

with order (R00 )

= 1 and order (R01 ) = 0 ,

S = (S00 , 0)

with order (S00 )

= 0.

(3.9.12)

−1 −1 Similarly, here for (3.9.5) one needs the existence of R01 and S00 . Then M and N are given by ( '   −1 0 , S∞ R00 , R01 (3.9.13) ; N = M= −1 −1 −1 S00 , 0 R01 , −R01 R00 S00

In general, the verification of the fundamental assumption is nontrivial (see e.g. Grubb [158, 159]), however, in most applications, the fundamental assumption is fulfilled. Because of (3.9.4) one may express the Cauchy data γu = (γ0 u, . . . , γ2m−1 u) by the given boundary functions ϕj in (3.9.3) and a set of complementary boundary functions λj via 

νj +1

Sjk γk−1 u = λj

with j = 1, . . . , m on Γ .

(3.9.14)

k=1

If we require (3.9.5) or normal boundary conditions, the Cauchy data γu can be recovered from ϕ and λ by the use of N , γ−1 u =

m 

Nj ϕj +

2m 

Np λp−m for  = 1, . . . , 2m ;

(3.9.15)

p=m+1

j=1

in short, γu = N1 ϕ + N2 λ

(3.9.16)

with rectangular tangential differential operators N1 and N2 from (3.9.15). Then (3.9.17) RN1 = I , RN2 = 0 , SN1 = 0 , SN2 = I since N is the right inverse. This enables us to insert the boundary data ϕ and λ with given ϕ into the representation formula (3.8.5), i.e., for x ∈ Ω we obtain:  E(x, y) f (y)dy (3.9.18) u(x) = Ω

−

2m−1  2m−−1  =0

p=0

Kp Pp++1

m  j=1

N+1,j ϕj +

m  j=1

 N+1,m+j λj (x) .

150

3. Representation Formulae

By taking the traces on both sides of (3.9.18) we arrive with (3.8.6) or (3.8.10) at       ϕ ϕ ϕ = γF + CΩ N = γF − QΩ PN γu = N λ λ λ 

where F (x) :=

E(x, y) f (y)dy .

Ω

Now, applying the boundary operator M, we find     ϕ ϕ . = MγF + MCΩ M−1 Mγu = λ λ  If f = 0, the right–hand side again is a projector for ϕλ = Mγu, which are generalized Cauchy data and boundary data to any solution u of the homogeneous equation P u = 0 in Ω. We call CΩ , defined by       R ϕ ϕ CΩ (N1 ϕ + N2 λ) , := MCΩ M−1 = (3.9.19) CΩ S λ λ the modified Calder´ on projector associated with P, Ω and M, which satisfies 2 = CΩ . CΩ

(3.9.20)

In terms of the finite–part integrals defined in (3.8.26), we have a relation corresponding to (3.8.28) for the modified Calder´on projector as well: 1 MDPM−1 χ = − CΩ χ + χ on Γ . 2

(3.9.21)

For the boundary value problem (3.9.1) and (3.9.3), ϕ is given. Consequently, for the representation of u in (3.9.18) the missing boundary datum λ is to be determined by the boundary integral equations     ϕ ϕ + MγF on Γ . (3.9.22) = CΩ λ λ As for the Laplacian, (3.9.22) is an overdetermined system for λ in the sense that only one of the two sets to determine λ. of equations is sufficient  Applying RM−1 = I0 and SM−1 = I0 to (3.9.22) and employing (3.9.21) leads to the following two choices of boundary integral equations in terms of finite part integral operators: Integral equations of the ”first kind”: AΩ λ := −RQΩ PN2 λ = ϕ + RQΩ PN1 ϕ − RγF = −RDPN2 λ

= 12 ϕ + RDPN1 ϕ − RγF on Γ .

(3.9.23)

3.9 Direct Boundary Integral Equations

151

Integral equations of the ”second kind”: BΩ λ := λ + SQΩ PN2 λ = −SQΩ PN1 ϕ + SγF = 12 λ + SDPN2 λ = −SDN1 ϕ + SγF on Γ .

(3.9.24)

Note that in the above equations the tangential differential operators R and S are applied to finite part integral operators. These operators can be shown to commute with the finite part integration (see Corollary 9.3.3). We now have shown how the boundary value problem could be reduced to each of these boundary integral equations. Consequently, the solution λ of one of the boundary integral equations generates the solution of the boundary value problem with the help of the representation formula (3.9.18). More precisely, we now establish the following equivalence theorem. This theorem assures that every solution of the boundary integral equation of the first kind generates solutions of the interior as well as of the exterior boundary value problem and that every solution of the boundary value problems can be obtained by solving the first kind integral equations. This equivalence, however, does not guarantee that the numbers of solutions of boundary value problems and corresponding boundary integral equations coincide. Theorem 3.9.1. Assume the fundamental assumption holds. Let f ∈ C ∞ (Ω) and ϕ ∈ C ∞ (Γ ). Then every solution u ∈ C ∞ (Ω) of the boundary value problem (3.9.1), (3.9.2) defines by (3.9.14) a solution λ of both integral equations (3.9.23) and (3.9.24). Conversely, let λ ∈ C ∞ (Γ ) be a solution of the integral equation of the first kind (3.9.23). Then λ together with ϕ defines via (3.9.18) a solution u ∈ C ∞ (Ω) of the boundary value problem (3.9.1), (3.9.2) and, in addition, λ solves also the the second kind equation (3.9.24). If λ ∈ C ∞ (Γ ) is a solution of the integral equation of the second kind (3.9.24) and if, in addition, the operator SDPN1 is injective, then the potential u defined by λ and ϕ via (3.9.18) is a C ∞ –solution of the boundary value problem (3.9.1), (3.9.2) and, in addition, λ solves also the first kind equation (3.9.23). Proof: i.) If u ∈ C ∞ (Ω) is a solution to (3.9.1) and (3.9.2) then the previous derivation shows that λ ∈ C ∞ (Γ ) and that λ satisfies both boundary integral equations. ii.) If λ ∈ C ∞ (Γ ) is a solution of the integral equation of the first kind (3.9.23) then the potentials in (3.9.18) define a C ∞ –solution u of the partial differential equation (3.9.1) due to Lemma 3.8.1 and (3.8.3), (3.8.4). Applying to both sides of (3.9.18) the trace operator yields   −1 ϕ on Γ . (3.9.25) γu = γF + CΩ M λ Applying R on both sides gives with (3.9.5)

152

3. Representation Formulae

Rγu = RγF + RCΩ (N1 ϕ + N2 λ) ; and (3.9.23) for λ yields with (3.8.11) Rγu = ϕ .  Hence, γu = M−1 ϕλ and the application of SM to (3.9.25) yields (3.9.24) for λ, too. iii.) If λ ∈ C ∞ (Γ ) solves (3.9.24), i.e. λ + SQΩ P(N1 ϕ + N2 λ) = SγF ,

(3.9.26)

then the potentials in (3.9.18) again define a C ∞ –solution u of the partial differential equation (3.9.1) in Ω, and for its trace holds (3.9.25) on Γ . Applying S yields with (3.9.26) and (3.8.11) the equation Sγu = λ

on Γ .

On the other hand, the solution u defined by (3.9.18) satisfies on Γ Mγu = MγF + CΩ Mγu whose second set of components reads λ = Sγu = SγF − SQΩ P(N1 Rγu + N2 λ) in view of (3.9.23) and (3.8.11). If we subtract (3.9.26) we obtain SCΩ N1 (Rγu − ϕ) = 0 . On the other hand, with (3.8.28), SCΩ N1 = S{ 12 I − DP}N1 = −SDPN1 due to (3.9.17). Therefore SDPN1 (ϕ − Rγu) = 0 which implies ϕ = Rγu due to the assumed injectivity. Now, the first equation (3.9.23) then follows as in ii).  Now let us consider the exterior boundary value problem for the equation (3.6.14), i.e., (3.9.27) P u = f in Ω c together with the boundary condition Rγc u = ϕ

on Γ

(3.9.28)

3.9 Direct Boundary Integral Equations

153

and the radiation condition — which means that M (x; u) defined by (3.6.15) is also given. Here, the solution can be represented by (3.7.13),  u(x) = E(x, y) f (y)dy + M (x; u) Ωc

+

2m−1  2m−−1   =0

p=0



∂  p E(x, y) ∂ny

 Pp++1 γc u(y)dsy .

Γ

As for the interior problem, we require the fundamental assumption and obtain with the exterior Cauchy data Rγc u = ϕ and Sγc u = λ, the modified representation formula for x ∈ Ω c , similar to (3.9.18): u(x) = Fc (x) + M (x; u) +

2m−1  2m−−1  =0

Kp Pp++1

m 

p=0

(3.9.29)   N+1,j ϕj + N+1,m+j λj (x)

j=1



where Fc (x) =

E(x, y) f (y)dy .

Ωc

In the same manner as for the interior problem, we obtain the set of boundary integral equations     ϕ ϕ  = CΩ c + Mγc Fc + Mγc M (•; u) (3.9.30) λ λ where

CΩ c = I − CΩ = MCΩ c M−1

(3.9.31)

and CΩ c is given in (3.8.17). These are the ordinary and modified exterior Calder´ on projectors, respectively, associated with P, Ω c and M. As before, we obtain two sets of boundary integral equations for λ. Integral equation of the ”first kind”: AΩ c λ := −RQΩ c PN2 λ = −ϕ + RQΩ c PN1 ϕ + Rγc Fc + Rγc M , = −RDPN2 λ = − 21 ϕ + RDPN1 ϕ + Rγc Fc + Rγc M .

(3.9.32)

Integral equation of the ”second kind”: BΩ c λ := λ − SQΩ c PN2 λ = SQΩ c PN1 ϕ + Sγc Fc + Sγc M , = 12 λ − SDPN2 λ = SDPN1 ϕ + Sγc Fc + Sγc M .

(3.9.33)

In comparison to equations (3.9.23) and (3.9.32), we see that AΩ and AΩ c are the same for interior and exterior problems. Hence, we simply denote this operator in the sequel by

154

3. Representation Formulae

A := AΩ = AΩ c = −RDPN2 . It is not difficult to see that under the same assumptions as for the interior problem, the equivalence Theorem 3.9.1 remains valid for the exterior problem (3.9.27), (3.9.28) with the radiation condition provided f ∈ C ∞ (Ω c ) and f has compact support in IRn . The following theorem shows the relation between the solutions of the homogeneous boundary value problems and the homogeneous boundary integral equations of the first kind. Theorem 3.9.2. Under the previous assumptions Γ ∈ C ∞ and aα ∈ C ∞ , we have for the eigenspaces of the interior and exterior boundary value problems and for the eigenspace of the boundary integral operator A on Γ , respectively, the relation: kerA

=

X

kerA

=

  {λ ∈ C ∞ (Γ )  Aλ = 0 on Γ }

where

and

(3.9.34)

  X = span {Sγu  u ∈ C ∞ (Ω) ∧ P u = 0 in Ω and Rγu = 0 on Γ }   ∪ {Sγc uc  uc ∈ C ∞ (Ω c ) ∧ P uc = 0 in Ω c , Rγc uc = 0 on Γ and the radiation condition M (x; uc ) = 0}

Proof: i.) Suppose λ ∈ kerA, i.e. Aλ = 0 on Γ . Then define the potential u(x) :=

2m−1  2m−−1  =0

Kp Pp++1 (N2 λ)

p=0

where (ψ) denotes the –th component of the vector ψ. Now, u is a solution of P u = 0 in Ω and in Ω c . Then we find in Ω Rγu

=

RQΩ PN2 λ = −Aλ = 0 and

Sγu

=

SQΩ PN2 λ .

(3.9.35)

Similarly, we find in Ω c , Rγc u

=

RQΩ c PN2 λ = −Aλ = 0 and

Sγc u

=

SQΩ c PN2 λ .

(3.9.36)

3.9 Direct Boundary Integral Equations

155

Subtracting (3.9.36) from (3.9.35) gives Rγu

=

Rγc u = 0

Sγu − Sγc u

=

S(QΩ − QΩ c )PN2 λ = SP −1 PN2 λ = λ .

and

This shows that λ ∈ X . ii.) Let u be the eigensolution of P u = 0 in Ω with Rγu = 0 on Γ . Then u admits the representation (3.9.18), namely u(x) = −

2m−1  2m−−1 

Kp Pp++1 {N2 Sγu} in Ω .

p=0

=0

By taking traces and applying the boundary operator R we obtain 0 = Rγu = A(Sγu) . This implies that Sγu ∈ kerA. In the same manner we proceed for an eigensolution of the exterior problem and conclude that Sγc u ∈ kerA as well. This completes the proof.  Corollary 3.9.3. The injectivity of the operator SDPN1 in the equivalence Theorem 3.9.1 is guaranteed if and only if the interior and exterior homogeneous boundary value problems Pu = Sγu =

0 in Ω 0

and and

P uc Sγc uc

= =

0 in Ω c , 0 on Γ with M (x; uc ) = 0

admit only the trivial solutions. Proof: For the proof we exchange the rˆ oles of R and S, λ and ϕ, respectively. Then the operator A will just be SDPN1 and Theorem 3.9.2 implies the Corollary.  3.9.2 Transmission Problems In this section we consider transmission problems of the following rather general type: Find u satisfying the differential equation P u = f in Ω ∪ Ω c and the transmission conditions

156

3. Representation Formulae

Rγu = ϕ− and [Rγu]Γ = (Rγc u − Rγu) = [ϕ] on Γ

(3.9.37)

where f ∈ C0∞ (IRn ) , ϕ− , [ϕ] ∈ C ∞ (Γ ) are given functions and where u satisfies the radiation condition (3.6.15). We begin with some properties of the modified Calder´on projectors. Obviously, they enjoy the same properties as the ordinary Calder´on projectors, namely C2 = CΩ , CΩ + CΩ c = I, C2 c = CΩ c . (3.9.38) Ω

Ω

In terms of the boundary potentials and boundary operators R and S, the explicit forms of CΩ and CΩ c are given by      RQΩ PN1 , RQΩ PN2 ϕ ϕ =− , CΩ λ λ SQΩ PN1 , SQΩ PN2      (3.9.39) RQΩ c PN1 , RQΩ c PN2 ϕ ϕ CΩ c = . λ λ SQΩ c PN1 , SQΩ c PN2 If we denote the jump of the boundary potentials u across Γ by [QPNj ] = QΩ c PNj − QΩ PNj then, as a consequence of (3.8.27), we have the following jump relations for any ϕ, λ ∈ C ∞ (Γ ) across Γ : [RQPN1 ϕ]

= ϕ , [RQPN2 λ]

= 0,

[SQPN1 ϕ]

= 0 , [SQPN2 λ]

= λ.

(3.9.40)

From the representation formulae (3.7.10), (3.7.11) and (3.7.13), (3.7.14) we arrive at the representation formula for the transmission problem,  E(x, y) f (y)dy + M (x; u) u(x) = IRn

+

2m−1  2m−−1   =0

 =

p=0

  ∂  p  E(x, y) Pp++1 γc u(y) − γ u(y) dsy ∂ny

Γ

E(x, y) f (y)dy + M (x; u)

(3.9.41)

IRn

+

2m−1  2m−−1  =0

Kp Pp++1 {N1 [ϕ] + N2 [λ]} for x ∈ Ω ∪ Ω c , x ∈ Γ .

p=0

Now, we apply γ and γc to both sides of (3.9.41) and obtain after applying M:

3.10 Remarks

Mγu

=

MγF + MγM + Mγ

2m−1  2m−−1  =0

= Mγc u =

2m−1  2m−−1  =0

=

Kp Pp++1 {N1 [ϕ] + N2 [λ]}

p=0

MγF + MγM + MQΩ P{N1 [ϕ] + N2 [λ]} , Mγc F + Mγc M + M

157

(3.9.42)

Kp Pp++1 {N1 [ϕ] + N2 [λ]}

p=0

Mγc F + Mγc M + MQ c P{N1 [ϕ] + N2 [λ]} ∼Ω

(3.9.43)

where [ϕ] = R(γc u − γu) and [λ] = S(γc u − γu) .

(3.9.44)

Inserting (3.8.27) into (3.9.42) and (3.9.43), we arrive at the boundary integral equations ϕ∓

=

1 RγF + RγM + RDP{N1 [ϕ] + N2 [λ]} ∓ [x] 2

(3.9.45)

λ∓

=

SγF + SγM + SDP{N1 [ϕ] + N2 [λ]} ∓ 12 [x] ,

(3.9.46)

and

where we use the properties γc F = γF and γc M = γM due to the C ∞ – continuity of F and M . Since ϕ− and [ϕ] in (3.9.45) are given we may use the boundary integral equation of the first kind for finding the unknown [λ]: A[λ]

:= =

−RDPN2 [λ]

(3.9.47) −

RγF + RγM (•; u) − ϕ −

1 2 [ϕ]

+ RDPN1 [ϕ] .

Once [λ] is known, λ− may be obtained from the equation (3.9.46). The solution u of the transmission problem is then given by the representation formula (3.9.41). We notice that in all the exterior problems as well as in the transmission problem, the corresponding boundary integral equations as (3.9.32), (3.9.33) and (3.9.47) contain the term M (x; u) which describes the behavior of the solution at infinity and may or may not be known a priori. In elasticity, M (x; u) corresponds to the rigid motions. In order to guarantee the unique solvability of these boundary integral equations, additional compatibility conditions associated with M (x; u) need to be appended.

3.10 Remarks In all of the derivations of Sections 3.4–3.9 we did not care for less regularity of Γ by assuming Γ ∈ C ∞ . It should be understood that with appropriate bookkeeping of the respective orders of differentiation, all of the calculations

158

3. Representation Formulae

presented in this chapter can be carried out under weaker assumptions for C k,κ boundaries according to the particular formulae and derivations. In order not to be overwhelmingly descriptive we leave these details to the reader. As far as generalizations to mixed and screen boundary value problems are concerned, as indicated in Section 2.5, one has to modify the representation formulae as e.g. (3.9.18) by splitting the generalized Cauchy data j in order to obtain systems of boundϕjD = ϕj + ϕ j and λjN = λj + λ ary integral equations instead of (3.9.23) and (3.9.26). Also, singularities of the charges at the collision points need to be incorporated which is crucial for the corresponding analysis. In the next chapter, we also include one of the very important applications of integral equations, namely those for the general Maxwell equations in electromagnetic theory, which have drawn much attention in recent years.

4. Sobolev Spaces

In order to study the variational formulations of boundary integral equations and their numerical approximations, one needs proper function spaces. The Sobolev spaces provide a very natural setting for variational problems. This chapter contains a brief summary of the basic definitions and results of the L2 –theory of Sobolev spaces which will suffice for our purposes. A more general discussion on these topics may be found in the standard books such as Adams [1], Grisvard [156], Lions and Magenes [270], Maz‘ya [288], Tartar [414], and also in McLean [290].

4.1 The Spaces H s (Ω) The Spaces Lp (Ω)(1 ≤ p ≤ ∞) We denote by Lp (Ω) for 1 ≤ p < ∞, the space of equivalence classes of Lebesgue measurable functions u on the open subset Ω ⊂ IRn such that |u|p is integrable on Ω. We recall that two Lebesgue measurable functions u, v on Ω are said to be equivalent if they are equal almost everywhere in Ω, i.e. u(x) = v(x) for all x outside a set of Lebesgue measure zero (see Kufner and Fuˇcik [249]). The space Lp (Ω) is a Banach space with the norm  1/p |u(x)|p dx . uLp (Ω) := Ω

In particular, for p = 2, we have the space of all square integrable functions L2 (Ω) which is also a Hilbert space with the inner product  (u, v)L2 (Ω) := u(x)v(x)dx for all u, v ∈ L2 (Ω) . Ω

A Lebesgue measurable function u on Ω is said to be essentially bounded if there exists a constant c ≥ 0 such that |u(x)| ≤ c almost everywhere, i.e. (a.e.) in Ω. We define ess sup |u(x)| = inf{c ∈ IR | |u(x)| ≤ c a.e.in Ω}. x∈Ω

© Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_4

160

4. Sobolev Spaces

By L∞ (Ω), we denote the space of equivalence classes of essentially bounded, Lebesgue measurable functions on Ω. The space L∞ (Ω) is a Banach space equipped with the norm  u L∞ (Ω) := ess sup |u(x)|. x∈Ω

We now introduce the Sobolev spaces H s (Ω). Here and in the rest of this chapter Ω ⊂ IRn is a domain. For simplicity we begin with s = m ∈ IN0 and define these spaces by the completion of C m (Ω)–functions. Alternatively, Sobolev spaces may be defined in terms of distributions and their generalized derivatives (or weak derivatives), (see, e.g. Adams [1], H¨ormander [187], McLean [290]). However, it is our belief that from a computational point of view. The following approach is more constructive. Let us first introduce the function space C∗m (Ω) := {u ∈ C m (Ω) | uW m (Ω) < ∞} where uW m (Ω) :=

  

|Dα u|2 dx

1/2 .

(4.1.1)

|α|≤m Ω

Then we define the Sobolev space of order m to be the completion of C∗m (Ω) with respect to the norm  · W m (Ω) . By this we mean that for every u ∈ W m (Ω) there exists a sequence {uk }k∈IN ⊂ C∗m (Ω) such that lim u − uk W m (Ω) = 0 .

k→∞

(4.1.2)

We recall that two Cauchy sequences {uk } and {vk } in C∗m (Ω) are said to be equivalent if and only if limk→∞ uk − vk W m (Ω) = 0. This implies that W m (Ω), in fact, consists of all equivalence classes of Cauchy sequences and that the limit u in (4.1.2) is just a representative for the class of equivalent Cauchy sequences {uk }. The space W m (Ω) is a Hilbert space with the inner product defined by   Dα uDα vdx . (4.1.3) (u, v)m := |α|≤m Ω

Clearly, for m = 0 we have W 0 (Ω) = L2 (Ω). The same approach can be used for defining the Sobolev space W s (Ω) for non–integer real positive s. Let s=m+σ

with m ∈ IN0

and 0 < σ < 1 ;

and let us introduce the function space C∗s (Ω) := {u ∈ C m (Ω) | uW s (Ω) < ∞}

(4.1.4)

4.1 The Spaces H s (Ω)

where uW s (Ω) :=



u2W m (Ω)

161

1/2    |Dα u(x) − Dα u(y)|2 + dxdy , |x − y|n+2σ |α|=m Ω Ω

(4.1.5) the Slobodetskii norm. Note that the second part in the definition (4.1.5) of the norm in W s (Ω) gives the L2 –version of fractional differentiability, which is compatible to the pointwise version in C m,α (Ω). In the same manner as for the case of integer order, the Sobolev space W s (Ω) of order s is the completion of the space C∗s (Ω) with respect to the norm  · W s (Ω) . Again, W s (Ω) is a Hilbert space with respect to the inner product (u, v)s :=

(4.1.6)  α    α α α  D u(x) − D u(y) D v(x) − D v(y) dx dy . (u, v)m + |x − y|n+2σ |α|=m Ω Ω

Clearly, for m = 0 we have W 0 (Ω) = L2 (Ω). Note, that all of the definitions above are also valid for Ω = IRn . In this case, Meyers and Serrin [294] have shown that the space C0∞ (IRn ) is dense in W s (IRn ) which implies that for Ω = IRn . The Sobolev spaces defined via distributions are the same as W s (IRn ). We therefore denote H s (IRn ) := W s (IRn ) for s ≥ 0 .

(4.1.7)

Instead of the functions in C m (Ω) let us now consider the function space of restrictions, C ∞ (Ω) := {u = u |Ω with u  ∈ C0∞ (IRn )} and introduce for s ≥ 0 the norm uH s (Ω) := inf{ uH s (IRn ) | u = u |Ω } .

(4.1.8)

Now we define H s (Ω) to be the completion of C ∞ (Ω) with respect to the norm ·H s (Ω) , which means that to every u ∈ H s (Ω) there exists a sequence {uk }k∈IN ⊂ C ∞ (Ω) such that lim u − uk H s (Ω) = 0 .

k→∞

Since C0∞ (Ω) ⊂ C ∞ (Ω) where for any u ∈ C0∞ (Ω) the trivial extension  s (Ω) for s ≥ 0 u  by zero outside of Ω is in C0∞ (IRn ), we define the space H ∞ to be the completion of C0 (Ω) with respect to the norm uH s (Ω) :=  uH s (IRn ) . This definition implies that

(4.1.9)

162

4. Sobolev Spaces

 s (Ω) ⊂ H s (Ω) for s ≥ 0. H For negative s, we define H s (Ω) by duality with respect to the inner product (·, ·)L2 (Ω) . More precisely, for s < 0 we define the norm by us =

sup  −s (Ω) 0=ϕ∈H

|(ϕ, u)L2 (Ω) | / ϕH −s (Ω) .

(4.1.10)

As usual, we denote the completion of L2 (Ω) with respect to (4.1.10) by  −s (Ω)) for s < 0 , H s (Ω) = (H

(4.1.11)

 −s (Ω). These are the Sobolev spaces of negative order. the dual space of H  s (Ω) for s < 0 by the Extending (4.1.11), we also define the spaces H 2 completion of L (Ω) with respect to the norm uH s (Ω) :=

sup 0=ψ∈H −s (Ω)

|(ψ, u)L2 (Ω) | / ψH −s (Ω) .

(4.1.12)

This completion is denoted by   s (Ω) = H −s (Ω)  , H

(4.1.13)

which is the dual of H −s (Ω). In fact, one can show that  s (Ω) = {u ∈ H s (IRn ) | supp u ⊂ Ω} , H

(4.1.14)

see H¨ormander [187]. These are also subspaces of the Sobolev spaces H s (Ω). For s > 0, we have the inclusions:  −s (Ω) ⊂ H −s (Ω)  s (Ω) ⊂ H s (Ω) ⊂ L2 (Ω) ⊂ H H

(4.1.15)

with continuous injections (i.e. injective linear mappings). We remark that the  −s (Ω), define elements of the Sobolev spaces of negative order, H −s (Ω) and H  s (Ω) and bounded, i.e. continuous, linear functionals on the Sobolev spaces H s H (Ω), respectively. As will be seen, these functionals can be represented in two different ways.  s (Ω) or H s (Ω) with In what follows, let V denote the Hilbert space H  0 ≤ s, and let V denote its dual. That is, we have V ⊂ L2 ⊂ V  . We set the value of the continuous linear functional  at v by , v := (v) for  ∈ V  and v ∈ V

with , v = (v, )L2

(4.1.16)

provided  ∈ L2 . Here the bilinear form , v is called the duality pairing on V  × V . If uV is the norm on V , the space V  is supplied with the dual norm |, v| . V  := sup 0=v∈V vV

4.1 The Spaces H s (Ω)

163

Since V is a Hilbert space, by the Riesz representation theorem, for each  ∈ V  there exists a unique λ ∈ V depending on  such that , v = (v) = (v, λ)V and V  = λV ,

(4.1.17)

where (v, λ)V denotes the inner product in V . This is one way to represent the element of V  , although it is not very practical, since this representation involves here the complicated inner product (•, •)V of the Hilbert space V . The other way to represent  ∈ V  is to identify  with an element u in L2 (Ω) and to consider the value (v) at v ∈ V as the L2 -inner product (v, u )L2 (Ω) provided  ∈ V  is also bounded on L2 . In other words, we express the bilinear form , v on V  ×V as an inner product (v, u )L2 (Ω) . In fact, the  s (Ω) definitions (4.1.10) and (4.1.12) for the dual spaces V  = H s (Ω) and H 2  for s < 0 are based on this representation. In case  ∈ L (Ω) ⊂ V , we simply let u =  ∈ L2 (Ω) and obtain   , v = (v, u )L2 (Ω) = v(x)u (x) dx = v(x)(x)dx Ω

Ω

for all v ∈ V . On the other hand, if  ∈ V  but  ∈ L2 (Ω), then we define  v(x)uk (x)dx (4.1.18) , v = lim (v, uk )L2 (Ω) = lim k→∞

k→∞

Ω

for all v ∈ V , where {uk } ⊂ L2 (Ω) is a sequence such that lim uk − V  = 0.

k→∞

We know that {uk } exists and (4.1.18) makes sense, since V  is the completion of L2 (Ω) with respect to the norm  · V  , defined by (4.1.10) and (4.1.11), respectively. The second way is more practical; and as will be seen, our variational formulations of boundary integral equations in Chapter 5 are based on this representation. In what follows, we write u, vL2 (Ω) = (u, v)L2 (Ω)

(4.1.19)

for the duality pairing of (u, v) ∈ V  × V ; and the L2 –inner product on the right–hand side is understood in the sense of (4.1.18) for u ∈ L2 (Ω). We remark that both representations are equivalent and are defined by the  −s (Ω) onto H s (Ω) same linear functional. Hence, the mapping  → λ from H is a well defined isomorphism. By an isomorphism we mean a continuous  −s (Ω) onto H s (Ω) whose inverse is also one–to–one linear mapping from H continuous. Now we consider the special case Ω = IRn . In this case, for any s ∈ IR, in addition we have

164

4. Sobolev Spaces

 s (IRn ) = H s (IRn ) . H

(4.1.20)

Moreover, these Sobolev spaces can be characterized via the Fourier transformation. For any u ∈ C0∞ (IRn ), the Fourier transformed u ˆ is defined by (3.1.12), and there holds  Fξ∗→x u * := u(x) = (2π)−n/2 eix·ξ u ˆ(ξ)dξ , (4.1.21) IRn

with the inverse Fourier transform of u ˆ. The following properties are well known (see e.g. Petersen [346, p.79]; i.e. the Parseval–Plancherel formula   u(x)v(x)dx= u ˆ(ξ)ˆ v (ξ)dξ = (ˆ u, vˆ)L2 (IRn ) ,(4.1.22) (u, v)L2 (IRn ) = n

IRn

 IR u(x)ˆ v (x)dx



=

IRn

u ˆ(ξ)v(ξ)dξ .

(4.1.23)

IRn

Formula (4.1.22) implies ˆ uL2 (IRn ) = uL2 (IRn )

(4.1.24)

for any u ∈ C0∞ (IRn ). A simple application of (4.1.24) to the identity   −n/2 −ix·ξ α |α| −n/2 (2π) e (D u(x))dx = (−1) (2π) (Dxα e−ix·ξ )u(x)dx IRn

IRn α

= (iξ) u ˆ(ξ) for any α ∈ INn0 shows that (ξ α u ˆ(ξ)) ∈ L2 (IRn )

for every α ∈ INn0 and u ∈ C0∞ (IRn ).

This implies that we have |u|2s := (2π)−n

 (1 + |ξ|2 )s |ˆ u(ξ)|2 dξ < ∞

(4.1.25)

IRn

for every s ∈ IR and u ∈ C0∞ (IRn ). Obviously, | · |s is a norm. Now let Hs (IRn ) be the completion of C0∞ (IRn ) with respect to the norm | · |s . Then the space Hs (IRn ) again is a Hilbert space with the inner product defined by  ((u, v))s := (2π)−n (1 + |ξ|2 )s u ˆ(ξ)ˆ v (ξ)dξ . (4.1.26) IRn

From Parseval’s formula, (4.1.24) it follows that

4.1 The Spaces H s (Ω)

165

H 0 (IRn ) = H0 (IRn ) = L2 (IRn ). In fact, one can show that H s (IRn ) is equivalent to Hs (IRn ) for every s ∈ IR, see e.g. Petersen [346, p.235]. Therefore, in what follows, we shall not distinguish between the spaces H s (IRn ) and Hs (IRn ) anymore. The corresponding norms  · s and | · |s will be employed interchangeably. As a further consequence of Parseval’s equality (4.1.24), we see that the Fourier transform Fu(ξ) := u ˆ(ξ) as a bounded linear mapping from C0∞ (IRn ) n into L2 (IR ) extends continuously to L2 (IRn ) and the extension defines an isomorphism L2 (IRn ) → L2 (IRn ). Here again, by an isomorphism we mean a continuous one–to–one mapping from L2 (IRn ) onto itself whose inverse is also continuous. In fact, the inverse F −1 is given by (4.1.21). Further properties of F on the Sobolev spaces and on distributions will be discussed in Chapter 7. Now we summarize some of the relevant results and properties of H s (Ω) without proofs. Proofs can be found e.g. in Adams [1], Kufner and Fuˇcik [249], McLean [290] and Petersen [346]. Lemma 4.1.1. Generalized Cauchy-Schwarz inequality (Adams [1, p.50]). The H s (Ω)–scalar product extends to a continuous bilinear form on s+t  s−t (Ω). Moreover, we have H (Ω) × H |(u, v)H s (Ω) | ≤ uH s+t (Ω) vH s−t (Ω)

(4.1.27)

 s−t (Ω) and all s, t ∈ IR. This inequality also for all (u, v) ∈ H s+t (Ω) × H n s−t  holds for Ω = IR where H (IRn ) = H s−t (IRn ). Lemma 4.1.2. (Petersen [346, Chap.4, Lemma 2.2]) For s < t, the inclusion H t (IRn ) ⊂ H s (IRn ) is continuous and has dense image. Theorem 4.1.3. Rellich’s Lemma (Petersen [346, Chap.4, Theorem 1.12]) Let s < t and Ω ⊂ IRn be a bounded domain. Then the imbedding c  t (Ω) → H s (IRn ) H

is bounded and compact. c

In the following, the symbol → will be used for compact imbedding. Theorem 4.1.4. Sobolev imbedding theorem (Petersen [346, Chap.4, Theorem 2.13]) Let m ∈ IN0 and s > m + n2 . Then for every u ∈ H s (IRn ) and every compact K  IRn we have u|K ∈ C m+α (K) and uC m,α (K) ≤ C(K, m, α)uH s (IRn ) if 0 < α ≤ s − m −

n 2

and α < 1.

166

4. Sobolev Spaces

Before we can state further properties we need some mild restrictions on the domain Ω under consideration. The domain Ω is said to have the uniform cone property if there exists a locally finite open covering {Uj } of ∂Ω = Γ and a corresponding sequence {Cj } of finite cones, each congruent to some fixed finite cone C such that: (i) For some finite number M , every Uj has diameter less than M . (ii) For some constant δ > 0 we have #∞ j=1 Uj ⊃ Ωδ := {x ∈ Ω| inf y∈Γ |x − y| < δ}. # (iii) For every index j we have x∈Ω∩Uj (x + Cj ) =: Qj ⊂ Ω. (iv) For some finite number R, every collection of R + 1 of the sets Qj has empty intersection. We remark that the uniform cone property is closely related to the smoothness of Γ . In particular, any C 2 –boundary Γ will possess the uniform cone property. It was shown by Gagliardo [132] and Chenais [66] that every Lipschitz domain has the uniform cone property. Conversely, Chenais [66] showed that every bounded domain having the uniform cone property, is a Lipschitz domain. Theorem 4.1.5. Strong extension property (Calderon and Zygmund [60], McLean [290], Wloka [439]) Let Ω be bounded and satisfy the uniform cone property. i. Let m ∈ IN0 . Then there exists a continuous, linear so–called extension operator FΩ : H s (Ω) → H s (IRn ) for m ≤ s < m + 1 satisfying FΩ uH s (IRn ) ≤ cΩ (s)uH s (Ω) for all u ∈ H s (Ω) .

(4.1.28)

ii. Let Γ be a C k,κ –boundary with k +κ ≥ 1 , k ∈ IN, and let 0 ≤ s < k +κ if k + κ ∈ IN or 0 ≤ s ≤ k + κ if k + κ ∈ IN. Then there exists a continuous, linear extension operator FΩ which is independent of s satisfying (4.1.28). As a consequence, whenever the extension operator FΩ exists, we have W s (Ω) = H s (Ω) and the norms uW s (Ω) and uH s (Ω)

(4.1.29)

are equivalent (Wloka [439, Theorem 5.3]). Theorem 4.1.6. Rellich’s Lemma, Sobolev imbedding Theorem Adams [1]: Let Ω be bounded and satisfy the uniform cone property. Then the following imbeddings are continuous and compact:

4.1 The Spaces H s (Ω)

i.

c  t (Ω)  s (Ω) → H H

167

for − ∞ < t < s < ∞ ,

(4.1.30)

H s (Ω) → H t (Ω) for − ∞ < t < s < ∞ .

(4.1.31)

ii.

c

iii. c

H s (Ω) → C m,α (Ω) For s = m + α +

n 2

for m ∈ IN0 , 0 ≤ α < 1, m + α < s −

n 2

. (4.1.32)

and 0 < α < 1, the imbeddding H s (Ω) → C m,α (Ω)

(4.1.33)

is continuous. Classical Sobolev spaces For m ∈ IN0 , let H0m (Ω) be the completion of C0m (Ω) with respect to the norm  · H m (Ω) . Then we have H0m (Ω)

=

H −m (Ω)

=

 m (Ω) H and    m (Ω) = (H m (Ω)) . H 0

(4.1.34)

 s (Ω) there are two cases to be considered. If In the case 0 < s ∈ IN0 , for H 1 s = m + σ with |σ| < 2 and m ∈ IN0 then there holds  s (Ω) = C m (Ω)·H s (IRn ) = H0s (Ω) := C m (Ω)·H s (Ω) , H 0 0

(4.1.35)

where the completion of the space C0m (Ω) with respect to the norms  · H s (Ω) and  · H s (IRn ) , respectively, is taken. s (Ω) The spaces H00 1  s (Ω) is strictly contained in H s (Ω): If s = m + 2 , then the space H 0 •  s (Ω) = C m (Ω)·H s (IRn ) ⊂ H0s (Ω) . H 0

(4.1.36)

 s (Ω) by using the norm In this case, Lions and Magenes [270] characterize H   12  1 s (Ω) = u2H s (Ω) + ρ− 2 Dα u2L2 (Ω) uH00

(4.1.37)

|α|=m

where ρ = dist (x, ∂Ω) for x ∈ Ω. It is shown in Lions and Magenes [270, p.66] that the norms

168

4. Sobolev Spaces

 s (Ω) and  uH00 uH s (IRn ) with u =

u 0

for x ∈ Ω , otherwise ,

are equivalent. In variational problems it is sometimes convenient to express equivalent norms in different forms. In the following we present a corresponding general theorem from which various well known equivalent norms can be deduced. Theorem 4.1.7. Various Equivalent Norms Triebel [421, Theorem 6.28.2] Let Ω be a bounded domain having the uniform cone property, m ∈ IN, and let q(v) be a nonnegative continuous quadratic functional on H m (Ω); i.e., q : H m (Ω) 0 ≤ q(v)

→

IR with q(λv) = |λ|2 q(v) for all λ ∈ C , v ∈ H m (Ω) ,

≤

q(℘)

>

cv2H m (Ω) and (4.1.38) 0 for all polynomials ℘ of degree less than m .

Then the norms vH m (Ω) and {

1

|α|=m

Dα v2L2 (Ω) + q(v)} 2 are equivalent on

H m (Ω). In particular, let us consider the following special choices of q:  2    β   i. q(v) :=  D vdx . Then we have the |β| 0. As we will show later on, Hs (Γ ) is actually a Hilbert space, although the inner product can not be deduced from the above definition (4.2.3). For s < 0, we can define the space Hs (Γ ) as the dual of H−s (Γ ) with respect to the L2 (Γ ) scalar product; i.e. the completion of L2 (Γ ) with respect to the norm sup | ϕ, uL2 (Γ ) |. (4.2.5) uHs (Γ ) := ϕH−s (Γ ) =1

These are the boundary spaces of negative orders. The Fichera trace spaces Hs (Γ ) Alternatively, one may define the Sobolev spaces on the boundary in terms of boundary norms due to Fichera [121], instead of the domain norms as in (4.2.3). We begin with the simplest case; for 0 < s < 1 we define Hs (Γ ) to be the completion of Cs 0 (Γ ) := {ϕ ∈ C 0 (Γ )| ϕHs (Γ ) < ∞} with respect to the norm ⎧ ⎫ 12   ⎨ ⎬ 2 |u(x) − u(y)| uHs (Γ ) := u2L2 (Γ ) + ds ds . x y ⎩ ⎭ |x − y|n−1+2s

(4.2.6)

Γ Γ

Again, Hs (Γ ) is a Hilbert space equipped with the inner product   (u(x) − u(y)) (v(x) − v(y)) dsx dsy . (4.2.7) (u, v)Hs (Γ ) := (u, v)L2 (Γ ) + |x − y|n−1+2s Γ Γ

Defining H (Γ ) for s ≥ 1, it is more involved. In the following, let m ∈ IN be fixed and let s

172

4. Sobolev Spaces

M

:=

{u ∈ C m (Ω) | (t(z) · ∇x )α u(x)|z=x∈Γ = 0 for all 0 ≤ |α| ≤ m and for all tangential vectors t|Γ } .

This defines an equivalence relation on C m (Ω). With M we can define the cosets of any u ∈ C m (Ω) by [u] = {v ∈ C m (Ω)| v − u ∈ M} . Now let us consider the quotient space C m (Ω)/M := {[u]|u ∈ C m (Ω)} . As usual, the quotient space C m (Ω)/M becomes a linear vector space by the proper extension of linear operations from C m (Ω) to C m (Ω)/M, i.e. α[u] + β[v] = [αu + βv] for all α, β ∈ C and u, v ∈ C m (Ω). On C m (Ω), we introduce the Hermitian bilinear form   ((u, v))m := Dα uDα v ds (4.2.8) |α|≤m Γ

and the associated semi–norm 1

2 . |u|m := ((u, u))m

(4.2.9)

If | · |m were a norm on C m (Ω), then the completion of the quotient space C m (Ω)/M, would provide us with the desired Hilbert space on Γ . However, since {u ∈ C m (Ω) | |u|m = 0} = {0}, we need to introduce an additional quotient space in which | · |m will become a norm. To be more precise, let N

:=

{u ∈ C m (Ω) | Dα u|Γ = 0 for all α with 0 ≤ |α| ≤ m}

=

{u ∈ C m (Ω) | |u|m = 0} .

Then N defines an additional equivalence relation on C m (Ω). Moreover, on the quotient space C m (Ω)/N , given by the collection of cosets, [u]N := {v ∈ C m (Ω) | |v − u|m = 0} , ((•, •))m and | • |m define, by construction, a scalar product and an associated norm. The corresponding completion B m (Γ ) := C m (Ω)/N

|·|m

then defines a Hilbert space with the scalar product (4.2.8) and norm (4.2.9); for any u, ˙ v˙ ∈ B m (Γ ) we find sequences un , vn ∈ C m (Ω) such that u˙ = limn→∞ [un ]N and v˙ = limn→∞ [vn ]N in B m (Γ ). Thus, we may define

4.2 The Trace Spaces H s (Γ )

173

(u, ˙ v) ˙ m := (u, ˙ v) ˙ Bm (Γ ) = lim ([un ]N , [vn ]N )Bm (Γ ) n→∞

= lim ((un , vn ))m ,

(4.2.10)

n→∞

u ˙ m := u ˙ Bm (Γ ) = lim [un ]N Bm (Γ ) = lim |un |m . (4.2.11) n→∞

n→∞

In order to identify C m (Ω)/M with a subspace of B m (Γ ), we map M into C m (Ω)/N by M  u −→ [u]N ∈ M/N , where M/N := {[u]N | u ∈ M} ⊂ C m (Ω)/N ⊂ B m . One can easily show that the quotient spaces  m C m (Ω)/M C (Ω)/N /(M/N ) are algebraically isomorphic, i.e., there exists a linear one–to–one correspondence between these two spaces. Now we denote by M the completion of M/N in B m (Γ ), and define the trace spaces by Hm (Γ ) := B m (Γ )/M .

(4.2.12)

For any u ∈ Hm (Γ ), by the definition of the quotient space (4.2.12) this means that u has the form u = [u] ˙ for some u˙ ∈ B m (Γ ), and the norm of u is defined by ˙ m. (4.2.13) uHm (Γ ) := inf u˙ + v v∈M ˙

Since M is complete, we can find some u˙ ∗ ∈ M such that uHm (Γ ) = u˙ + u˙ ∗ m = inf u˙ + v ˙ m.

(4.2.14)

v∈M ˙

Moreover, for ⊥

u := u˙ + u˙ ∗ ∈ M ⊂ B m (Γ ) there holds (u, v) ˙ m = 0 for all v˙ ∈ M . o

o

Therefore, Hm (Γ ) and M

⊥

are isomorphic. o

By the definition of completion, there exists a sequence un ∈ C m (Ω) such that o o lim [un ]N − um = 0 n→∞

and o

o

o

o

lim [un ]N − [uk ]N m = lim | un − uk |m = 0 .

n,k→∞

In particular, we see that

n,k→∞

(4.2.15)

174

4. Sobolev Spaces o

uHm (Γ ) = lim | un |m .

(4.2.16)

n→∞ o

⊥

o

Similarly, for v ∈ Hm (Γ ) we can find v ∈ M and a sequence v n ∈ C m (Ω) with the corresponding properties. Then we can define the inner product by o o

o

o

(u, v)Hm (Γ ) := (u, v)m = lim ((un , v n ))m n→∞

(4.2.17)

so that Hm (Γ ) becomes a Hilbert space. Remark 4.2.1: As we mentioned earlier, the space Hm (Γ ) may be considered as the completion of the quotient space C m (Ω)/M with respect to the norm [u]m = inf |u + ϕ|m ϕ∈M

(4.2.18)

corresponding to (4.2.14). However, in order to define the inner product (·, ·)m in (4.2.17), it is necessary to introduce the quotient space C m (Ω)/N . In addition, note that, in general, the infimum in (4.2.18) will not be attained in M since M is not complete. To define Hs (Γ ) for s ∈ IN, s > 0, we write s = m + σ where − 12 ≤ σ < ∈ IN, and proceed in a similar manner as before. More precisely, we first replace the scalar product (4.2.8) by 1 2, m

((u, v))s := ((u, v))[s]    (Dα u(x) − Dα u(y))(Dα v(x) − Dα v(y)) + dsx dsy |x − y|n−1+2(s−[s])

(4.2.19)

|α|=[s] Γ Γ

where [s] := max{ ∈ Z |  ≤ s} denotes the Gaussian bracket. Correspondingly, we define the associated semi–norm 1

|u|s := ((u, u))s2  s (Ω) which is a subspace of C m (Ω) defined by on C ∗  s (Ω) = {u ∈ C m (Ω) | |u|s < ∞} . C ∗ The subspace M is now replaced by Ms

=

 s (Ω) | (t(z) · ∇x )α u(x)|z=x∈Γ = 0 {u ∈ C ∗ for all |α| ≤ m and all tangent vectors t|Γ } ;

 s (Ω)/Ms . Furthermore, let and the quotient space is replaced by C ∗  s (Ω) | |u|s = 0} ; N s := {u ∈ C ∗

(4.2.20)

4.2 The Trace Spaces H s (Γ )

175

 s (Ω)/N s , given by the cosets and the completion of the quotient space C ∗  s (Ω) | |u − v|s = 0} , [u]N s := {v ∈ C ∗ s

is denoted by B s (Γ ). We denote by M the completion of Ms /N s := {[u]N s | u ∈ Ms } ⊂ B s (Γ ) in the Hilbert space B s (Γ ). Then, finally we define Hs (Γ ) := B s (Γ )/Ms

(4.2.21)

as the new boundary Hilbert space. Note that any element u ∈ Hs (Γ ) can o  s (Ω) ; and the details again be approximated in Hs (Γ ) by sequences un ∈ C ∗ are exactly the same as in the case of integer s = m. For negative s, we define Hs (Γ ) by duality with respect to the L2 (Γ )– inner product, (·, ·)L2 (Γ ) . Similar to (4.1.10), the norm is given by uHs (Γ ) :=

sup 0=ϕ∈H−s (Γ )

|(ϕ, u)L2 (Γ ) | / ϕH−s (Γ ) ;

(4.2.22)

and the completion of L2 (Γ ) with respect to (4.2.22) is denoted by Hs (Γ )

= =



 H−s (Γ ) , { : H−s (Γ ) → C |  is linear and continuous } ,

(4.2.23)

i.e., the dual space of H−s (Γ ). We write again , v = (v, )L2 (Γ ) for  ∈ Hs (Γ ) and v ∈ H−s (Γ ), and the L2 (Γ )-inner product (·, ·)L2 (Γ ) is understood to be similar in a sense to (4.1.18) for  ∈ L2 (Γ ). Remark 4.2.2: In principle, to replace the cosets, one can define Hs (Γ ) in terms of functions on Γ with covariant derivatives of orders α with |α| ≤ m from differential geometry which we tried to avoid here. Nevertheless, properties concerning the geometry of Γ will be discussed later. Standard trace spaces Hs (Γ ) In this approach we identify the boundary Γ with IRn−1 by means of the local parametric representations of the boundary. Roughly speaking, we define the trace space to be isomorphic to the Sobolev space H s (IRn−1 ). This approach is the one which can be found in standard text books in connection with the trace theorem see Aubin [17, p.198ff]. Let us recall the parametric representation (3.3.3) of Γ , namely  x = x(r) + T(r) y  , a(r) (y  ) for y  ∈ Q, r = 1, . . . , p .

176

4. Sobolev Spaces

Then, for s with 0 ≤ s < k + κ or for noninteger k + κ or 0 ≤ s ≤ k + κ for integer k + κ we define here the Sobolev space Hs (Γ ) by the boundary space  {u ∈ L2 (Γ ) | u x(r) + T(r) (y  , a(r) (y  )) ∈ H s (Q), r = 1, . . . , p} equipped with the norm uHs (Γ ) :=

p 

 12  u x(r) + T(r) (y  , a(r) (y  )) 2H s (Q) .

(4.2.24)

r=1

This space is a Hilbert space with the inner product (u, v)Hs (Γ ) :=

p  

u(x(r) + T(r) (y  , a(r) (y  )), v(x(r) + T(r) (y  , a(r) (y  ))

H s (Q)

.

r=1

(4.2.25) Note that the above restrictions of s are necessary since otherwise the differentiations with respect to y  required in (4.2.24) and (4.2.25) may not be well defined. Also note that Hs (Q) is the Sobolev space in the (n − 1)–dimensional domain Q as in Section 4.1. In an additional step, these definitions, (4.2.24) and (4.2.25), can be rewritten in terms of the Sobolev space H s (IRn−1 ) by using the partition of unity on Γ subordinate to the open covering B(r) . More precisely, we take the functions α(r) ∈ C0∞ (IRn ), supp α(r)  B(r) introduced previously, with p  α(r) (x) = 1 (4.2.26) r=1

in some neighbourhood of Γ . For u given on Γ , we define the extended function on IRn−1 by   (α(r) u) x(r) + T(r) (y  , a(r) (y  )) for y  ∈ Q ,  (α (r) u)(y ) := 0 otherwise . n−1 s Then Hs (Γ ) is given by all u on Γ for which α ), r = 1, . . . , p. (r) u ∈ H (IR The corresponding norm now reads

uHs (Γ ) := {

p 

1

2 2 (α (r) u)H s (IRn−1 ) }

(4.2.27)

r=1

and is associated with the scalar product (u, v)Hs (Γ ) :=

p 

(α  (r) u, α (r) v)H s (IRn−1 ) .

(4.2.28)

r=1

Clearly, for 1 ≤ s ≤ k+κ, the integrals in (4.2.24)–(4.2.28) contain derivatives of u on Γ with respect to the local coordinates, i.e. derivatives of

4.2 The Trace Spaces H s (Γ )

177

 u x(r) + T(r) (y  , a(r) (y  )) with respect to yj , j = 1, · · · , n−1. These derivatives are, in fact, the covariant derivatives of u with respect to Γ (see Bishop and Goldberg [32] and Millman and Parker [306]). The particular charts, the pushed forward functions α (r) u are defined on IRn−1 having compact supports in Q and in (4.2.27), (4.2.28) and using H s (IRn−1 ), we can introduce via L2 –duality the whole scale of Sobolev spaces < s < k + κ. for Hs (Γ ) all s with −k − κ ( − ) (−) Up to now, we have introduced three different boundary spaces. The connection between these boundary spaces and the Sobolev spaces on the domain Ω are given through the trace theorem in Ω. To this end, let us introduce the trace operators γj , j = 0, 1, · · · , m − 1, 1 ≤ m ∈ IN, γj : C m−1 (Ω) → C m−1 (Γ ) defined by γj u =

∂j u |Γ := (n · ∇)j u|Γ , ∂nj

j = 0, 1, · · · , m − 1 .

In particular, we have, as before, for u ∈ C o (Ω), γ0 u = u|Γ , and γ0 is a linear operator which acts on a continuous function u ∈ C o (Ω) to produce its restriction to the boundary Γ as a continuous function γ0 u on Γ . What we are really interested in, is the problem of how to define γ0 u when u belongs to L2 (Ω) or, more generally, to one of the Sobolev spaces H m (Ω). As we mentioned earlier, functions belonging to H m (Ω) are only defined uniquely on Ω and not on Ω since Γ is a set of n–dimensional measure zero and functions in H m (Ω) differing on a set of measure zero are regarded as being identical. However, the trace theorem enables us to define unambiguously γ0 u (or more generally γj u), provided that u is smooth enough, e.g. in H 1 (Ω) ( or u ∈ H m (Ω)) and Γ is sufficiently smooth. We now state the Trace Theorem. Theorem 4.2.1. Trace Theorem (Costabel [80], Gagliardo [131], Grisvard [156], Lions [268], Miyazaki [312], Neˇcas [326, Chap.2,5], Ding [102], Wloka [439, p.130], Dindos and Mitrea [101], Mikhailov [297], Mitrea and Wright [310], Wendland [433] and on manifolds Große and Schneider [157]) i. Let Ω ∈ C k,κ where k + κ ≥ 1 and s ∈ IR with 12 < s < max{k + κ, 32 } for non integer k + κ or 12 < s ≤ k + κ for integer k + κ. Then there exists a linear continuous trace operator γ0 with 1

γ0 : H s (Ω) → H s− 2 (Γ ) which is an extension of

(4.2.29)

178

4. Sobolev Spaces

for u ∈ C 0 (Ω) .

γ0 u = u|Γ The adjoint operator 1

γ0∗ : H −s+ 2 (Γ ) → H −s (Rn ) given by

(4.2.30)

(γ0∗ σ, Φ)L2 (Rn ) = σ, γ0 ΦL2 (Γ )

satisfies c0 σ

1

H −s+ 2 (Γ )

≤ γ0∗ σH −s (Rn ) ≤ c1 σ

1

H −s+ 2 (Γ )

(4.2.31)

with some positive constants c0 and c1 (depending on Γ and s). ii. Let Ω ∈ C k,1 and s ∈ IR and j, k ∈ IN0 with 12 + j < s ≤ k + 1 for non integer k + κ or 12 + j < s ≤ k + κ for integer s. Then there exists a linear continuous trace operator γj with 1

γj : H s (Ω) → H s−j− 2 (Γ )

(4.2.32)

which is an extension of γj u =

∂j u |Γ ∂nj

for u ∈ C  (Ω)

with s + j ≤  ∈ IN .

Moreover, for these s the three boundary spaces are equivalent: 1

Hs− 2 (Γ )

1

1

Hs− 2 (Γ )

Hs− 2 (Γ )

(4.2.33)

In view of the trace theorem, for Ω ∈ C k,1 we identify once for all the boundary spaces H s (Γ ) := Hs (Γ ) = Hs (Γ ) = Hs (Γ )

(4.2.34)

for 0 < |s| ≤ k + 12 , take H 0 (Γ ) = L2 (Γ ) and call H s (Γ ) the trace spaces. We note that for 0 < s ≤ k + 12 , the equivalence is due to the trace theorem. For −k − 12 ≤ s < 0 the equivalence follows from the common definition via L2 (Γ )–duality. Since in these cases the boundary spaces are equivalent to the spaces H s (IRn−1 ), we have with the imbedding theorem in IRn−1 also on Γ : Theorem 4.2.2. Let Γ be a C k,1 boundary, k ∈ IN0 and let |t| , |s| ≤ k + 12 . Then the imbeddings c

H s (Γ ) → H t (Γ ) s

c

H (Γ ) → C are compact.

m,α

for t < s

(Γ ) for m + α < s −

(4.2.35) n 2

+

1 2

, m ∈ IN0 , 0 ≤ α < 1 (4.2.36)

4.2 The Trace Spaces H s (Γ )

179

Remarks 4.2.3: i. In case of a Lipschitz domain, i.e. k = 0, κ = 1, we see that for the trace operator γ0 exists with

1 2

< s ≤ 1,

1

γ0 : H s (Ω) → H s− 2 (Γ ) and γ0 u = u|Γ for u ∈ C 0 (Ω). However, as shown by Costabel [80], this even extends to 12 < s < 32 and in (Wendland [433]) for 12 < s < k + 32 , k ∈ N, which indicates that the equivalence (4.2.33) can also be extended for 0 ≤ |s| < k + κ. However, as shown in (Wloka [439, p.130]), for 0 < κ < 1 it seems that one needs the stronger restrictions 12 + j < s < k + κ − 1 for (4.2.32) to hold. ii. For Ω ∈ C k,1 and fixed s with 12 < s ≤ k + 1 let m be the largest integer in IN with m < s + 12 . Here, we may collect the trace operators γj u for j = 0, · · · , m − 1 and define γu := (γ0 u, . . . , γm−1 u) . Then γ : H s (Ω) →

m−1 ,

(4.2.37)

1

H s−j− 2 (Γ )

(4.2.38)

j=0

is continuous and linear and is an extension of  ∂u ∂ m−1 u  γu = u, , . . . m−1 |Γ for u ∈ C m−1 (Ω) . ∂n ∂n The continuity of γ can be expressed through the inequality m−1 

γj u

1

H s−j− 2 (Γ )

≤ cuH s (Ω)

j=0

with a constant c independent of u. 1 iii. The first approach shows that the spaces Hs− 2 (Γ ) are just the quotient spaces 1 Hs− 2 (Γ ) = H s (Ω)/H0s (Ω) . (4.2.39) Here, H0s (Ω) denotes the completion of C0∞ (Ω) in H s (Ω). Hence, due to (4.2.34) we also have with c2 , c1 > 0, c1 u

1

H s− 2 (Γ )

≤ inf vH s (Ω) ≤ c2 u γ0 v=u

1

H s− 2 (Γ )

.

(4.2.40)

p,p iv. The trace theorem for γ0 : Lps (Ω) → B1− 1 with 1 < p < ∞ and 1 p

p

< s < 1 + p1 and Lps - and Besov spaces can be found in (Jerison et al., [216], Mikhailov [297], Mitrea et al. [310]).

180

4. Sobolev Spaces

From the trace theorem, it is clear that the concept of the trace on Γ is a natural generalization of the concept of the restriction on Γ . We shall adopt the convention that the trace of the function u will be denoted by the symbol u|Γ or simply by the same symbol u. If ϕ = ϕ(x) is a function defined on Γ , then the statement: u = ϕ on Γ in the sense of traces will express the fact that γ0 u = ϕ on Γ . In addition to the trace theorem, we also have the inverse trace theorem. Theorem 4.2.3. Inverse Trace Theorem (McLean [290], Ne¸cas [326, p.104], Wloka [439]). For Ω ∈ C k,κ let s be fixed with 12 < s ≤ k + κ. Let m be the largest integer with m < s + 12 . Then there exists a linear bounded (i.e. continuous) right inverse Z to γ with Z:

m−1 ,

1

H s−j− 2 (Γ ) → H s (Ω) and γ(Zϕ) = ϕ

(4.2.41)

j=0

for all ϕ ∈

m−1 ,

1

H s−j− 2 (Γ ) .

j=0

The continuity of Z can be expressed through the inequality ZϕH s (Ω) ≤ c

m−1 

ϕj 

1

H s−j− 2 (Γ )

(4.2.42)

j=0

with c independent of ϕj , the components of ϕ. Remark 4.2.4: The trace theorem for Ω ∈ C k,1 together with the inverse trace theorem implies that the trace operator γ in (4.2.38) is surjective. The kernel of γ is H0s (Ω), dense in H 0 (Ω) = L2 (Ω). The proof of the trace theorem usually is reduced to the case when Ω = IRn+ = {x ∈ IRn | xn > 0} and Γ = IRn−1 = {x ∈ IRn | xn = 0} is its boundary (Aubin [17, p.198 ff.]), which corresponds to our spaces Hs (Γ ). Note that in this case one also has s γH s (Ω) = γHloc (Ω c ) .

Remark 4.2.5: For s > k + 1 and Ω ∈ C k,1 , the equivalence of our three boundary spaces in (4.2.34) is no longer valid since the trace theorem does 1 not hold anymore. However, the definition of the boundary spaces Hs− 2 (Γ ) provides the trace theorem by definition and we therefore define 1

1

H s− 2 (Γ ) := Hs− 2 (Γ ) for s > k + 1 to be the trace spaces.

4.2 The Trace Spaces H s (Γ )

181

1

For these s, we can see that the trace space H s− 2 (Γ ) is in fact the same as the subspace of L2 (Γ ) defined by {u ∈ L2 (Γ ) | There exists (at least) one extension  = u} u  ∈ H s (Ω) with γ0 u

(4.2.43)

equipped with the norm u

1

H s− 2 (Γ )

:= inf  uH s (Ω) ; γ0 u =u

(4.2.44)

since C 0 (Γ ) is dense in L2 (Γ ) and s > 1, we have H s (Ω) ⊂ H 1 (Ω) ∩ C 0 (Ω). 1  ∈ H 2 (Γ ) ⊂ L2 (Γ ). Consequently, Moreover, for any u  ∈ H s (Ω) we have γ0 u every u  ∈ H s (Ω) is admissible in (4.2.43) as well as in (4.2.3).

4.2.1 Trace Spaces for Periodic Functions on a Smooth Curve in IR2 In the special case n = 2 and Γ ∈ C ∞ , the trace spaces can be identified with spaces of periodic functions defined by Fourier series. For simplicity, let Γ be a simple, closed curve. Then Γ admits a global parametric representation Γ : x = x(t) for t ∈ [0, 1] with x(0) = x(1)

(4.2.45)

satisfying

 dx    ≥ γ0 > 0 for all t ∈ IR . dt Clearly, any function on Γ can be identified with a function on [0, 1] and its 1–periodic continuation to the real axis. In particular, x(t) in (4.2.45) can be extended periodically. Then any function f on Γ can be identified with f ◦ x which is 1–periodic. Since Γ ∈ C ∞ , we may use either of the definitions of the trace spaces defined previously. In particular, for s = m ∈ IN0 , the norm of u ∈ Hm (Γ ) in the standard trace spaces is given by (4.2.27) with n = 2; namely p      1 (α(r) u)(β) (t)2 dt 2 . uH m (Γ ) = r=1 0≤β≤mIR

For 1–periodic functions u(t), this norm can be shown to be equivalent to a weighted norm in terms of the Fourier coefficients of the functions. As is well known, any 1–periodic function can be represented in the form u (x(t)) =

+∞ 

u *j e2πijt , t ∈ IR

j=−∞

where u *j are the Fourier coefficients defined by

(4.2.46)

182

4. Sobolev Spaces

1 u *j =

e−2πijt u (x(t)) dt, j ∈ Z .

(4.2.47)

0

By using the Parseval equality, the following Lemma can be established. Lemma 4.2.4. Let u(t) be 1–periodic: Then, for m ∈ IN0 , the norms   12 |* u(k)|2 (k + δk0 )2m uH m (Γ ) and uS m := k∈Z

are equivalent. Proof: i. The definition (4.2.27) implies that u2H m (Γ )

≤c



p 



 (β) 2 |u x(t) | dt,

0≤β≤m r=1 supp(α(r) )

from which by making use of the 1–periodicity of u ◦ x, we arrive at u2H m (Γ )

≤c

1



 (β) 2 |u x(t) | dt .

0≤β≤m 0

An application of the Parseval equality then yields 

1



 (β) 2 |u x(t) | dt =



|2πj|2β |* u(j)|2

0≤β≤m j∈Z

0≤β≤m 0

≤

(2π)2m (m + 1)



|j + δ0j |2m |* u(j)|2

j∈Z

which gives uH m (Γ ) ≤ cuS m . ii. Conversely, again by the Parseval equality, we have 1  u2S m = |* u(0)|2 + |j2π|2m |* u(j)|2 (2π)m j∈Z

 1 =

(u ◦ x)dt



2

=

≤

c

|(u ◦ x)(m) |2 dt 0

0 p  1 

1

+



2 α(r) udt

0 r=1 p  

+

1

p 

α(r) |(u ◦ x)(m) |2 dt

0 r=1



r=1 0≤β≤m IR

|(α(r) u)(β) (t)|2 dt .

4.2 The Trace Spaces H s (Γ )

183

This shows that uS m ≤ cuH m (Γ ) .  In general, for 0 < s ∈ IN, we write s = m + σ with 0 < σ < 1. We now define for periodic functions the Fourier series norm,  1 |* u(k)|2 |k + δk0 |2s } 2 . (4.2.48) uS s := { k∈Z

Following Kufner and Kadlec [250, p.250ff], we can again show the equivalence between the norms  · H s (Γ ) and  · S s . Lemma 4.2.5. Let u(t) be 1–periodic. Then for 0 < s ∈ IN, the norms uH s (Γ ) and uS s are equivalent. Proof: In view of (4.2.27), (4.1.5) with Ω = IR and Lemma 4.2.4, it suffices to prove that the double integral 1 1 I(u) := 0 0

|u(m) (t) − u(m) (τ )|2 dtdτ | sin π(t − τ )|1+2σ

converges if and only if u(m) S σ is bounded. Of course, we may restrict ourselves to m = 0, since the case m ∈ IN reduces to m = 0, for v := u(m) . Using periodicity and interchanging the order of integration we obtain 1 1 I(v)

= 0 0

1 =

|v(t + τ ) − v(t)|2 dtdτ | sin πτ |1+2σ

|w(τ )|2 | sin πτ |−1−2σ dτ

0

where w(τ ) = v(• + τ ) − v(•)L2 (0,1) . With the identity (v(• + τ ) − v(•))*= −* v (j)(1 − e2πijτ ) and the Parseval equality one finds  |* v (j)|2 (sin πjτ )2 . w2 (τ ) = 4 j∈Z

Therefore,

184

4. Sobolev Spaces

I(v)

=

4



1 |* v (j)|2

j∈Z

=

8



0

(sin πjτ )2 dτ | sin πτ |1+2σ

1

2 |* v (j)|2

j∈Z

(sin πjτ )2 dτ . (sin πτ )1+2σ

0

Since for τ ∈ (0, 12 ) we have τ ≤ sin πτ ≤ πτ , 2 it follows that 1

8π −1−2σ

2

1

2

(sin πjτ )2 dτ τ 1+2σ

≤

8

0

(sin πjτ )2 dτ (sin πτ )1+2σ

0 1

2 ≤

8 · 21+2σ

(sin πjτ )2 dτ . τ 1+2σ

0

We now rewrite the last integral on the right-hand side in the form: 1

2

π

(sin πjτ )2 dτ τ 1+2σ

j 2 =

(πj)

2σ

0

sin2 s ds s1+2σ

0 2σ

(πj) cσj ,

=: and we see that

∞ cσ1 ≤ cσj ≤ cσ∞ =

sin2 s ds < ∞ . s1+2σ

0

As a consequence, we find π −1 · 8 · cσ1



|* v (j)|2 |j|2σ

≤ I(v) ≤ 8 · 21+2σ · π 2σ cσ∞



|* v (j)|2 |j|2σ

which shows the desired equivalence of vH σ (Γ ) and uS σ . With u(m) = v and Lemma 4.2.4, the desired equivalence of uH s (Γ ) and uS s for s = m+σ and 0 < σ < 1 follows easily.  For s ≥ 0, the trace space H s (Γ ) can also be considered to be the completion of C ∞ (Γ ) with respect to the Fourier series norm  • S s which, in fact, is generated by the scalar product

4.2 The Trace Spaces H s (Γ )

(u, v)S s :=



|j + δj0 |2s u *(j)* v (j) .

185

(4.2.49)

j∈Z

By applying the Schwarz inequality to the right–hand side of (4.2.49), we immediately obtain the inequality   12    12 |(u, v)| ≤ |j + δj0 |2(s+) |* u(j)|2 |j + δj0 |2(s−) |* v (j)|2 (4.2.50) j∈Z

j∈Z

for any real . In terms of the norms (4.2.48), the last inequality can be written as |(u, v)S s | ≤ uS s+ vS s− (4.2.51) which shows that the spaces H s+ (Γ ) and H s− (Γ ) form a duality pair with respect to the inner product (4.2.49), if these spaces are supplied with the norms  · S s+ and  · S s− , respectively, for 0 ≤  ≤ s. Moreover, the inequality (4.2.50) for any  ∈ IR indicates that the norm of the dual space gives for negative s the norm of the dual space of H −s (Γ ) with respect to the (•, •)S 0 –inner product. Therefore, for negative s, we can also use the norm (4.2.48) for the dual spaces of H −s (Γ ). Corollary 4.2.6. Let u be a 1–periodic distribution. Then, for s ∈ IR, the norms uH s (Γ ) and uS s are equivalent. Remarks 4.2.6: Since for 1–periodic functions on IR, the Fourier series representation coincides with (4.1.22), the Fourier transformation of u on IR, the S s –norm (4.2.48) is identical to the one in terms of Fourier transform (4.1.25). From the explicit expression (4.2.48) of the norm one can obtain elementarily the Sobolev imbedding theorems (4.1.31), (4.1.32) for the trace spaces where n = 1 and Ω is replaced by Γ .

4.2.2 Trace Spaces on Curved Polygons in IR2 Let Γ ∈ C 0,1 be a curved polygon which is composed of N simple C ∞ –arcs Γj , j = 1, · · · , N such that their closures Γ j are C ∞ . The curve Γ j+1 follows Γ j according to the positive orientation. We denote by Zj the vertex being the end point of Γj and the starting point of Γj+1 . s (Γj ) be the standard Sobolev spaces on the pieces For s ∈ IR let H ∼ Γj , j = 1, · · · , N which are defined as follows. Without loss of generality, we assume that for each of the Γj we have a parametric representation x = x(j) (t) for t ∈ Ω j = [aj , bj ] ⊂ IR with x(j) (aj ) = Zj−1 , x(j) (bj ) = Zj , Ωj = (aj , bj ), j = 1, · · · , N where x(j) ∈ C ∞ (Ω). Then, we define the space

186

4. Sobolev Spaces

 s H (Γj ) = {ϕ | ϕ x(j) (•) ∈ H s (Ωj )} ∼ to be equipped with the norm ϕH s := ϕ ◦ x(j) H s (Ωj ) , (Γ ) ∼ j s

where  · H s (Ωj ) is defined as in Section 4.1. Then H (Γj ) is a Hilbert space ∼ with inner product (ϕ, ψ)H s = (ϕ ◦ x(j) , ψ ◦ x(j) )H s (Ωj ) . (Γ ) ∼ j

(4.2.52)

From the formulation of the trace theorem, for s = m + σ with m ∈ IN and |σ| < 12 , we now introduce the boundary spaces 1

Tm+σ− 2 (Γ ) :=

N m−1 , , =0 j=1

H ∼

m−j+σ− 12

(Γ ).

We note that the restriction of a smooth function in IR2 to the curve Γ automatically satisfies compatibility conditions of the derivatives at each vertex 1 zj , and this is not the case generally for functions in Tm+σ− 2 (Γ ). Here we see that m−1 , 1 1 Tm+σ− 2 (Γ ) = Hm−j+σ− 2 (Γ ), j=0

since the latter, i.e. the natural trace space, does implicitly include the appropriate compatibility conditions by construction. The compatibility conditions can be found in Grisvard’s book [156]. These are the following conditions: Let Pα =



α

aα D =

|α|≤d

α 

P,j

j=0

∂j |Γ ∂nj 

be any partial differential operator in IR2 of order d having constant coefficients aα where the sums on the right–hand side correspond to the local decomposition of Pd in terms of the normal derivatives and the tangential differential operators P,j of orders d − j along Γ for  = 1, · · · , N . Further let Pd denote the class of all these operators of order d. Then the compatibility conditions read for all Pd ∈ Pd , d

(C1 )

d

P,j f,j (z ) = j=0

P+1,j f+1,j (z ) j=0

for d ≤ m − 2 if σ ≤ 0 or d ≤ m − 1 if σ > 0,

4.2 The Trace Spaces H s (Γ )

δ

(C2 )

0

|

d

P,j f,j (x() (t)) − P+1,j f+1,j (x+1 (t)) |2

j=0

187

dt 0. Here, the functions f,j ∈ H ∼

m−j+σ− 12

(Γ ) for j = 0, . . . , m − 1 . 1

By requiring the functions in Tm+σ− 2 (Γ ) to satisfy the compatibility conditions we now have the following version of the trace theorem. Theorem 4.2.7. Trace theorem Grisvard [156, Theorem 1.5-2-8]: Let 1

1

P m− 2 +σ (Γ ) := {f ∈ Tm−σ− 2 (Γ )|f = (f0 , · · · , fm−1 ) ∧ fj|Γ =: f,j } satisfy (C1 ) and for σ = 0 also (C2 ). Then m−1 ,

1

1

Hm−j− 2 +σ (Γ ) = P m− 2 +σ (Γ ).

j=0

Alternatively, the mapping u −→ γu|Γ = (u,

∂um−1 ∂u , · · · , m−1 )|Γ ∂n ∂n 1

is a linear continuous and surjective mapping from H m+σ (Ω) onto P m− 2 +σ (Γ ). In order to illustrate the idea of the compatibility conditions (C1 ) and (C2 ), we now consider the special example for the polygonal region consisting of only two pieces Γ1 and Γ2 at the vertex z as shown in Figure 4.2.2. We consider two cases m = 1 and m = 2: 1. m = 1, σ > 0. Then d = m − 1 = 0, Pd = P0 = {f0 = a0 D0 | a0 ∈ IR} and P,0 = a0 ,  = 1, 2. The condition (C1 ) takes the form P1,0 f1,0 (z1 ) = P2,0 f2,0 (z1 ) which reduces simply to the condition f1,0 (z1 ) = f2,0 (z1 ). 2. m = 2, σ > 0. Then d ≥ m − 1 = 1 and Pd = P1 = {P1 = a0 D0 + a10 ∂1 + a01 ∂2 | a0 , a10 , a01 ∈ IR}. The differential operator P1 restricted to Γ1 can be written in terms of tangential and normal derivatives,

188

4. Sobolev Spaces

P1 | Γ 1

= = =

  ∂ ∂ ∂  ∂  − sin α + a01 sin α + cos α a0 + a10 cos α ∂s ∂n ∂s ∂n ∂ ∂ + (−a10 sin α + a01 cos α) a0 + (a10 cos α + a01 sin α) ∂s ∂n ∂ P1,0 + P1,1 |Γ1 , ∂n

where P1,j , j = 0, 1 are the differential operators tangential to Γ1 , explicitly given by P1,0

=

a0 + (a10 cos α + a01 sin α)

P1,1

=

(−a10 sin α + a01 cos α) .

∂ |Γ , ∂s 1

The differential operator P1 restricted to Γ2 is P1 |Γ2 = a0 + a10

∂ ∂ ∂ |Γ − a01 |Γ2 = P2,0 + P2,1 |Γ2 ∂s 2 ∂n ∂n

with P2,0

=

a0 + a10

P2,1

=

−a01 .

∂ |Γ , ∂s 2

The compatibility condition (C1 ) now reads P1,0 f1,0 (z1 ) + P1,1 f1,1 (z1 ) = P2,0 f2,0 (z1 ) + P2,1 f2,1 (z1 ) which reduces explicitly to the relation: a0 (f1,0 (z1 ) − f2,0 (z1 ))   ∂ ∂ f2,0 (z1 ) +a10 cos α f1,0 (z1 ) − sin αf1,1 (z1 ) − ∂s ∂s   ∂f1,0 (z1 ) + cos αf1,1 (z1 ) + f2,1 (z1 ) = 0. +a01 sin α ∂s Since a0 , a10 , a01 are arbitrary, this yields three linearly independent conditions: f1,0 (z1 ) − f2,0 (z1 ) ∂f1,0 ∂f2,0 cos α (Z1 ) − sin αf1,1 (z1 ) − (z1 ) ∂s ∂s ∂f1,0 (z1 ) + cos αf1,1 (z1 ) + f2,1 (z1 ) sin α ∂s

=

0,

=

0,

=

0.

4.3 The Trace Spaces on an Open Surface

189

Γ1

α Z1

Γ2

Fig. 4.1. Polygonal vertex

4.3 The Trace Spaces on an Open Surface •

In some applications we need trace spaces on an open connected part Γ0 ⊂ Γ of a closed surface Γ . To simplify the presentation let us assume Γ ∈ C k,1 • with k ∈ IN0 . In the two–dimensional case Γ0 ⊂ Γ = ∂Ω with Ω ⊂ IR2 , the boundary of Γ0 denoted by γ = ∂Γ0 consists just of two endpoints γ = {z1 , z2 }. In the three–dimensional case let us assume that the boundary ∂Γ0 of Γ0 is a closed curve γ. In what follows we assume s satisfying |s| ≤ k + 1. In this case, all the trace spaces H s (Γ ) coincide. Similar to Section 4.1 let us introduce the spaces of trivial extensions from Γ 0 to Γ of functions u defined . Then we define on Γ 0 by zero outside of Γ 0 which will be denoted by u   s  s (Γ0 ) := {u ∈ H s (Γ ) u  H |Γ \Γ 0 = 0} = {u ∈ H (Γ ) supp u ⊂ Γ 0 } (4.3.1) as a subspace of H s (Γ ) equipped with the corresponding norm uH s (Γ0 ) =  uH s (Γ ) .  s (Γ0 ) ⊂ H s (Γ ). By definition, H For s ≥ 0 we also introduce the spaces  H s (Γ0 ) = {u = v|Γ0  v ∈ H s (Γ )}

(4.3.2)

(4.3.3)

equipped with the norm uH s (Γ0 ) =  s (Γ0 ) ⊂ H s (Γ0 ). Clearly, H

inf

v∈H s (Γ )∧v|Γ0 =u

vH s (Γ ) .

(4.3.4)

190

4. Sobolev Spaces

For s < 0, the dual spaces H s (Γ0 ) with respect to the L2 (Γ0 )–inner product are well defined by the completion of L2 (Γ0 ) with respect to the norm  uH s (Γ0 ) := sup | ϕ¯ uds| /ϕH s (Γ0 ) . (4.3.5)  −s (Γ0 ) 0=ϕ∈H

Γ0

 s (Γ0 ) with the norm Correspondingly, we also have H  uH s (Γ0 ) := sup | ϕuds | /ϕH −s (Γ ) . 0=ϕ∈H −s (Γ0 )

(4.3.6)

Γ0

The following lemma shows the relations between these spaces in full analogy to (4.1.11) and (4.1.13). Lemma 4.3.1. Let Γ ∈ C k,1 , k ∈ IN0 . For |s| ≤ k + 1 we have (H s (Γ0 )) and



   s (Γ0 ) H

=

 −s (Γ0 ) H

(4.3.7)

=

H −s (Γ0 ) .

(4.3.8)

We also have the inclusions for s > 0:  s (Γ0 ) ⊂ H s (Γ0 ) ⊂ L2 (Γ0 ) ⊂ H  −s (Γ0 ) ⊂ H −s (Γ0 ) . H

(4.3.9)

s (Ω) in Section 4.1, the inclusions in (4.3.9) are strict Similar to the spaces H00 1 for s ≥ 2 and  s (Γ0 ) = H s (Γ0 ) for |s| < 1 . H 2  s (Γ0 ) satisfies u| = 0. Hence, we can introduce For s > 12 , we note that u ∈ H γ s  s (Γ0 ) with respect to the norm the space H0 (Γ0 ) as the completion of H  s (Γ0 ) can also be defined by  · H s (Γ ) . Then the spaces H 0

  s (Γ0 ) = H

H0s (Γ0 )

for s = m + 12 , m ∈ IN0 ,

s H00 (Γ0 )

for s = m + 12 , m ∈ IN0 . m+ 12

Here, as in Section 4.1, the space H00 m+ 12

H00

m+ 12

(Γ0 ) = {u | u ∈ H0

(4.3.10)

(Γ0 ) is defined by 1

(Γ0 ), − 2 Dα u ∈ L2 (Γ0 ) for |α| = m} ,

where  = dist (x, γ) for x ∈ Γ0 , and Dα stands for the covariant derivatives in Γ (see e.g. Berger and Gostiaux [30]). It is worth pointing out that again 1

1

 m+ 12 (Γ0 ) = H m+ 2 (Γ0 ) = H m+ 2 (Γ0 ) . H 00 0

(4.3.11)

4.4 Weighted Sobolev Spaces

191

4.4 The Weighted Sobolev Spaces H m (Ω c ; λ) and H m (IRn ; λ) In applications one often deals with exterior boundary-value problems. In order to ensure the uniqueness of the solutions to the problems, appropriate growth conditions at infinity must be incorporated into the solution spaces. For this purpose, a class of weighted Sobolev spaces in the exterior domain Ω c := IRn \ Ω or in the whole IRn is often used. For simplicity we shall confine ourselves only to the case of the weighted Sobolev spaces of order m ∈ IN. We begin with the subspace of C m (Ω c ),  C m (Ω c ; λ) := {u ∈ C m (Ω c )uH m (Ω c ;λ) < ∞} , where λ ∈ IN0 is given. Here,  · H m (Ω c ;λ) is the weighted norm defined by   α 2 −(m−|α|−λ) −1 uH m (Ω c ;λ) := 0 D uL2 (Ω c ) 0≤|α|≤κ

+



−(m−|α|−λ) Dα u2L2 (Ω c )

 12

(4.4.1) ,

κ+1≤|α|≤m

where  = (|x|) and 0 = 0 (|x|) are the weight functions defined by 1

(|x|) = (1 + |x|2 ) 2 , 0 (|x|) = log(2 + |x|2 ), x ∈ IRn , and the index κ in (4.4.1) is chosen depending on n and λ such that  m − ( n2 + λ) if n2 + λ ∈ {1, . . . , m} , κ= −1 otherwise . The weighted Sobolev space H m (Ω c ; λ) is the completion of C m (Ω c ; λ) with respect to the norm  · H m (Ω c ;λ) . Again, H m (Ω c ; λ) is a Hilbert space with the inner product  α −(m−|α|−λ) −1 α (−(m−|α|−λ) −1 0 D v)L2 (Ω c ) (u, v)H m (Ω c ;λ) := 0 D u,  0≤|α|≤κ



+

(−(m−|α|−λ) Dα u, −(m−|α|−λ) Dα v)L2 (Ω c ) . (4.4.2)

κ+1≤|α|≤m

In a complete analogy to H0m (Ω), we may define H0m (Ω c ; λ) = C0m (Ω c ; λ) where

C0m (Ω c ; λ)

 · H m (Ω c ;λ)

(4.4.3)

is the subspace

 C0m (Ω c ; λ) = {u ∈ C0m (Ω c )  uH m (Ω c ;λ) < ∞} .

Then the following result can be shown (see Nedelec [327, Lemma 2.2], [328, p.16ff.])

192

4. Sobolev Spaces

Lemma 4.4.1. (a) The semi–norm |u|H m (Ω c ;λ) defined by |u|H m (Ω c ;λ) :=

 

λ Dα u2L2 (Ω c )

 12

(4.4.4)

|α|=m

is a norm on H0m (Ω c ; λ), and is equivalent to the norm uH m (Ω c ;λ) on H0m (Ω c ; λ). (b) In general, |u|H m (Ω c ;λ) is a norm on the quotient space H m (Ω c ; λ)/Pη , and is equivalent to the quotient norm u ˙ H m (Ω c ;λ) := inf u + pH m (Ω c ;λ) p∈Pη

(4.4.5)

for any representative u˙ in the coset. Here Pη is the set of all polynomials of degree less than or equal to η = min{η  , m − 1} where η  is the non–negative integer defined by  m − n2 − λ for n2 + λ ∈ {0, 1, · · · , m} ,  η = otherwise . [m − n2 − λ] We remark that the definition of H m (Ω c ; λ) carries over to cases when Ω is the entire IRn , in which case we have the weighted Sobolev space H m (IRn ; λ). It is also worth pointing out that the space H m (Ω c ; λ) is contained in the space c

m Hloc (Ω c ) := {u ∈ H m (K) |

for every K  Ω c } ,

(4.4.6)

the latter can also be characterized by the statement m (Ω c ) iff ϕu ∈ H m (Ω c ) for every ϕ ∈ C0∞ (IRn ). u ∈ Hloc

Consequently, the trace theorem can also be applied to ϕu and, hence, to all these spaces. To conclude this section, we now consider two special cases for H m (Ω c ; λ). For n = 2, we let m = 1 and λ = 0. Then we see that the weighted Sobolev space H 1 (Ω c ; 0) is equipped with the norm  1 Dα u2L2 (Ω c ) } 2 , (4.4.7) uH 1 (Ω c ;0) := {ρ−1 −1 0 uL2 (Ω c ) + |α|=1

and the corresponding inner product is −1 −1 0 v)L2 (Ω c ) + (u, v)H 1 (Ω c ;0) = (−1 −1 0 u, 



(Dα u, Dα v)L2 (Ω c ) . (4.4.8)

|α|=1

Similarly, for n = 3, λ = 0, m = 1, the norm · H 1 (Ω c :0) is given by

4.4 Weighted Sobolev Spaces



uH 1 (Ω c ;0) := {−1 u2L2 (Ω 2 ) +

1

Dα u2L2 (Ω c ) } 2

193

(4.4.9)

|α|=1

and the associated inner product is 

(u, v)H 1 (Ω c ;0) = (−1 u, −1 v)L2 (Ω c ) +

(Dα u, Dα v)L2 (Ω c ) .

(4.4.10)

|α|=1

As will be seen, these are most frequently used weighted Sobolev spaces which are connected with the energy spaces of potential functions for the two– and three–dimensional Laplacian, respectively. An Exterior Equivalent Norm In analogy to the interior Friedrichs inequality (4.1.40) we now give the corresponding version for m = 1 and the weighted Sobolev space H 1 (Ω c ; 0). Lemma 4.4.2. Nedelec [328] Let Ω c be an unbounded domain with Lipschitz boundary. Then the norms  1 uH 1 (Ω c ; 0) and {∇u2L2 (Ω c ) + |v|2 ds} 2 are equivalent . (4.4.11) Γ

Moreover, the exterior Friedrichs–Poincar´e inequality   u2H 1 (Ω c ; 0) ≤ c ∇u2L2 (Ω c ) + u2H 1 (Ω c )

(4.4.12)

R0

c is valid where ΩR := {x ∈ Ω c | |x| < R0 } and R0 > 0 is chosen large enough 0 to guarantee Ω ⊂ {x ∈ IRn | |x| < R0 }.

Proof: Let ϕ ∈ C ∞ (IRn ) be a cut–off function with 0 ≤ ϕ(x) ≤ 1 and ϕ(x) = 0 for |x| ≤ R0 , ϕ(x) = 1 for |x| ≥ 2R0 . c Let BR := {x ∈ IRn | |x| > R0 } . Then 0

uH 1 (Ω c ; 0) ≤ ϕuH 1 (BRc Since ϕu ∈ (4.4.1),

H01 (Ω c

0

; 0)

c + cuH 1 (Ω2R ). 0

; 0), we can apply Lemma 4.4.1 part (a) and obtain with

uH 1 (Ω c ; 0) ≤ c∇(ϕu)L2 (BRc

0

)

+ cu2H 1 (Ω c

2R0 )

.

Because of the properties of ϕ the product rule implies u2H 1 (Ω c ; 0) ≤ c1 ∇u2L2 (Ω c ) + c2 u2H 1 (Ω c

2R0 )

c , the norms {∇u2L2 (Ω c ) +u2H 1 (Ω c In Ω2R 0

2R0 )

are equivalent due to (4.1.40) which yields

.

 1 1 } 2 and {∇uL2 (Ω c ) + |u|2 ds} 2 Γ

194

4. Sobolev Spaces

u2H 1 (Ω c ; 0)

≤c





∇u2L2 (Ω c )

+

 |u|2 ds ;

(4.4.13)

Γ

and with the trace theorem, (4.2.32) for ψu with ψ(x) = 1 − ϕ we find  |u|2 ds ≤ ψuH 1 (Ω c ; 0) ≤ cuH 1 (Ω c ; 0) , Γ



i. e.

|u|2 ds ≤ cu2H 1 (Ω c ; 0) .

∇u2L2 (Ω c ) + Γ

Hence, uH 1 (Ω c ; 0) and (4.4.11) are equivalent. The inequality (4.4.12) follows from (4.4.13) with the trace theorem, c .  (4.2.32) for ΩR 0

4.5 Function Spaces H( div , Ω) and H( curl , Ω) In this section, we will give a brief summary of some vector function spaces and related topics which are needed for treating Maxwell’s equations in Chapters 6 and 12. For a detailed description of the subject, we refer to the fundamental papers Buffa and Ciarlet [49] and [50] for Lipschitz polyhedral domains and the seminal paper Buffa, Costabel and Sheen [52] in the case of general Lipschitz boundaries (see also Claeys and Hiptmair [71]). For smooth domains, most of relevant results are available in the monographs Cessenat [63] and Dautray and Lions [96]. Unless otherwise specified, through out the section, we assume that Ω ⊂ R3 is a bounded domain with a Lipschitz–continuous boundary Γ and a unit outward normal to Ω defined almost everywhere on Γ = ∂Ω. We begin with two basic function spaces for Maxwell’s equations: H(div , Ω) := {u ∈ L2 (Ω) := L2 (Ω)3 | div u ∈ L2 (Ω)}

(4.5.1)

endowed with the norm 1/2  . u||H(div ,Ω) := u2L2 (Ω) + div u2L2 (Ω) and H(curl , Ω) := {u ∈ L2 (Ω)| curl u ∈ L2 (Ω)}

(4.5.2)

with the norm 1/2  , uH(curl ,Ω) := u2L2 (Ω) + curl u||2L2 (Ω) Clearly, both H(div , Ω) and H(curl , Ω) are Hilbert spaces. But note that their imbeddings into L2 (Ω) are not compact as was shown by Murat [318]. Consequently, the intersection space

4.5 Function Spaces H( div , Ω) and H( curl , Ω)

X(Ω) := H(div, Ω) ∩ H(curl , Ω) → L2 (Ω)3

195

(4.5.3)

with the norm 1

uX(Ω) := {u2H(div,Ω)+ u2H(curl ,Ω) } 2 is in L2 (Ω)3 not compactly imbedded. However, the subspace H10 (Ω) = {u ∈ H 1 (Ω)3 with γ0 u = 0 on Γ } is compactly imbedded into L2 (Ω). And in Amrouche, Ciarlet and Mardare [11] it is shown that the Poincar´e inequality (4.1.39) is equivalent to the right–invertibility of the div–operator on H10 (Ω). Lemma 4.5.1. The mapping  divH10 (Ω) → L20 (Ω) = {f ∈ L2 (Ω) with

f dx = 0}

(4.5.4)

Ω

is onto. Consequently, for each f ∈ L2 Ω) there exists a unique element uf satisfying (uf , v 0 )L2 (Ω) = 0 for all v 0 ∈ Ker(div). If Ω is starlike with respect to an open ball B ⊂ Ω then  with a cut–off function ψ ∈ C0∞ (R3 ) with supp ψ ⊂ B , ψ(x) ≥ 0 and ψ(x)dx = 1, the Bogovskii operator [35]  B f (x) :=

f (y) Ω

x−y  |x − y|3

∞

|x−y|

 x−y  2  r dr dy ψ y+r |x − y|

(4.5.5)

defines a right–inverse to the div–operator on L20 (see Costabel and McIntosh [86]). Since a Lipschitz domain Ω has the strong cone property the bounded Lipschitz domain Ω ⊂ R3 can be decomposed into a finite number of subdomains where each of the subdomains is starlike to some ball and on each of the subdomains the corresponding Bogovskii operators provide a local right– inverse to the div–operator. In particular, we are interested in the traces of functions in the spaces H(div, Ω) and H(curl , Ω) because of boundary integral formulations of solutions of Maxwell equations. We begin with u ∈ H(div , Ω) and ϕ ∈ H 1 (Ω). Then from generalized Green’s formula, we see that  (u, grad ϕ) + (div u, ϕ) = (u|Γ · n)ϕ|Γ dΓ (4.5.6) Γ =: (u · n)|Γ , ϕ|Γ 

196

4. Sobolev Spaces

where ·, · denotes the duality pairing between H −1/2 (Γ ) and H 1/2 (Γ ) (see McLean [290, Chapt. 3]). This means we may define a normal trace operator γn : H(div , Ω) −→ H −1/2 (Γ ) , u → γn (u) := u|Γ · n.

(4.5.7)

Indeed, based on the density argument, the following theorem is available in Dautray and Lions [96] and Girault and Raviart [147]. Theorem 4.5.2. (Trace Theorem for H(div , Ω)). Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary Γ . Then (a) The space C ∞ (Ω)3 is dense in H(div , Ω). (b) The normal trace map γn : u → u|Γ · n defined on C ∞ (Ω)3 extends by continuity to a continuous linear map — still denoted by γn — from H(div , Ω) onto H −1/2 (Γ ). (c) The Green’s formula (4.5.6) holds for functions u ∈ H(div , Ω) and ϕ ∈ H 1/2 (Γ ). (d) The kernel Ker (γn ) of this mapping γn is the space   H0 (div , Ω) := u ∈ H(div , Ω) : u · n|Γ = 0 . (4.5.8) On the other hand, for the space H(curl , Ω) two tangential traces of functions are involved. We collect some of basic results, give here an overview of their developments, and point out the corresponding relevant mathematical ingredients employed. In the following, let   L2t (Γ ) := μ ∈ L2 (Γ ) : μ · n = 0 . We define the tangential trace operator: γt : C ∞ (Ω)3 −→ L2t (Γ ) , u → γt u := n × u|Γ

(4.5.9)

and the tangential projection operator: πt : C ∞ (Ω)3 −→ L2t (Γ ) : u → πt (u) := −n × (n × u|Γ ) = u|Γ − (u|Γ · n)n.

(4.5.10)

Note that various notations have been adopted for the tangential trace operator γt and for the tangential projection operator πt ; these notions are not uniform. Geometrically, γt u is tangent to Γ , while πt u is the projection onto the tangent plane to Γ of the restriction u|Γ of u to Γ (i.e. onto the tangent bundle). These traces are rotated ninety degrees about the normal vector field with respect to each other. Remark 4.5.1: Let γ0 be the standard trace operator such that γ0 : H1 (Ω) := H 1 (Ω)3 −→ H1/2 (Γ ) := H 1/2 (Γ )3 , γ0 : u → γ0 (u) = u|Γ and with the adjoint  1  −1 (Ω) : σ → γ ∗ (σ) = σ ⊗ δΓ . γ0∗ : H− 2 (Γ ) → H 0

4.5 Function Spaces H( div , Ω) and H( curl , Ω)

197

Since γ0 is surjective, let Z be one of its right inverses. Then the composite operator πt ◦ Z (resp. γt ◦ Z ) may also be used to act only on the trace functions (see, e.g Buffa, Costabel and Sheen [52]). By density of C ∞ (Ω)3 |Γ into L2 (Γ ), the operators πt and γt can be extended to linear continuous operators in L2 (Γ ). Tangential trace spaces for H1 (Ω) A simple integration by parts shows that  (u, curl v)L2 (Ω) − (curl u, v)L2 (Ω) = (n × u) · v dΓ

(4.5.11)

Γ

= γt u, πt vt ,

for all u, v ∈ H1 (Ω),

where the bracket in the right hand side of (4.5.11) is the inner product of L2t (Γ ). This motives us to define the spaces   1/2 H⊥ (Γ ) := γt (H1 (Ω)) = ϕ ∈ L2t (Γ ) : ϕ = γt u , u ∈ H1 (Ω) , (4.5.12)   1/2 H (Γ ) := πt (H1 (Ω)) = ϕ ∈ L2t (Γ ) : ϕ = πt u , u ∈ H1 (Ω) . (4.5.13) These spaces are dense in L2t (Γ ) and they are different for non–smooth boundaries. The spaces are equipped with the norms:   ϕH1/2 (Γ ) := inf uH1 (Ω)) : γt u = ϕ ϕ and ⊥   ϕH1/2 (Γ ) := inf uH1 (Ω)) : πt u = ϕ , ϕ ||

1/2

which ensure the mappings γt : H1 (Ω)) → H⊥ (Γ ) and πt : H1 (Ω)) → 1/2 H|| (Γ ) are bounded and surjective. We may also define their dual spaces : −1/2

H⊥

(Γ ) := H⊥ (Γ ) , 1/2

−1/2

and H||

(Γ ) := H (Γ ) , 1/2

(4.5.14)

equipped, respectively with the norms: λH−1/2 (Γ ) := λ ⊥

λ , ϕ t | |λ , ϕ ϕ  1/2 1/2 H (Γ ) 0 =ϕ ∈H (Γ ) sup

⊥

λH−1/2 (Γ ) λ ||

⊥

λ , ϕ t | |λ := sup . ϕH1/2 (Γ ) ϕ 1/2 0 =ϕ ∈H (Γ ) ||

||

Here the bracket  , t denotes the two possible extensions of the L2t (Γ ) inner −1/2 1/2 −1/2 1/2 product to the duality pairings H⊥ (Γ )×H⊥ (Γ ) and H|| (Γ )×H|| (Γ ). In order to see their extensions to the tangential traces for the space H(curl , Ω), we need some informations concerning various differential operators defined on the surface Γ .

198

4. Sobolev Spaces

Tangential differential operators (see Buffa and Ciarlet [49, 50], Buffa, Costabel and Sheen [52] ) In the following we need differential operators grad Γ and curl Γ as well as their corresponding adjoint operators defined on the surface Γ , that is, a closed Lipschitz surface without boundary. For definiteness, we define the operators : grad Γ := (πt grad ) ◦ Z −1 : H 1 (Γ ) −→ L2t (Γ ) and

(4.5.15)

−1

(4.5.16)

curl Γ := −(γt grad ◦ Z

1

: H (Γ ) :−→

L2t (Γ ).

The adjoint operators of −grad Γ and curl Γ are respectively, divΓ : L2t (Γ ) → H∗−1 (Γ ),

and curlΓ : L2t (Γ ) → H∗−1 (Γ ),

(4.5.17)

which are defined by the following duality pairings :  1 0 divΓ λ , ϕ = − λ · grad Γ ϕ dΓ and 0



1

Γ

λ · curl Γ ϕ dΓ

curlΓ λ, ϕ = Γ

for all ϕ ∈ H 1 (Γ ) and for all λ ∈ L2t (Γ ), i.e. div∗Γ = −grad Γ and curl∗Γ = curl Γ . In the equation (4.5.17) we denote H∗−1 (Γ )



= ϕ∈H

−1

 (Γ )|

 ϕdΓ = 0 .

Γ

The operators grad Γ and curl Γ can be restricted to more regular spaces. If H 3/2 (Γ ) = γ0 (H 2 (Ω)), and H −3/2 (Γ ) = H 3/2 (Γ ) , then we see that 1/2

grad Γ : H 3/2 (Γ ) −→ H|| (Γ ),

1/2

and curl Γ : H 3/2 (Γ ) −→ H⊥ (Γ ). (4.5.18)

Thus, by L2t (Γ ) duality −1/2

(Γ ) −→ H −3/2 (Γ ),

−1/2

(Γ ) −→ H −3/2 (Γ ). (4.5.19) Now in order to define suitable extensions of the operators πt and γt , we make use of the Green formula: For all u ∈ H( curl , Ω) and for all v ∈ H1 (Ω), we see that    0 1 (curl u) · v − u · curl v dΩ = γt (u), πt (v) t (4.5.20) Ω 0 1 = − πt (u), γt (v) t . div Γ : H||

and curlΓ : H⊥

4.5 Function Spaces H( div , Ω) and H( curl , Ω) 1/2

199 1/2

Since the operators πt : H1 (Ω) −→ H|| (Γ ) and γt : H1 (Ω) −→ H⊥ (Γ ) are surjective, based on (4.5.20) and the closed graph theorem (see Bishop [32] and Ammari [8], Ammari and Nedelec [9]), we obtain that the operators −1/2

γt : H( curl , Ω) −→ H

−1/2

(Γ ) and πt : H( curl , Ω) −→ H⊥

(Γ ) (4.5.21) are linear and continuous. We are now in a position to describe the ranges of these operators. By making use of (4.5.20), this shows that for all u ∈ −1/2 H( curl , Ω) there holds γt (u) ∈ H (Γ ) and  divΓ γt (u) = −(curl u) · n ∈ H −1/2 (Γ ),   divΓ γt (u) H −1/2 (Γ ) ≤ cuH( curl ,Ω) ,

(4.5.22)

(curl u) · n ∈ H −1/2 (Γ ) since curl u ∈ H(div, Ω). From (4.5.20) again, we −1/2 obtain similarly for all u ∈ H( curl , Ω) that πt (u) ∈ H⊥ (Γ ) and   curlΓ πt (u) = −divΓ n × πt (u) = −divΓ (n × u),   curlΓ πt (u) = curl u · n ∈ H −1/2 (Γ ),   (4.5.23)  curlΓ πt (u) H −1/2 (Γ ) ≤ cuH( curl ,Ω) . −1/2

Thus, we arrive at two trace spaces, namely H||

(divΓ , Γ ) and

−1/2 H⊥ (curlΓ

, Γ ) for the space H( curl , Ω) in the following main theorem due to Buffa, Costabel and Sheen in [52]. We state the theorem and refer details of the proof to [52]. Theorem 4.5.3. (Trace Theorem for H (curl , Ω )) Let   −1/2 −1/2 λ) ∈ H −1/2 (Γ ) , H|| (divΓ , Γ ) : = λ ∈ H|| (Γ ) | divΓ (λ

(4.5.24)

and −1/2

H⊥

  −1/2 λ) ∈ H−1/2 (Γ ) , (curlΓ , Γ ) := λ ∈ H⊥ (Γ )| curlΓ (λ

(4.5.25)

be the trace spaces for H( curl , Ω) equipped with graph norms: λH−1/2 (div ,Γ ) := λ λH−1/2 (Γ ) + divΓ λ H −1/2 (Γ ) , λ Γ ||

λH−1/2 (Γ ) +  curlΓ λ H−1/2 (Γ ) , λH−1/2 ( curl ,Γ ) := λ λ Γ ⊥

and

(4.5.26)

||

⊥

(4.5.27)

respectively, Then the operators −1/2

γt : H( curl , Ω) → H||

−1/2

πt : H( curl , Ω) → H⊥

(divΓ , Γ ) and ( curlΓ , Γ )

(4.5.28)

200

4. Sobolev Spaces

are linear, continuous, surjective and possess continuous right inverses. Moreover, the L2t (Γ ) inner product can be extended to define a duality pairing  ,  −1/2 −1/2 between the spaces H|| (divΓ , Γ ) and H⊥ (curlΓ , Γ ). Remark 4.5.2: In Theorem 4.5.3, we have replaced the original notations −1/2 Vπ and Vγ adopted in Buffa, Costabel and Sheen [52] by H|| (Γ ) and −1/2

H⊥

(Γ ).

As a consequence of the theorem, Green’s formula (4.5.20) can be extended to functions u, v ∈ H( curl , Ω), if the boundary integral of the right–hand side is interpreted as γt (u), πt (v)t , that is, the duality pair−1/2 −1/2 ing between H⊥ ( curl Γ , Γ ) and H|| (divΓ , Γ ) with respect to the pivot space index duality pairing L2t (Γ ). The extension of L2t (Γ ) dualities to the  −1/2 −1/2 pair H (divΓ , Γ ) , H⊥ (curlΓ , Γ ) is a surprising and deep result since  we were used to dualities for pairs of Sobolev spaces like Hs (Γ ), H−s (Γ )) with 0 ≤ s ≤ 1. We follow Buffa and Ciarlet [49, 50] and Buffa, Costabel and Sheen [52] to proceed; first based on a Hodge decomposition of functions H( curl , Ω), and then based on Hodge decompositions of trace spaces −1/2 −1/2 H|| (divΓ , Γ ) and H⊥ ( curlΓ , Γ ). Hodge decomposition of vector fields We recall that there exists the decomposition: H( curl , Ω) = H1 (Ω) + grad H 1 (Ω) (see, e.g. Amrouche, Bernardi, Dauge and Girault [10], Buffa and Ciarlet [49], [50]). Then for u, v ∈ H( curl , Ω), we assume that u = φ + grad p,

and v = ψ + grad q

(4.5.29)

where φ , ψ ∈ H1 (Ω) and p, q ∈ H 1 (Ω). We obtain       ( curl u) · v − u · curl v dΩ = ( curl φ ) · ψ − φ · curl ψ dΩ Ω

Ω





curl φ · grad q dΩ −

+ Ω

grad p · curl ψ dΩ. Ω

Applying formula (4.5.20) three times yields    ( curl u) · v − u · curl v dΩ = γt (u), πt (v)t , Ω

where the boundary term can be defined as

(4.5.30)

4.5 Function Spaces H( div , Ω) and H( curl , Ω)

0

γt (u), πt (v)

1 t

  =

201

 0 1 φ) · πt (ψ ψ ) d Γ + grad Γ q, γt (φ φ) γt (φ

Γ

0 1 ψ) − curl Γ p, πt (ψ    0 1 φ) · πt (ψ ψ ) d Γ − divΓ (γt (φ φ)), γ0 q = γt (φ Γ

0 1 ψ ), γp − curlΓ πt (ψ

(4.5.31)

Hence, the integration by parts formula (4.5.30) holds for any fields u, v ∈ H( curl , Ω) and from the surjectivity of the trace operators in Theorem 4.5.3, it is then possible to define the following duality for any u, v ∈ H( curl , Ω): 0 1 0 1 φ), πt (ψ ψ ) L2 (Γ ) γt (u), πt (v) t = γt (φ t (4.5.32) 0 1 0 1 φ)), γ q − curlΓ (πt (ψ ψ )), γ p − divΓ (γt (φ These three dualities on the right–hand side are well defined and continuous with respect to the H( curl , Ω) norms. However, this duality pairing based on a decomposition of vector fields in H( curl , Ω) is by no means an intrinsic −1/2 −1/2 characterization of the spaces H|| (divΓ , Γ ) and H⊥ ( curlΓ , Γ ). −1/2

We now consider the duality pairing between H||

(divΓ , Γ ) and

−1/2 H⊥ ( curlΓ , Γ )

based on Hodge decompositions of tangential vector fields in the trace spaces directly. Hodge decompositions of tangential vector fields The following orthogonal Hodge decompositions of the tangential spaces have been introduced in Buffa and Ciarlet [50]: −1/2

H||

(divΓ , Γ ) = grad Γ H(Γ ) ⊕ curl Γ H 1/2 (Γ ) and

−1/2 H⊥ ( curlΓ , Γ )

= curl Γ H(Γ ) ⊕ grad Γ H

1/2

(4.5.33)

(Γ ),

(4.5.34)

where

  H(Γ ) := α ∈ H 1 Γ ) with ΔΓ α ∈ H −1/2 (Γ ) and α, 1 = 0 . −1/2

A duality can now be defined between H|| −1/2 H (divΓ , Γ )

−1/2

(divΓ , Γ ) and H⊥

−1/2 H⊥ ( curlΓ , Γ )

since the pair of spaces and pairing. That is, there exists a bounded sesquilinear form −1/2

 ·, · t : H||

−1/2

(divΓ , Γ ) × H⊥

( curlΓ , Γ )

form a duality

( curlΓ , Γ ) → C

that extends the L2t (Γ ) inner product to the duality pairing of the two spaces such that the following inequalities hold:

202

4. Sobolev Spaces

λH−1/2 (div ,Γ ) ≤ c1 λ Γ ||

sup −1/2 0=η ∈H⊥ ( curlΓ

λ , η t | |λ η η  −1/2 H ( curlΓ ,Γ ) ,Γ ) ⊥

λH−1/2 (div ,Γ ) ≤ c2 λ Γ ||

and c3 ηη H−1/2 ( curlΓ ,Γ ) ≤ ⊥

sup −1/2 0=λ∈H (divΓ

λ , η t | |λ λ λ  −1/2 H (divΓ ,Γ ) ,Γ ) 

≤ c4 ηη H−1/2 ( curlΓ ,Γ ) , ⊥

where ci are positive constants, independent of λ and η . We now state without proof the following theorem: −1/2

Theorem 4.5.4. Let u ∈ H||

−1/2

(divΓ , Γ ) and v ∈ H⊥

there exist αu , αv ∈ H(Γ ) and βu , βv ∈ H decompositions hold:

1/2

( curlΓ , Γ ). Then

(Γ ) such that the following

u = grad Γ αu + curl Γ βu , v = curl Γ αv + grad Γ βv .

(4.5.35)

Let us define u, vt := −ΔΓ αu , βv t − ΔΓ αv , βu t .

(4.5.36)

Then there holds the integration by parts formula:  0 1 { curl u · v − u · curl v}dΩ = γt (u), πt (v) t

(4.5.37)

Ω

for all u, v ∈ H( curl , Ω), and the formula (4.5.37) is consistent with the definition of u, vt in (4.5.36). The proof of this theorem can be found in Buffa and Ciarlet [50] in the case of Lipschitz polyhedra and in Buffa, Costabel and Sheen [52] in the case of general Lipschitz domains. We remark that the integration by parts formula (4.5.30) is consistent with the definition of u, vt in (4.5.36). The duality pairing −1/2 −1/2 (4.5.38) H|| (divΓ , Γ ) = (H⊥ ( curlΓ , Γ )) with L2t (Γ ) as a pivot space has already been stated in the case of a smooth domain with C 1,1 boundary Γ in Cessenat [63, p.40]. Note that from (4.5.37), we have 1 0 | γt (u), πt (v) t | ≤ uH( curl ,Ω) vH( curl ,Ω) and since the right–hand side of (4.5.37) depends only on the traces, this shows 1 0 | γt (u), πt (v) t | ≤ γt (u)H−1/2 (divΓ ,Γ ) πt (v)H−1/2 ( curlΓ ,Γ ) , ||

⊥

4.5 Function Spaces H( div , Ω) and H( curl , Ω)

203

which implies that the duality is continuous. Furthermore, we have the duality property 1 0 1 0 (4.5.39) γt (u), πt (v) t = − γt (v), πt (u) t , ∀ u, v ∈ H( curl , Ω), as a consequence of the Green formula (4.5.37). grad Γ αu is already in γt u see (4.5.35), and grad Γ αu , curl Γ αv t = curlΓ grad Γ αu , αv t = 0, also curl Γ βu , grad Γ βv t = 0, (see V(A) or V(3) in Appendix B). The basic function spaces for the solutions of Maxwell’s equations are H( curl , Ω) and H(div , Ω). However, as is well–known, one can not establish weak solutions of Maxwell equations by applying the Fredholm alternative directly to their variational formulations from either PDEs or BIEs, because of non–compactness of the imbeddings H(div, Ω) → L2 (Ω) and H( curl , Ω) → L2 (Ω) as well as H(div, Ω) ∩ H( curl , Ω) → L2 (Ω) (see, e.g. Amrouche, Bernardi, Dauge and Girault [10]). Fortunately, these difficulties can be circumvented by introducing the idea of the before mentioned Hodge decompositions (or Helmholtz decompositions) of solution spaces of the PDEs or the trace spaces for the BIEs to reveal hidden compactness properties of the formulations as will be demonstrated in Chapter 12. It is worth mentioning that for the same space there may be different decompositions. To understand the developments for the Hodge decompositions of spaces, we need the concept of known as de Rham diagram in differential geometry to characterize the range and the kernel of the operators grad , curl and div in the space of square integrable fields in an open bounded domain Ω ⊂ R3 with boundary Γ . In the exact sequence of the de Rham diagram, the co–domain of each previous operator fills out the kernel of the next operator in the sequence. This topic is out of the scope of our book. For interested readers, we refer to the references Bossavit [38, pp. 132–134], Cessenat [63, Appendix] and Dautray and Lions [96, pp.202, 313]. We comment that as mentioned in Dautray and Lions [96, p.209], it is important to make precise the connection between the spaces H(div, Ω), H( curl , Ω) and the Sobolev space H1 (Ω). We now state the following theorem: Theorem 4.5.5. Let Ω be a bounded domain in R3 with a C 1,1 boundary Γ . Then the Sobolev spaces: ⎧ ⎨ H1t (Ω) := {u ∈ H1 (Ω)|γn (u) = 0}; (4.5.40) ⎩H1 (Ω) := {u ∈ H1 (Ω)|γ (u) = 0} t n are such that

⎧ ⎨ H1t (Ω) = H( curl , Ω) ∩ H0 (div, Ω); ⎩H1 (Ω) = H ( curl , Ω) ∩ H(div, Ω) 0 n

(4.5.41)

204

4. Sobolev Spaces

where H0 (curl , Ω) := {u ∈ H(curl , Ω) and γt u = 0},

and

H0 (div, Ω) := {u ∈ H(div, Ω) and γn u = 0}. On these spaces, the norm uH1 (Ω) is equivalent to the norm: 1/2  (| curl u|2 + |div u|2 + |u|2 )dΩ . u :=

(4.5.42)

Ω

The proof of Theorem 4.5.5 is given in Dautray and Lions [96, Theorem 3]. In both cases, based on a Green formula after a tedious manipulation, it shows that there holds the following inequality:      | grad u|2 dΩ ≤ | curl u|2 + |div u|2 dΩ + c |u|2 dsΓ Ω

Ω

Γ

where c is a positive constant. Then to complete the proof, Peetre’s lemma Peetre [345] is applied to the Sobolev spaces H1t (Ω) as well as to H1n (Ω). As a consequence of Theorem 4.5.5, one has Corollary 4.5.6. Under the hypotheses of Theorem 4.5.5 on Ω , the spaces  ⎫ u ∈ H( curl , Ω) ∩ H(div, Ω)|γn (u) ∈ H 1/2 (Γ ) ⎪ ⎬ (4.5.43)  ⎪ 1/2 ⎭ u ∈ H( curl , Ω) ∩ H(div, Ω)|γt (u) ∈ H (Γ ) are identified with the Sobolev space H1 (Ω). Moreover, there exist positive constants c1 and c2 such that ⎧  ⎪ uH1 (Ω) ≤ c1 uL2 (Ω) +  curl uL2 (Ω) ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ + div uL2 (Ω) + γn (u)H 1/2 (Γ )  (4.5.44) ⎪ ⎪ 1 (Ω) ≤ c2 uL2 (Ω) +  curl uL2 (Ω) u|| ⎪ H ⎪ ⎪ ⎪  ⎪ ⎪ ⎩ + div uL2 (Ω) + γt (u)H1/2 (Γ ) ⊥

for all ∈ H (Ω). 1

In closing this section, we include the following crucial results: Theorem 4.5.7. The embeddings of H1t (Ω) and H1n (Ω) into L2 (Ω)3 are compact (see Theorem 2.8 in Amrouche, Bernardi, Dauge and Girault [10] and Picard, Weck and Witsch [348]). This Theorem is proven by Ch. Weber and P. Werner [430]. As a consequence, there holds

4.5 Function Spaces H( div , Ω) and H( curl , Ω)

205

Theorem 4.5.8. The intersection space H1t (Ω) ∩ H1n (Ω) coincides with 3  1 H0 (Ω) (see Theorem 2.5 in Amrouche, Bernardi, Dauge and Girault [10]). Theorem 4.5.8 is proven by V. Girault and P.A. Raviart [147].

5. Variational Formulations

In this chapter we will discuss the variational formulation for boundary integral equations and its connection to the variational solution of partial differential equations. We collect some basic theorems in functional analysis which are needed for this purpose. In particular, Green’s theorems and the Lax–Milgram theorem are fundamental tools for the solvability of boundary integral equations as well as for elliptic partial differential equations. We will present here a subclass of boundary value problems for which the coerciveness property for some associated boundary integral operators follows directly from that of the variational form of the boundary and transmission problems. In this class, the solution of the boundary integral equations will be established with the help of the existence and regularity results of elliptic partial differential equations in variational form. This part of our presentation goes back to J.C. Nedelec and J. Planchard [332] and is an extension of the approach used by J.C. Nedelec [328] and the ”French School” (see Dautray and Lions [95, Vol. 4]). It also follows closely the work by M. Costabel [79, 80, 84] and our works [90, 201, 401].

5.1 Partial Differential Equations of Second Order Variational methods proved to be very successful for a class of elliptic boundary value problems which admit an equivalent variational formulation in terms of a coercive bilinear form. It is this class for which we collect some of the basic results. We begin with the second order differential equations which are the simplest examples in this class. In this section let Ω ⊂ IRn be a strong Lipschitz domain with strong Lipschitz boundary Γ , and let Ω c = IRn \ Ω denote the exterior domain. We consider the second order elliptic p × p system P u := −

n  j,k=1

∂ ∂xj



∂u ajk (x) ∂xk

 +

n  j=1

bj (x)

∂u + c(x)u = f (x) ∂xj

(5.1.1)

in Ω and/or in Ω c . The coefficients ajk , bj , c are p × p matrix–valued functions which, for convenience, are supposed to be C ∞ –functions. For ajk we © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_5

208

5. Variational Formulations

require the uniform strong ellipticity condition (3.6.2). We begin with the first Green’s formula for (5.1.1) in Sobolev spaces. Throughout this chapter we assume that a fundamental solution E(x, y) to the system (5.1.1) is available (see Lemma 3.6.1 and Section 7.3). First Green’s Formula ( Neˇcas [326, Chap.3]) Multiplying (5.1.1) by a test function v and applying the Gauss divergence theorem one obtains   (P u) vdx = aΩ (u, v) − (∂ν u) vds (5.1.2) Ω

Γ

with the energy–sesquilinear form aΩ (u, v) :=

   n  Ω

  ∂u  ∂v ∂u  + v + (cu) v dx bj (x) ∂xk ∂xj j=1 ∂xj n

ajk (x)

j,k=1

(5.1.3) and the conormal derivative ∂ν u :=

n 

nj ajk γ0

j,k=1

∂u ∂xk

(5.1.4)

provided u ∈ H 2 (Ω) and v ∈ H 1 (Ω) (see Neˇcas [326]); or in terms of local coordinates, inserting (3.4.35) into (5.1.4), by ∂ν u =

n   j,k=1

nj ajk nk

 ∂u ∂n

+

n−1 n   j,k=1 μ,=1

nj ajk

∂yk μ ∂u γ . ∂σμ ∂σ

(5.1.5)

For further generalization we introduce the function space   −1 (Ω)} H 1 (Ω, P ) := {u ∈ H 1 (Ω)  P u ∈ H 0 equipped with the graph norm uH 1 (Ω,P ) := uH 1 (Ω) + P uH −1 (Ω) .  −1 = {f ∈ H −1 (R ) with supp  −1 (Ω) is a subspace embedded into H Here H 0 f1 ⊆ Ω} which consists of all functions in the form:  −1 with supp f1  Ω f = f1 + f2 where f1 ∈ H  t (Ω) , for t ≥ − 1 , and f2 ∈ H 2 equipped with the norm inf{f1 H −1 (Ω) + f2 H t (Ω) }. Clearly, the following inclusions hold:

(5.1.6)

5.1 Partial Differential Equations of Second Order

209

 −1 (Ω) ⊂ H −1 (Ω) . H01 (Ω) ⊂ H 1 (Ω) ⊂ L2 (Ω) ⊂ H 0 It is understood, that u = (u1 , . . . , up ) ∈ (H 1 (Ω, P ))p =: H 1 (Ω, P ). Here and in the sequel, as in the scalar case, we omit the superindex p. Then first Green’s formula (5.1.2) remains valid in the following sense. Lemma 5.1.1 (Generalized First Green’s Formula). For fixed u ∈ H 1 (Ω, P ), the mapping  v → τ u, vΓ := aΩ (u, Zv) − (P u) Zvdx

(5.1.7)

Ω 1

is a continuous linear functional τ u on v ∈ H 2 (Γ ) that coincides for u ∈ 1 H 2 (Ω) with ∂ν u, i.e. τ u = ∂ν u. The mapping τ : H 1 (Ω, P ) → H − 2 (Γ ) with u → τ u is continuous. Here, Z is a right inverse to the trace operator γ0 (4.2.41). In addition, there holds also the generalized first Green’s formula  (P u) vdx = aΩ (u, v) − τ u, γ0 vΓ (5.1.8) Ω

for u ∈ H (Ω, P ) and v ∈ H 1 (Ω) where 1



τ u, γ0 vΓ = (τ u, γ0 v)L2 (Γ ) =

(τ u) γ0 vds

(5.1.9)

Γ

with τ u ∈ H

− 12

(Γ ).

Note that the resulting operator τ u = ∂ν u in (5.1.7) does not depend on the special choice of the right inverse Z in (4.2.41); and that ·, ·Γ denotes the duality pairing (5.1.9) with respect to L2 (Γ ). If f is given in (5.1.6) then the definition of the generalized conormal derivative τ in (5.1.7) corresponds to T + in Mikhailov [297, Definition 3] and Costabel [80, Lemma 2.3].   −1 set in (5.1.8) For f ∈ H (P u)Zvdx = f, ZuΩ and approximate Γ  −1 (Ω) by distributions in H −1 (Ω), characterized in u in H 1 (Ω) and f in H 0 (5.1.6). Then r ∈ H −1/2 (Ω) is the well defined limit but now depends not  −1 (Ω) only on u in Ω but also on f , since the extension of P u ∈ H −1 to f ∈ H is not unique (see McLean [290, Lemma 4.3] and Mikhailov [297, Lemma 6]). Proof: First, let u ∈ C ∞ (Ω) be any given function. Then for any test 1 function v ∈ H 2 (Γ ) and its extension Zv ∈ H 1 (Ω) (see (4.2.41)) with Z a right inverse of γ0 , the first Green’s formula (5.1.2) yields  (5.1.10) ∂ν u, vΓ = aΩ (u, Zv) − (P u) Zvdx , Ω

210

5. Variational Formulations

where ∂ν u is independent of Z. Moreover, by applying the Cauchy–Schwarz inequality and the duality argument on the right–hand side of (5.1.10), we obtain the estimate |∂ν u, vΓ | ≤ cuH 1 (Ω) ZvH 1 (Ω) + P uH −1 (Ω) ZvH 1 (Ω) ≤

cZ {uH 1 (Ω) + P uH −1 (Ω) }v

1

H 2 (Γ )

= cZ uH 1 (Ω,P ) v

1

H 2 (Γ )

where cZ is independent of u and v. This shows that the right–hand side of 1 (5.1.10) defines a bounded linear functional on v ∈ H 2 (Γ ) which for every u ∈ C ∞ (Ω) is given by ∂ν u satisfying ∂ν u

1

H − 2 (Γ )

=

sup v

|∂ν u, vΓ | ≤ cZ uH 1 (Ω,P ) ,

(5.1.11)

=1 1 H 2 (Γ )

where cZ is independent of u. This defines a linear mapping from u ∈ 1 C ∞ (Ω) ⊂ H 1 (Ω, P ) onto the linear functional on H 2 (Γ ) given by τ u = 1 ∂ν u ∈ C ∞ (Γ ) ⊂ H − 2 (Γ ). This mapping u → τ u can now be extended continuously to u ∈ H 1 (Ω, P ) due to the estimate (5.1.11) by the well known completion procedure since C ∞ (Ω) is dense in H 1 (Ω, P ). Namely, if u ∈ H 1 (Ω, P ) is given then choose a sequence uk ∈ C ∞ (Ω) such that u − uk H 1 (Ω,P ) → 0 for k → ∞. Then the sequence τ uk = ∂ν uk defines 1 a Cauchy sequence in H − 2 (Γ ) whose limit defines τ u. This procedure also proves the validity of the generalized Green’s formula (5.1.7).  Note that the generalized first Green’s formula (5.1.8) in particular holds  −1 (Ω). for the H 1 –solution u of P u = f with f ∈ H 0 We also present the generalized first Green’s formula for the exterior domain Ω c . Here we use the local space  1 1 −1 (Ω c , P ) := {u ∈ Hloc (Ω c )  P u ∈ Hcomp (Ω c )} Hloc as defined in (4.1.44). The following version of Lemma 5.1.1 is valid in Ω c . Lemma 5.1.2 (Exterior Generalized First Green’s Formula). 1 Let u ∈ Hloc (Ω c , P ). Then the linear functional τc u given by the mapping  1 τc u, γc0 vΓ := −aΩ c (u, Zc v) + (P u) Zc vdx for all v ∈ H 2 (Γ ) Ωc 1

belongs to H − 2 (Γ ) and coincides with τc u = ∂cν u =

n  j,k=0

nj ajk γc0

∂u ∂xk

(5.1.12)

5.1 Partial Differential Equations of Second Order

211

provided u ∈ C ∞ (Ω c ). Moreover, τc : u → τc u is a linear continuous map1 1 (Ω c , P ) into H − 2 (Γ ). In addition, there holds the exterior ping from Hloc generalized first Green’s formula  (P u) vdx = aΩ c (u, v) + τc u, γc0 vΓ (5.1.13) Ωc 1 for u ∈ Hloc (Ω c , P ) and for every

  1 1 v ∈ Hcomp (Ω c ) := {v ∈ Hloc (Ω c ) v has compact support in IRn } . Here Zc denotes some right inverse to γc0 (cf. (4.2.41)) which maps v ∈ 1 1 (Ω c ) where supp(Zc v) is contained in some fixed H 2 (Γ ) into Zc v ∈ Hcomp n c compact set ΩR ⊂ IR containing Γ . Note that the distributions belonging −1 (Ω c ) cannot have a singular support on Γ due to the definition to Hcomp in (4.1.44). The sesquilinear form aΩ c (u, v) is defined as in (5.1.3) with Ω replaced by Ω c .

Proof: The proof resembles word by word the proof of Lemma 5.1.1 when c in the corresponding norms there.  replacing Ω by ΩR Before we formulate the variational formulation for the boundary value problems for P we return to the sesquilinear form aΩ (u, v) given by (5.1.3). In the sequel, the properties of aΩ and aΩ c will be used extensively. Let us begin with the general definition of sesquilinear forms. The sesquilinear form a(u, v) (Lions [268, Chap.II], see also Stummel [410] and Zeidler [444].) We recall that a(u, v) is called a sesquilinear form on the product space H×H, where H is a Hilbert space with scalar product (·, ·)H and u2H = (u, u)H , iff: a: H × H → C ; a (•, v) is linear for each fixed v ∈ H; i.e. for all c1 , c2 ∈ C, u1 , u2 ∈ H: a (c1 u1 + c2 u2 , v) = c1 a(u1 , v) + c2 a(u2 , v); a (u, •) is antilinear for each fixed u ∈ H; i.e. for all c1 c2 ∈ C, v1 , v2 ∈ H: a (u, c1 v1 + c2 v2 ) = c1 a(u, v1 ) + c2 a(u, v2 ). The sesquilinear form a(·, ·) is said to be continuous, if there exists a constant M such that for all u, v ∈ H: |a(u, v)| ≤ M uH vH .

(5.1.14)

212

5. Variational Formulations

5.1.1 Interior Problems With the sesquilinear form aΩ (u, v) available, we now are in a position to present the variational formulation of boundary value problems. For simplicity, we begin with the standard Dirichlet problem. The interior Dirichlet problem  −1 (Ω) and ϕ ∈ H 12 (Γ ), find u ∈ H 1 (Ω) with γ0 u = ϕ on Γ such Given f ∈ H 0 that (5.1.15) aΩ (u, v) = f (v) := f, vΩ for all v ∈ H01 (Ω) , where aΩ (u, v) is the sesquilinear form (5.1.3) and the right–hand side is a given bounded linear functional on H01 (Ω). We notice that the standard boundary condition γ0 u = ϕ on Γ is imposed in the set of admissible functions for the solution. For this reason, the Dirichlet boundary condition is referred to as an essential boundary condition. It can be replaced by the slightly more general form 1

Bγu = B00 γ0 u = ϕ ∈ H 2 (Γ )

(5.1.16)

with order (B00 ) = 0 corresponding to (3.9.10) with Rγu = Bγu where B00 is an invertible smooth matrix–valued function on Γ . For this boundary condition, in order to formulate the variational formulation similar to (5.1.15), one only needs to modify the last term in the first Green’s formula (5.1.8) according to (5.1.16): τ u, γ0 vΓ = N γu, B00 γ0 vΓ

(5.1.17)

if we choose the complementary operator S = N of (3.9.10) as −1 ∗ ) τ u = N00 γ0 u + N01 N γu := (B00

∂u |Γ ∂n

(5.1.18)

with N00

:=

−1 ∗ (B00 )

n n−1   

nj ajk

j,k=1 μ,ρ=1

N01

:=

−1 ∗ (B00 )

n  

∂yk μρ  ∂ γ , ∂σμ ∂σρ

 nj ajk nk

j,k=1

satisfying the conditions required in (3.9.5) because of the strong ellipticity of P . This leads to the variational formulation: The general interior Dirichlet problem 1 For given f ∈ L2 (Ω) and ϕ ∈ H 2 (Γ ), find u ∈ H 1 (Ω) with B00 γ0 u = ϕ on Γ such that

5.1 Partial Differential Equations of Second Order

213

aΩ (u, v) = f (v) := f, vΩ for all v ∈ H := {v ∈ H 1 (Ω) | Bγv = 0 on Γ } . (5.1.19) Clearly, H = H01 (Ω) since Bγv = B00 γ0 v = 0 for v ∈ H 1 (Ω) iff γ0 v = 0 because the smooth matrix–valued function B00 is assumed to be invertible. The interior Neumann problem  −1 (Ω) and ψ ∈ H − 12 (Γ ), find u ∈ H 1 (Ω) such that Given f ∈ H 0 aΩ (u, v) = f,ψ (v) := f, vΩ + ψ, vΓ for all v ∈ H 1 (Ω) .

(5.1.20)

where aΩ (u, v) again is the sesquilinear form (5.1.3) and the right–hand side is a given bounded linear functional on H 1 (Ω) defined by f and ψ. In contrast to the Dirichlet problem, the Neumann boundary condition τ u = ψ is not required to be imposed on the solution space. Consequently, the Neumann boundary condition is referred to as the natural boundary condition. Both formulations (5.1.15) and (5.1.20) are derived from the first Green’s formula (5.1.2). The more general Neumann–Robin condition with Rγu = N γu in (3.9.13) is given by 1 ∂u N γu := N00 γ0 u + N01 = ψ ∈ H − 2 (Γ ) . (5.1.21) ∂n Here N01 is now a smooth, invertible matrix–valued function and N00 is a tangential first order differential operator of the form N00 γ0 u =

n−1  ρ=1

c0ρ

∂ γ0 u + c00 γ0 u on Γ ∂σρ

(5.1.22)

with smooth matrix–valued coefficients c0ρ and c00 . In contrast to the standard Neumann problem, we now require additionally that Γ is more regular than Lipschitz, namely Γ ∈ C 1,1 . The latter guarantees the continuity of the 1 mapping N γ : H 1 (Ω, P ) → H − 2 (Γ ). In (5.1.8) we now insert (5.1.5) with (5.1.21) and (5.1.22) to obtain τu =

n   j,k=1

and

n−1 n    ∂yk μρ ∂ −1 nj ajk nk N01 (ψ − N00 γ0 u) + nj ajk γ γ0 u ∂σ ∂σρ μ μ,ρ=1 j,k=1

 γ0 u, γ0 vΓ , τ u, γ0 vΓ = N γu, SγvΓ + N

(5.1.23)

where the complementary boundary operator is now chosen as −1 ∗ ) Sγv := Bγu := (N01

n   j,k=1

 is therefore defined by and N

nj ajk nk

∗

γ0 v

(5.1.24)

214

5. Variational Formulations

 γ0 u := N



−

n   j,k=1

n−1 n    ∂yk μρ ∂  −1 nj ajk nk N01 N00 + nj ajk γ γ0 u . ∂σμ ∂σρ μ,ρ=1 j,k=1

This operator can be written with (5.1.22) as  γ0 u = N

n−1 

 c0ρ

ρ=1

where  c0 =

n n−1  

∂ γ0 u +  c00 γ0 u on Γ , ∂σρ

nj ajk

 ∂y

j,k=1 μ=1

k

∂σμ

(5.1.25)

 −1 γ μ − nk N01 c0ρ .

Following an idea of Weinberger (see the remarks in Agmon [3, p.147ff.], Fichera [121]), the corresponding term in (5.1.23) will now be incorporated into aΩ (u, v) generating an additional skew–symmetric bilinear form. For fixed ρ, (3.3.7) gives, with the chain rule,   ∂yk ∂nk  ∂u ∂u = + σn ∂σρ ∂σρ ∂σρ ∂xk n

near Γ .

k=1

By extending yk and nj to all of Ω smoothly, e. g. by zero sufficiently far away from Γ , we define the skew–symmetric real–valued coefficients   ∂yj ∂yk = −βjkρ . − nj (5.1.26) βkjρ := nk ∂σρ ∂σρ With

n 

n2k = 1 and

k=1

n  k=1

nk

∂yk |Γ = 0 ∂σ

we have n n−1     ∂yj ∂u  γ0 vds  c0ρ ∂σρ ∂xj j=1 ρ=1 Γ

=

  n n−1      ∂yj ∂yk ∂u − nj γ0 vds  c0ρ nk nk ∂σρ ∂σρ ∂xj =1

k,j=1

Γ

   n n−1  ∂u =  c0ρ βjkρ nk γ0 vds ∂xj ρ=1 Γ

j,k=1

and integration by parts shows that

5.1 Partial Differential Equations of Second Order

215

n n−1     ∂yj ∂u  γ0 vds  c0ρ ∂σρ ∂xj j=1 ρ=1 Γ

=

  n−1 n   ∂u  ∂   c0ρ βkjρ vdx ∂xk ρ=1 ∂xj

Ω j,k=1

+

  n  n−1 

Ω j,k=1

 c0ρ βkjρ

ρ=1

∂u  ∂v dx . ∂xj ∂xk

Since the last term vanishes because of (5.1.26), this yields  γ0 u, γ0 vΓ = N

n n−1    ∂u  ∂¯ v dx  c0ρ βkjρ ∂xj ∂xk ρ=1

j,k=1

Ω

n   n n−1  ∂u      ∂  ∂  + (nj  c00 u) v¯ +  c0ρ βkjρ v¯ dx ∂xj ∂xk ∂xj ρ=1 j=1 k=1

Ω

(5.1.27) which motivates us to introduce the modified bilinear form  aΩ (u, v) := aΩ (u, v) −

n   n−1   j,k=1 Ω

 c0ρ βjkρ

ρ=1

 ∂u  ∂¯ v dx ∂xj ∂xk

  n  n   ∂u    n−1 ∂  ∂ − (nj  c00 u) v +  c0ρ βkjρ v¯ dx . ∂xj ∂xk ∂xj ρ=1 j=1 Ω

k=1

(5.1.28) Equations (5.1.23) and (5.1.28) in connection with the first Green’s formula (5.1.8) lead to the variational formulation of the general Neumann–Robin problem. The general interior Neumann–Robin problem  −1 (Ω) and ψ ∈ H − 12 (Γ ) in (5.1.21), find Let Γ ∈ C 1,1 . Given f ∈ H 0 1 u ∈ H (Ω) such that  aΩ (u, v) = f, vΩ + ψ, BγvΓ

for all v ∈ H 1 (Ω)

(5.1.29)

where  aΩ (u, v) is defined by (5.1.28) and the right–hand side is a bounded linear functional on H 1 (Ω) defined by f and ψ.

The combined Dirichlet–Neumann problem A class of more general boundary value problems combining both Dirichlet and Neumann conditions can be formulated as follows. First let us introduce a symmetric, real (p × p) projection matrix π on Γ which has the properties:

216

5. Variational Formulations

rank π(x) = r ≤ p for all x ∈ Γ , π = π  and π 2 = π ,

(5.1.30)

where r is⎛constant⎞ on Γ and π(x) ∈ C 1 (Γ ). As a simple example consider 1 0 0 π = T (x) ⎝0 0 0⎠ T  (x) for p = 3 where T (x) is any smooth orthogonal 0 0 0 matrix on Γ . The combined interior Dirichlet–Neumann problem reads:  −1 (Ω) , ϕ1 ∈ πH 12 (Γ ) and ψ2 ∈ (I − π)H − 12 (Γ ) find u ∈ For given f ∈ H 0 H 1 (Ω, P ) satisfying P u = f in Ω , πBγu = ϕ1 and (I − π)N γu = ψ2 on Γ .

(5.1.31)

Here I denotes the identity matrix on Γ . The corresponding variational formulation then takes the form: Find u ∈ H 1 (Ω) with πBγu = ϕ1 on Γ such that   aΩ (u, v) = f  vdy + ψ2 , (I − π)BγuΓ (5.1.32) Ω

for all v ∈ HπB := {v ∈ H 1 (Ω) | πBγv|Γ = 0} . Note that in the special case π = I, (5.1.32) reduces to the interior Dirichlet problem (5.1.16), while in the case π = 0, (5.1.32) becomes the interior Neumann–Robin problem (5.1.29). Note that in the special case B00 = I and N00 = 0, for the treatment of (5.1.31) we only need Γ ∈ C 0,1 , i.e., to be Lipschitz. 5.1.2 Exterior Problems For extending our variational approach also to exterior problems, we need to incorporate the radiation conditions. For motivation we begin with the ”classical” representation formula (3.6.17) in terms of our boundary data γc0 u and τc u = ∂cν u (see (5.1.12)), respectively,    u(x) = E (x, y)f (y)dy − E  (x, y)τc udsy (5.1.33) y∈Ω c

 

∂cνy E(x, y) +

+ Γ

Γ n 

nj (y)bj (y)E(x, y)



γc0 udsy + M (x; u)

j=1

−1 provided f ∈ Hcomp (Ω c ) has compact support in IRn (see (4.1.44)). Here, M is defined by

5.1 Partial Differential Equations of Second Order

  M (x; u)

:=

E  (x, y)τc uds

lim

R→∞

 −

217

(5.1.34)

|y|=R

 ∂νy E(x, y) +

n 

nj (y)bj (y)E(x, y)



 γc0 udsy .

j=1

|y|=R

From this definition, we see that M must satisfy the homogeneous differential equation PM = 0 (5.1.35) in all of IRn . Its behaviour at infinity is rather delicate, heavily depending on the nature of P there. In order to characterize M more systematically we require that M is a tempered distribution, and we further simplify P by assuming the following: Conditions for the coefficients: c ≡ 0 for all x , ajk (x) = ajk (∞) = const. and bj (x) = 0 for |x| ≥ R0 . (5.1.36) In addition, we require that for P there holds the following Unique continuation property: (See H¨ormander [188, Chapter 17.2].) If P u = 0 in any domain ω ⊂ IRn and u ≡ 0 in some ball Bρ (x0 ) := {x with |x − x0 | < ρ} ⊂ ω , ρ > 0 then u ≡ 0 in ω . (5.1.37) Under these assumptions together with the restriction that uniqueness holds for both Dirichlet problems with P in Ω as well as in Ω c , we now show that every tempered distribution M ∈ S  (IRn ) satisfying (5.1.35) admits the form  αβ (u)pβ (x) , (5.1.38) ML (x; u) = |β|≤L

where αβ are linear functionals on u and pβ (x) are generalized polynomials behaving at infinity like homogeneous polynomials of degree |β|. (For the constant coefficient case see, e.g., Dautray and Lions [95, pp. 360ff.], [97, p. 119] and Miranda [307, p. 225].) Let M be a tempered distribution satisfying (5.1.35). Because of assumption (5.1.36) we have M (x; u) = p∞ (x) for |x| ≥ R0 where p∞ is a tempered distribution satisfying P∞ p∞ (x) :=

n  j,k=1

ajk (∞)

∂2 p∞ (x) = 0 . ∂xj ∂xk

(5.1.39)

By taking the Fourier transform of (5.1.39) we obtain n  j,k=1

∞ (ξ) = 0 for all ξ ∈ IRn \ {0} . ξj ajk (∞)ξk p6

(5.1.40)

218

5. Variational Formulations

Since for each ξ ∈ IRn \ {0} the strong ellipticity (3.6.2) implies that n

(

ξj ajk (∞)ξk )−1 exists, it follows that

j,k=1 ∞ (ξ) = 0 for all ξ ∈ IRn \ {0} . p6

(5.1.41)

∞ (ξ) is a distribution with only support {0}. The Hence, every component of p6 ∞ (ξ) then Schwartz theorem (Dieudonn´e [100, Theorem 17.7.3]) implies that p6 is a finite linear combination of Dirac distributions at the origin. This yields:

Lemma 5.1.3. For a strongly elliptic operator P with (5.1.36) in IRn , every tempered distributional solution of (5.1.39) in IRn is a polynomial. To see that ML is of the form (5.1.38), we first decompose every generalized polynomial of degree |β| as β (x) pβ (x) = p∞ β (x) + p

(5.1.42)

and require P pβ (x) = 0 in IRn and pβ (x) = 0 for all |x| ≥ R0 .

(5.1.43)

In order to find pβ (x) we need the following additional general Assumption: The Dirichlet problem P w = 0 in BR0 := {x ∈ IRn | |x| < R0 }

(5.1.44)

with w ∈ H01 (BR0 ) admits only the trivial solution w ≡ 0. (For a more detailed discussion involving more general unbounded domains see Nazarov [325]). Remark 5.1.1: In case that we have the unique continuation property (5.1.37) one can show that this assumption can always be fulfilled for some R0 > 0 (see Miranda [307, Section 19]). Now we solve the following variational problem: Find pβ ∈ H01 (BR0 ) such that   aIRn ( pβ , v) = − (P p∞ ¯dx for all v ∈ H01 (BR0 ) . β ) v

(5.1.45)

BR 0

This variational problem has a unique solution pβ ∈ H01 (BR0 ) which can 1 (IRn ). With the constructed pβ , be extended by zero to all of IRn and pβ ∈ Hloc ∞ every polynomial ββ generates a generalized polynomial pβ (x) via (5.1.42). This justifies the representation of ML in (5.1.38). On the other hand, since E(x, y) represents in physics the potential of a point charge at y observed at x, the representation formula (5.1.33) suggests

5.1 Partial Differential Equations of Second Order

219

that one may assume for the behaviour of u(x) at infinity a behaviour similar to that of E(x, 0). This motivates us, based on (5.1.33), to require, with given L and ML (x; u), Dα (u(x) − ML (x; u)) = Dα E  (x, 0)q + o(1) for |x| → ∞ and with |α| ≤ 1 , (5.1.46) where q is a suitable constant vector. Hence, we need to have some information concerning the growth of the fundamental solution at infinity. (We shall return to the question of fundamental solutions in Section 7.3.) Since for |x| ≥ R0 , the equation (3.6.4) reduces to the homogeneous equation with constant coefficients, here the corresponding fundamental solution has the explicit form (see John [219, (3.87)]):  1 Ejk (x, y) = Δ P jk (ξ)|(x − y) · ξ|2 log |(x − y) · ξ|dωξ (5.1.47) y 8π 2 |ξ|=1

for n = 2 and |x| ≥ R0 , where the integration is taken over the unit circle |ξ| = 1 with arc length element dωξ and where P jk (ξ) denotes the inverse of the matrix ajk (ξ) = ajk (∞)ξj ξk , (( ajk (ξ)))n×n with  which is regular for all ξ ∈ IR2 \ {0} due to the uniform ellipticity. For n = 3 and |x| ≥ R0 , one has (see John [219, (3.86)])  1 Ejk (x, y) = − Δ P jk (ξ)|(y − x) · ξ|dωξ , y 16π 2

(5.1.48)

|ξ|=1

where dωξ now denotes the surface element of the unit sphere. For the radiation condition let us consider the case n = 2 first. Here, (5.1.47) implies (see [219, (3.49)])   1 α Ejk = O(log |x|) and D Ejk = O for |α| = 1 as |x| → ∞ . |x| Since ML (x; u) is of the form (5.1.38), no logarithmic terms belong to ML . Consequently, it follows from (5.1.33) by collecting the coefficients of E  that the constant vector q in (5.1.46) is related to u by    q= f (y)dy − (τc u − (b · n) γc0 u)ds . (5.1.49) Ωc

Γ

The proper radiation conditions for u now will be   1 for |α| ≤ 1 as |x| → ∞ . Dα u = Dα E  (x, 0)q+Dα ML (x; u)+O |x|1+|α| (5.1.50)

220

5. Variational Formulations

The exterior Dirichlet problem Now we can formulate the exterior Dirichlet problem in two versions. 1 −1 (Ω c ) with compact support in IRn and ϕ ∈ H 2 (Γ ) we For given f ∈ Hcomp 1 require that u belongs to Hloc (Ω c ) ∩ S  (IRn ) and satisfies Pu γc0 u

= =

f ϕ

in Ω c , on Γ ,

(5.1.51)

−1 (Ω c ) is detogether with the radiation conditions (5.1.50). The space Hcomp fined as in (4.1.44) where Ω is replaced by Ω c = IR \ Ω. (5.1.51) allows the following two interpretations of the radiation conditions and leads to two different classes of problems.

First version: In addition to f and ϕ, the constant vector q and the integer L in (5.1.50) are given a–priori. Find u as well as ML (x; u) to satisfy (5.1.51) and (5.1.50). Note that modifying the solution by adding the term E  (x, 0)q with given q, this problem can always be reduced to the case where q = 0. In the latter case, the corresponding variational formulation reads: 1 Find a tempered distribution u in Hloc (Ω c ) ∩ S  (IRn ) with γc0 u = ϕ on Γ which behaves as (5.1.50) with q = 0 and given ML and satisfies 1 (Ω c ) with γc0 v = 0 on Γ , aΩ c (u, v) = f, vΩ c for all v ∈ Hcomp (5.1.52) where aΩ c (u, v) is defined by (5.1.3) with Ω replaced by Ω c . We remark that, in order to obtain a unique solution one requires side conditions.

Second version: In addition to f and ϕ, the function ML (x; u) in the form (5.1.38) is given a–priori. Find u as well as q to satisfy (5.1.51) and (5.1.50). We may modify the solution by subtracting the given ML and reduce the problem to the case ML = 0. Under this assumption, the variational formulation of the second version now reads: 1 (Ω c ) ∩ S  (IRn ) with γc0 u = ϕ on Γ Find a tempered distribution u ∈ Hloc which behaves at infinity as (5.1.50) and satisfies 1 (Ω c ) with γc0 v = 0 on Γ aΩ c (u, v) = f, vΩ c for all v ∈ Hcomp

(5.1.53)

subject to the constraint ML (x; u) = 0 for x ∈ IRn . In view of (5.1.34) and our assumption (5.1.36), the radiation condition is      0 = lim E (x, y)τR udsy − (∂νy E(x, y)) γ0 u(y)dsy (5.1.54) R→∞

|y|=R

|y|=R

5.1 Partial Differential Equations of Second Order

221

for every x ∈ Ω c . Alternatively, for ML (x; u) = 0, in view of (5.1.50) and (5.1.49) we obtain from (5.1.33) together with the asymptotic behaviour of E(x, y) for large |x|,  $  % f (y)dy − (τc u − (b · n)γc0 u) ds = 0 . lim u(x) − E  (x, 0) |x|→∞

Ωc

Γ

(5.1.55) Since the representation formula (5.1.33) holds for every exterior domain, we   may replace Ω c by {y ∈ IR2 |y| > R ≥ R0 } with supp(f ) ⊂ {y |y| ≤ R} and find instead of (5.1.55) the radiation condition    τR uds = 0 (5.1.56) lim u(x) + E  (x, 0) |x|→∞

|y|=R

for every sufficiently large R ≥ R0 . In contrast to the interior Dirichlet problem, the above formulations in terms of local function spaces are not suitable for the standard Hilbert space treatment unless one can find an equivalent formulation in some appropriate Hilbert space. In the following, we will present such an approach for the first version only which is based on the finite energy of the potentials. The same approach, however, cannot directly be applied to the second version since the corresponding potentials do not, in general, have finite energy. However, the following approach will be used for analyzing the corresponding integral equations for problems formulated in the second version. We will pursue this in Section 5.6. The Hilbert space formulation for the first version of the general exterior Dirichlet problem In view of the growth condition (5.1.50), in the first version with q = 0 we have to choose Hilbert spaces combining growth and M (x; u) appropriately. Let us consider first the case of self–adjoint equations (5.1.1) with a symmetric energy–sesquilinear form aΩ c (u, v) where, in addition to the assumptions (5.1.36), bj (x) = 0 for all x ∈ Ω c . Then aΩ c (u, v) = aΩ c (v, u) and aΩ c (v, v) ≥ 0 for all u, v ∈ C0∞ (IRn ) due to the strong ellipticity condition (3.6.2). For a fixed chosen L denote by PL all the generalized polynomials ML (x) of the form  p(x) = αβ pβ (x) |β|≤L

satisfying (5.1.43). By EL let us denote the subspace of PL of generalized polynomials with ”finite energy”, i. e.

222

5. Variational Formulations

EL := {p ∈ PL | |aΩ c (p, p)| < ∞} .

(5.1.57)

Because of (5.1.35) one finds aΩ (p, v) = 0 for all v ∈ H01 (Ω) and further

aIRn (p, v) = 0 for all v ∈ C0∞ (IRn ) .

(5.1.58)

Now define the pre–Hilbert space HE∗ (Ω c )

:=

1 {v = v0 + p | p ∈ EL and v0 ∈ Hcomp (Ω c )

(5.1.59)

satisfying aΩ c (p, v0 ) = 0 for all p ∈ EL } equipped with the scalar product  (u, v)HE := aΩ c (u, v) +

γc0 u γc0 vds .

(5.1.60)

Γ

(See also [205], Leis [263, p.26].) Note that C0∞ (Ω c ) ⊂ HE∗ (Ω c ) because of (5.1.58). For convenience, we tacitly assume Θ = 1 in the uniform strong ellipticity condition (3.6.2). Then (u, v)HE has all the properties of an inner product; in particular, (u, u)HE ≥ 0 for every u ∈ HE∗ (Ω c ). Now, suppose  (v, v)HE = aΩ c (v0 + p, v0 + p) + |v0 + p|2 ds = 0 , Γ

which implies (v0 + p)|Γ = 0 since aΩ c (v0 + p, v0 + p) ≥ 0. The former allows us to extend v0 by −p into Ω and to define   v0 in Ω c 1 ∈ Hcomp v0 := (IRn ) , where ( v0 + p)|Ω = 0 . −p in Ω Then v0 + p, v0 + p) 0 = aΩ c (v0 + p, v0 + p) = aIRn ( and (5.1.58) implies v0 + p, v0 + p) = aIRn (p, p) = 0 . aIRn ( Here, our assumption (5.1.44) together with the strong ellipticity of P implies o

the H01 (BR1 )–ellipticity of aIRn (·, ·) for any R1 > 0, in particular for B R1 ⊃ supp( v0 ). Hence, v0 = 0 in IRn . Consequently, p = 0 in Ω and P p = 0 in n IR . The unique continuation property then implies p = 0 in IRn . Hence, (v, v)HE = 0 implies v = 0 in HE . Consequently, 1

2 vHE := (v, v)H E

(5.1.61)

5.1 Partial Differential Equations of Second Order

223

defines a norm. Now, HE (Ω c ) is defined by the completion of HE∗ (Ω c ) with respect to the norm  · HE . In the more general case of a non–symmetric sesquilinear form aΩ c (u, v) (and for general exterior Neumann problems later-on) we take a slightly more general approach by using the symmetric part of aΩ c , i. e.   aSΩ c (u, v) := 12 aΩ c (u, v) + aΩ c (v, u) (5.1.62) together with the requirement aSΩ c (v, v) ≥ 0 for all v ∈ C0∞ (IRn ) . Note that again (u, v)HSE := aSΩ c (u, v) has all the properties of an inner product as before in view of condition (5.1.36). The corresponding space is now defined by S 1 HE∗ (Ω c ) := {v0 + pS | pS ∈ ELS and v0 ∈ Hcomp (Ω c )

satisfying aSΩ c (pS , v0 ) = 0 for all pS ∈ ELS } where ELS now denote all generalized polynomials of degree ≤ L with finite energy aSIRn (pS , pS ) < ∞ satisfying aSIRn (pS , v) = 0 for all v ∈ C0∞ (IRn ) . Then by following the same arguments as in the symmetric case, one can S again show that (u, v)HSE∗ is an inner product on the space HE∗ (Ω c ). For the formulation of the exterior Dirichlet problem we further introduce the closed subspace  HE,0 (Ω c ) := {v ∈ HE (Ω c )γc0 v = 0 on Γ } . (5.1.63) If u ∈ HE (Ω c ) and v ∈ HE,0 (Ω c ) then u = u0 + p and v = v0 + r where p, r ∈ EL . Moreover, by applying Green’s formula to the annular region Ω∩BR with BR := {x ∈ IRn | |x| < R} for R ≥ R0 sufficiently large, we obtain the identity    c (P u) v¯dx + (τR p) γ0R rdsR aΩ ∩BR (u, v) = Ω c ∩BR



∂BR

(P u) v¯dx + aBR (p, r) .

= Ω c ∩BR

For R → ∞ we obtain



aΩ c (u, v) = Ωc

(P u) v¯dx + aIRn (p, r) .

(5.1.64)

224

5. Variational Formulations

  With a basis {qs }L s=1 of EL where L = dim EL , the last term can be written

more explicitly with v = v0 + r = v0 +

aIRn (p, r) =

 L 

 L ρ=1

κρ qρ as

κρ aIRn (p, qρ ) .

ρ=1

Now, for the first version we formulate:  = The general exterior Dirichlet problem. For given q = 0, L and L 1 2 c  2 dim EL , ϕ ∈ H (Γ ) , f ∈ Lcomp (Ω ) , dρ ∈ C for ρ = 1, . . . , L, find u = 1 u0 + p ∈ HE with Bγc0 u = ϕ ∈ H 2 (Γ ) on Γ satisfying aΩ c (u, v) = f, vΩ c +

 L 

d ρ κρ

for all v = v0 +

ρ=1

 L 

κρ qρ ∈ HE,0 (5.1.65)

ρ=1

subject to the side conditions . aIRn (p, qρ ) = dρ , ρ = 1, . . . , L In this formulation, we need in addition to the assumptions (5.1.36) that B00 = I for |x| ≥ R0 where I is the identity matrix

.

(5.1.66)

We note that in this formulation we have replaced the function space {v ∈ HE | Bγc v = 0 on Γ } by HE,0 . Since det (B00 )|Γ = 0, they are equivalent. Remark 5.1.2: In our Hilbert space formulation, for the Laplacian in IR2 and the special choice of L = 0, and in IR3 and EL = {0} = ML , the energy space HE (Ω) is equivalent to the weighted Sobolev space H 1 (Ω c ; 0) corresponding to W01 (Ω  ) in Nedelec [328]. Hence, in these cases, one may also formulate the variational problems in the weighted Sobolev spaces as in Dautray and Lions [97, Chap.XI], Giroire [148] and Nedelec [328]. The exterior Neumann problem The exterior Neumann problem is defined by Pu

=

f

in Ω c ,

τc u

=

ψ

on Γ .

(5.1.67)

Again we require the solution to behave like (5.1.50) for |x| → ∞. Since τc u = ψ is given, the equation (5.1.49) yields    q = { f dy − ψds + (b · n) γc0 uds} . Ωc

Γ

Γ

5.1 Partial Differential Equations of Second Order

225

For the special case b · n = 0 on Γ , the constant q is determined explicitly by the given data from (5.1.67). In order to find a solution in HE (Ω c ) for n = 2, the constant q in the growth condition (5.1.50) must vanish. This yields the necessary compatibility condition   (5.1.68) q = 0 = f (y)dy − ψds for n = 2 Ωc

Γ

and for the givendata  in (5.1.67). Then the solution will take the form u = u0 with u0 (x) = O

1 |x|

for |x| → ∞. In the three–dimensional case n = 3, the

compatibility condition (5.1.68) is not needed to guarantee a solution of finite energy. The behaviour at infinity motivates us to consider the following variational problem in the Hilbert space HE (Ω c ). For the solution we assume the form u = u0 + p + E  (x, 0)q

in Ω c

(5.1.69)

where u0 + p ∈ HE and q ∈ Cp to be determined. The exterior Neumann variational problem Find u0 + p ∈ HE and q ∈ Cp satisfying aΩ c (u0 , v)

=



f, vΩ c − ψ − τc E (x, 0)q, vΓ + for all v = v0 +

and

 q= Ωc

 f (y)dy −

 L ρ=1

 L 

d ρ κρ

ρ=1

(5.1.70)

κρ q ρ ∈ H E

  ψds + (b · n) γc0 (u0 + p + E  (x, 0)q)ds

Γ

Γ

subject to the side conditions . aIRn (p, qρ ) = dρ for ρ = 1, . . . , L This formulation shows that the given data f, ψ do not have to satisfy the standard compatibility condition (5.1.68) and, in general, the total solution u in (5.1.69) does not have finite energy. If the latter is required, i.e., q = 0 in (5.1.70) then the compatibility condition between f and ψ can only be satisfied in an implicit manner resolving the bilinear equation with respect to u0 + p and with q = 0 and inserting p in the second equation of (5.1.70) with q = 0. The general exterior Neumann–Robin problem

226

5. Variational Formulations

Here we extend the boundary conditions to N01 = B00 = I and N00 = 0 on the artificial boundary ΓR for R ≥ R0 . Then we can proceed in the same manner as for the general interior Neumann–Robin problem except for the radiation conditions which need to be incorporated. In this case, the constant vector q in the radiation condition (5.1.50) is related to the solution u by    n −1 f (y)dy − ( nj ajk nk )N01 ψds q = Ωc

b·n+

+

j,k=1

Γ

  Γ

  ∂ c00 γc0 (u0 + p + E  (•, 0)q)ds  c0ρ −  ∂σρ ρ=1

n−1 

c00 are given by (5.1.25), which modifies (5.1.49) of the stanwhere  c0ρ and  dard Neumann problem. For the special case b·n+

n−1  ρ=1

∂  c0ρ −  c00 = 0 on Γ , ∂σρ

we recover, similar to (5.1.68), the necessary compatibility condition   −1 q = 0 = f (y)dy − (Σnj ajk nk )N01 ψds Ωc

Γ

for the given data f and ψ and for a solution with finite energy. The exterior combined Dirichlet–Neumann problem We begin with the formulation of the boundary value problem in distribu1 1 −1 tional form: Given f ∈ Hcomp (Ω c ) , ϕ1 ∈ πH 2 (Γ ) and ψ2 ∈ (I − π)H − 2 (Γ )  find u = u0 + p + E  (x; 0)q with on Γ and dρ ∈ C , ρ = 1, . . . , L, c p u0 + p ∈ HE (Ω ) and q ∈ C as the distributional solution of Pu = πBγc u =

f in Ω c ϕ1 and (1 − π)N γc u = ψ2 on Γ

(5.1.71)

subject to the side conditions  where EL = span{q }L , aIRn (p, qρ ) = dρ for ρ = 1, . . . , L =1 and the constraint





q={ Ωc

f (y)dy −

(τc u − (b · n) γc0 u)ds} .

Γ

The application of Lemma 5.1.2 then yields:

5.1 Partial Differential Equations of Second Order

227

The variational formulation for the exterior problem Find u = u0 + p + E  (•; 0)q with u0 + p ∈ H(Ω c ) and πBγc u = ϕ1 on Γ satisfying  f  v¯dx − ψ2 , (I − π)Bγc vΓ (5.1.72)  aΩ c (u0 + p, v) = Ωc

+ (I − π)N γc E  (•, 0)q , (I − π)Bγc vΓ +

 L 

d s κρ

ρ=1  L

for all v = v0 + ρ=1 κρ qρ ∈ HEπB := {v ∈ HE (Ω c ) | πBγc v = 0 on Γ }, subject to the side conditions  where EL = span{q }L , aIRn (p, qρ ) = dρ for ρ = 1, . . . , L =1

(5.1.73)

and the constraint   q = { f (y)dy − (τc − (b · n) γc0 )(u0 + p + E  (•; 0)q)ds} .

(5.1.74)

Ωc

Γ

−1 −1 Since τc u = N01 (N γc u − N00 γc0 u) due to (5.1.21) and γc0 u = B00 Bγc u due to (5.1.16), in view of the given boundary conditions in (5.1.71), the last constraint (5.1.74) can be reformulated as the linear equation of the form      −1 −1 −1 q = N01 f (y)dy − (1 − π)ψ2 + (N01 N00 − (b · n))B00 πϕ1 ds Ωc

−

Γ

 

−1 −1 −1 N01 πN γc + (N01 N00 − (b · n))B00 (1 − π)Bγc



Γ

(u0 + p + E  (·; 0)q)ds .

(5.1.75)

Remark 5.1.3: In the special case π = I, one verifies that the combined problem reduces to the general Dirichlet problem (5.1.65). For π = 0, we recover the general exterior Neumann–Robin problem.

5.1.3 Transmission Problems The combined Dirichlet–Neumann conditions As we can see in the previous formulations of the boundary value problems, the boundary operator R is always given while the complementing boundary operator S could be chosen according to the restrictions (3.9.4) and (3.9.5). As generalization of these boundary value problems, we consider transmission

228

5. Variational Formulations

problems where both boundary operators are given to form a pair of mutually complementary boundary conditions. We will classify the transmission problems in two main classes. In both −1  and classes we are given f ∈ Hcomp (IRn \ Γ ) and dρ ∈ C , ρ = 1, . . . , L 1 we are looking for a distributional solution u with u|Ω ∈ H (Ω, P ) and u|Ωc = u0 + p + E  (•; 0)q with u = u0 + p ∈ HE (Ω c ) and q ∈ Cp satisfying  ; (5.1.76) P u = f in IRn \ Γ and aIRn (p, qρ ) = dρ for ρ = 1, . . . , L   and q = f dy − (τc u − (b · n)γc0 u)ds (5.1.77) Ωc

Γ

subject to the following additional transmission conditions on Γ ; EL =  span{a }L =1 . In the first class these transmission conditions are 1.

1

1

[πBγu] = ϕ1 ∈ πH 2 (Γ ) and [πN γu] = ψ1 ∈ πH − 2 (Γ )

(5.1.78)

where ϕ1 and ψ1 are given on Γ together with one of the following conditions: 1

1.1

(I − π)Bγu = ϕ20 ∈ (I − π)H 2 (Γ ) and 1 (I − π)N γc u = ψ21 ∈ (I − π)H − 2 (Γ )

1.2

(I − π)Bγc u = ϕ21 ∈ (I − π)H 2 (Γ ) and 1 (I − π)N γu = ψ20 ∈ (I − π)H − 2 (Γ ) .

(5.1.79)

1

(5.1.80)

For the corresponding variational formulations, for simplicity we confine ourselves to the cases with dρ = 0 and p(x) ≡ 0 in (5.1.69), (5.1.70). The 0 (Ω c ). Then the corresponding exterior energy space will be denoted by HE variational equations for this class of transmission problems can be formulated as: 

aΩ (u, v) +  aΩ c (u, v)  aIRn \Γ (u, v) := 

=

f  v¯dx + N γc E  (·; 0)q, Bγc vΓ + (¯ v)

(5.1.81)

IRn \Γ

for all test functions v as characterized in Table 5.1.1, and where the linear functional (¯ v ) = N γu, BγvΓ − N γc u, Bγc vΓ will be specified depending on the particular transmission conditions (5.1.78)– (5.1.80), whereas q ∈ Cp must satisfy the constraint (5.1.77), i.e.,    q = f dy − τc u − (b · n) γc0 u ds . (5.1.82) Ω

Ω

In this class with (5.1.78), the corresponding variational formulations now read:

5.1 Partial Differential Equations of Second Order

229

0 Find u ∈ H 1 (Ω) × HE (Ω c ) with the enforced constraints given in Table 5.1.1 satisfying the variational equation (5.1.81) with (5.1.82) for all test 0 (Ω c ) characterized in Table 5.1.1 functions v in the subspace of H 1 (Ω) × HE with the functional (¯ v ) also specified in Table 5.1.1.

Table 5.1.1. Relevant data for the variational equations in the first class Boundary conditions

Constraints for u

(¯ v)

Subspace conditions

(5.1.78)

[πBγu] = ϕ1 ,

−ψ1 , πBγv Γ

[πBγv] = 0,

(5.1.79)

(I − π)Bγu = ϕ20

−ψ21 , (I − π)[Bγv] Γ

(I − π)Bγv = 0

(5.1.78)

[πBγu] = ϕ1 ,

−ψ1 , πBγv Γ

[πBγv] = 0,

(5.1.80)

(I − π)Bγc u = ϕ21

+ψ20 , (I − π)Bγv Γ

(I − π)Bγc v = 0

We remark that in all classes π, r and (I − π), (p − r) can be interchanged throughout due to the properties (5.1.30). In the second class, the transmission conditions are 1

1

[πBγu] = ϕ1 ∈ πH 2 (Γ ) and [(I − π)N γu] = ψ2 ∈ (I − π)H − 2 (Γ ) (5.1.83) together with one of the following conditions: 2.

1

2.1

πBγu = ϕ10 ∈ πH 2 (Γ ) and 1 (I − π)N γu = ψ20 ∈ (I − π)H − 2 (Γ ) ;

2.2

πBγc u = ϕ11 ∈ πH 2 (Γ ) and 1 (I − π)N γc u = ψ21 ∈ (I − π)H − 2 (Γ ) ;

2.3

πBγu = ϕ10 ∈ πH 2 (Γ ) and 1 (I − π)N γc u = ψ21 ∈ (I − π)H − 2 (Γ ) ;

2.4

πBγc u = ϕ11 ∈ πH 2 (Γ ) and 1 (I − π)N γu = ψ20 ∈ (I − π)H − 2 (Γ ) ;

2.5

(I − π)Bγu = ϕ20 ∈ (I − π)H 2 (Γ ) and 1 πN γu = ψ10 ∈ πH − 2 (Γ ) ;

2.6

(I − π)Bγc u = ϕ21 ∈ (I − π)H 2 (Γ ) and 1 πN γc u = ψ11 ∈ πH − 2 (Γ ) .

(5.1.84)

1

(5.1.85)

1

(5.1.86)

1

(5.1.87)

1

(5.1.88)

1

(5.1.89)

We remark that in all the above cases, one may rewrite the equations (5.1.77) for q in terms of the given transmission and boundary conditions, accordingly. Similar to the variational formulations for the first class of transmission problems, we may summarize these formulations for the cases 2.1–2.4 in Table 5.1.2.

230

5. Variational Formulations

Table 5.1.2. Relevant data for the variational equations in the second class Boundary conditions

Constraints for u

(¯ v)

Subspace conditions

(5.1.83)

πBγu = ϕ10 ,

−ψ2 , (1 − π)Bγc v Γ

πBγv = 0,

(5.1.84)

πBγc u = ϕ1 + ϕ10

−ψ20 , (1 − π)[Bγv] Γ

πBγc v = 0

(5.1.83)

πBγc u = ϕ11 ,

−ψ2 , (1 − π)Bγv Γ

πBγv = 0,

(5.1.85)

πBγu = ϕ11 − ϕ1

−ψ21 , (1 − π)[Bγv] Γ

πBγc v = 0

(5.1.83)

πBγu = ϕ10 ,

−ψ2 , (1 − π)Bγv Γ

πBγv = 0,

(5.1.86)

πBγc u = ϕ1 + ϕ10

−ψ20 , (1 − π)[Bγv] Γ

πBγc v = 0

(5.1.83)

πBγc u = ϕ11 ,

−ψ2 , (1 − π)Bγc v Γ

πBγv = 0,

(5.1.87)

πBγu = −ϕ1 + ϕ11

−ψ20 , (1 − π)[Bγv] Γ

πBγc v = 0

We note that for all the cases in Table 5.1.2 one could solve the transmission problems equally well by solving the interior and the exterior problems independently. However, in the remaining two cases 2.5, 2.6 we can solve the transmission problems only by solving first one of the interior or exterior problems since one needs the corresponding resulting Cauchy data for solving the remaining problem by making use of the transmission conditions in (5.1.83). To illustrate the idea we now consider case 2.5. Here, first find u ∈ H 1 (Ω) with (1 − π)Bγu = ϕ20 satisfying the variational equation   aΩ (u, v) = f  v¯dx + ψ10 , πBγvΓ Ω

for all v ∈ H 1 (Ω) with (1 − π)Bγv = 0 . Then, with the help of (5.1.83), we have the data (1 − π)N γc u = ψ3 := ψ2 + (1 − π)N γu and πBγc u = ϕ3 := ϕ1 + πBγu for the exterior variational problem (5.1.72). In the case 2.6, the exterior problem is to be solved first for providing the necessary missing Cauchy data for the interior problem. Remark 5.1.4: We note that for the problems above we have restricted ourselves to the case ML = 0 if exterior Neumann problems are involved. For the case ML = 0, the formulation is more involved and will not be presented here.

5.2 Abstract Existence Theorems for Variational Problems

231

5.2 Abstract Existence Theorems for Variational Problems The variational formulation of boundary value problems for partial differential equations (as well as that of boundary integral equations) leads to sesquilinear variational equations in a Hilbert space H of the following form: Find an element u ∈ H such that a(u, v) = (v)

for all v ∈ H .

(5.2.1)

Here a(v, u) is a continuous sesquilinear form on H and (v) is a given continuous linear functional. In order to obtain existence results, we often require further that a satisfies a G˚ arding inequality in the form Re{a(v, v) + (Cv, v)H } ≥ α0 v2H

(5.2.2)

with a positive constant α0 and a compact linear operator C from H into H. We recall the definition of a linear compact operator C between two Banach spaces X and Y. Definition 5.2.1. A linear operator C : X → Y is compact iff the image CB of any bounded set B ⊂ X is a relatively compact subset of Y; i.e. every sequence of CB contains a convergent sub–sequence in Y. G˚ arding’s inequality plays a fundamental rˆ ole in the variational methods for the solution of partial differential equations and boundary integral equations as well as for the stability and convergence analysis in finite and boundary elements [204] and Dautray and Lions [97], Nedelec [328], Sauter and Schwab [371] and Steinbach [399]. In this section we present the well known Lax–Milgram theorem which provides an existence proof for the H–elliptic sesquilinear forms. Since these theorems are crucial in the underlying analysis of our subjects we present this fundamental part of functional analysis here. Definition 5.2.2. A continuous sesquilinear form a(u, v) is called H– elliptic if it satisfies the inequality |a(v, v)| ≥ α0 v2H for all v ∈ H

(5.2.3)

with α0 > 0. Remark 5.2.1: In literature the term of H–ellipticity is some times defined by the slightly stronger condition of strong H–ellipticity (see Stummel [410]), Re a(v, v) ≥ α0 v2H for all v ∈ H with α0 > 0. (See also Lions and Magenes [270, p.201].)

(5.2.4)

232

5. Variational Formulations

Clearly, (5.2.4) implies (5.2.3); in words, strong H–ellipticity implies H– ellipticity since |a(v, v)| ≥ Re a(v, v) ≥ α0 v2H . Hence, when C = 0 in G˚ arding’s inequality (5.2.2) then a(u, v) is strongly H–elliptic. In most applications, however, C = 0 but compact.

5.2.1 The Lax–Milgram Theorem In this section we confine ourselves to the case of H–ellipticity of a(u, v). We begin with two elementary but fundamental theorems. Theorem 5.2.1 (Projection theorem). Let M ⊂ H be a closed subspace of the Hilbert space H and let M⊥ := {v ∈ H  (v, u) = 0 for all u ∈ M}. Then every u ∈ H can uniquely be decomposed into the direct sum ⊥ ⊥ u = u0 + u⊥ 0 with u0 ∈ M and u0 ∈ M .

(5.2.5)

The operator defined by πM : u → u0 =: πM u is a linear continuous projection. In this case one also writes H = M ⊕ M⊥ . Proof: The proof is purely based on geometric arguments in connection with the completeness of a Hilbert space. As shown in Figure 5.2.1, u0 will be the closest point to u in the subspace M.

u u0 M

Figure 5.2.1: H = M ⊕ M⊥

Let d := inf u − vH . v∈M

5.2 Abstract Existence Theorems for Variational Problems

233

Then there exists a sequence {vk } ⊂ M with d = lim u − vk H .

(5.2.6)

k→∞

With the parallelogram equality (u − vk ) + (u − v )2H + (u − vk ) + (u − v )2H = 2(u − vk )2H + 2(u − v )2H we obtain together with (vk − v )2H

v k + v ∈ M the estimate 2

=

2(u − vk )2H + 2(u − v )2H − 4u −

≤

2(u − vk )2H + 2(u − v )2H − 4d2 .

vk + v  2  2

Hence, from (5.2.6) we obtain that vk − v 2H → 0 for k,  → ∞ . Therefore, the sequence {vk } is a Cauchy sequence in H. Thus, it converges to u0 ∈ H due to completeness. Since M is closed, we find u0 = limk→∞ vk ∈ M. If vk → u0 and vk → u0 are two sequences in M with u − vk H → d and u − vk H → d then with the same arguments as above we find vk − vk H ≤ 2u − vk H + 2u − vk H − 4d2 → 0 for k → 0. Hence, u0 = u0 ; i.e. uniqueness of u0 and, hence, of u⊥ 0 , too. Thus, the mapping π is well defined. Since for u0 ∈ M, u0 = πu0 + 0, one finds π 2 u = πu0 = u0 = πu for all u. Hence, π 2 = π. The linearity of π can be seen from ⊥ ⊥ (απu1 + βπu2 ) ∈ M and αu⊥ 1 + βu2 ∈ M

for every u1 , u2 ∈ H and α, β ∈ C, as follows. Since the decomposition ⊥ ⊥ ⊥ αu1 + βu2 = α(πu1 + u⊥ 1 ) + β(πu2 + u2 ) = (απu1 + βπu2 ) + (αu1 + βu2 )

is unique, there holds π(αu1 + βu2 ) = απu1 + βπu2 . From u2H = (u, u)H = (u0 + u⊥ , u0 + u⊥ )H = u0 2H + u⊥ 2H , one immediately finds πuH = u0 H ≤ uH and πH,H = 1 because of πu0 = u0 for u0 ∈ M.



234

5. Variational Formulations

Theorem 5.2.2 (The Riesz representation theorem). To every bounded linear functional F (v) on the Hilbert space H there exists exactly one element f ∈ H such that F (v) = (v, f )H for all v ∈ H . (5.2.7) Moreover, f H = sup |F (v)| . vH ≤1

Remark 5.2.2: For a Hilbert space H with the dual space H∗ defined by all continuous linear functionals equipped with the norm F H∗ = sup |F (v)| , vH ≤1

the Riesz mapping j : H∗ → H defined by jF := f is an isometric isomorphism from H∗ onto H. ∗ Proof: i.) Existence: We first show  the existence of f for any given F ∈ H . The linear space ker(F ) := {v ∈ H  F (v) = 0} is a closed subspace of H since F is continuous. Then either ker(F ) = H and f = 0, or H \ker(F ) = ∅. In the latter case there is a unique decomposition of H due to the Projection Theorem 5.2.1:

H

=

u =

ker(F ) ⊕ {ker(F )}⊥ and u0 + u⊥ with u0 ∈ ker(F ) and (u⊥ , v)H = 0 for all v ∈ ker(F ) .

Since H \ ker(F ) = ∅, there is an element g ∈ {ker(F )}⊥ with F (g) = 0 defining 1 z := g. F (g) Then F (z) = 1 and for every v ∈ H we find F (v − F (v)z) = F (v) − F (v)F (z) = 0 , i.e. v − F (v)z ∈ ker(F ) . Moreover, (v, z)H = (v − F (v)z + F (v)z, z)H = (F (v)z, z)H = F (v)z2H . 1 z will be the desired element, since z2   z 1 (v, f )H = v, = (v, z)H = F (v) for all v ∈ H . z2 H z2

Hence, f :=

5.2 Abstract Existence Theorems for Variational Problems

235

This implies also F H∗ = sup |F (v)| = sup |(v, f )H | = f H . vH ≤1

vH ≤1

ii.) Uniqueness: Suppose f1 , f2 ∈ H are two representing elements for F ∈ H∗ . Then F (v) = (v, f1 )H = (v, f2 )H for all v ∈ H implying (v, f1 − f2 )H = 0 for all v ∈ H . The special choice of v = f1 − f2 yields 0 = f1 − f2 2H = (f1 − f2 , f1 − f2 )H , i.e. f1 = f2 .  Theorem 5.2.3 (Lax–Milgram Lemma). Let a be a continuous H–elliptic sesquilinear form. Then to every bounded linear functional (v) on H there exists a unique solution u ∈ H of equation (5.2.1). Proof: For any fixed u ∈ H, the relation (Au)(v) := a(v, u) defines a bounded linear functional Au ∈ H∗ operating on v ∈ H. By the Riesz representation theorem 5.2.2, there exists a unique element jAu ∈ H with (v, jAu)H = a(v, u) for all v ∈ H . The mappings A : u → Au ∈ H∗ and Bu := jA : H → H are linear and bounded, since a(v, u) is a continuous sesquilinear form (see(5.1.14)). Moreover, M = BH,H = jAH,H = AH,H∗ :=

sup AuH∗ =

uH ≤1

sup u=v=1

|a(v, u)| .

Obviously, the H–ellipticity implies the invertibility of the operator B on its range. So, B : H → range (B) is one–to–one and onto and there B −1 is bounded with B −1 range(B),H ≤ α10 . This implies that range (B) is a closed subset of H. Next we show that range (B) = H. If not then there existed z ∈ H \ range(B) with z = 0 and there  ⊥ was z0 ∈ range(B) with z = 0 due to the Projection Theorem 5.2.1, but a(z0 , z0 ) = (Bz0 , z0 )H = 0 . Then the H–ellipticity implied z0 = 0, a contradiction. Consequently, range (B) = H and the equation

236

5. Variational Formulations

a(u, v) = (Bu, v)H = (v) = (j, v)H for all v ∈ H has a unique solution

u = B −1 j . 

Note that the solution u then is bounded satisfying uH ≤

1 α0 jH

=

1 ∗ α0 H

.

(5.2.8)

As an immediate consequence we have Corollary 5.2.4. Let a(u, v) be a continuous strongly H–elliptic sesquilinear form on H × H, i.e., satisfying (5.2.4) has a unique solution u ∈ H. Clearly, since a is also H–elliptic, we obtain u via the Lax–Milgram lemma, Theorem 5.2.3. However, for the strongly H–elliptic sesquilinear form, we can reduce the problem to Banach’s fixed point principle and present an alternative proof. Proof: The equation (5.2.1) has a solution u if and only if u is a fixed point, u = Q(u) , of the mapping Q defined by Q(w) := w − ρ(jAw − f ) for w ∈ H 0 with the parameter ρ ∈ (0, 2α M 2 ), where α0 is the ellipticity constant in (5.2.4). α0 With this choice of ρ (preferably ρ = M 2 ), Q is a contractive mapping, since

Q(w) − Q(v)2H = (w − v − ρjA(w − v), w − v − ρjA(w − v))H = w − v2H − 2ρRe a(w − v, w − v) + ρ2 jA(w − v)2H ≤ {1 − 2ρα0 + ρ2 M 2 }w − v2H = q 2 w − v2H , for q 2 := (1 − 2ρα0 + ρ2 M 2 ) < 1. Consequently, the successive approximation defined by the sequence {wk }, w0 = 0 and wk+1 := Q(wk ) , k = 0, 1, . . . , converges in H to the solution u = lim wk . Moreover, we see that the solution k→0

u is unique. Since if u1 and u2 are two solutions, then u1 − u2 = Q(u1 ) − Q(u2 ) implies that

5.2 Abstract Existence Theorems for Variational Problems

237

u1 − u2 H ≤ qu1 − u2 H . That is with 0 < q < 1, 0 ≤ (1 − q)u1 − u2 H ≤ 0 , 

which implies that u1 = u2 .

In applications one often obtains H–ellipticity of a sesquilinear form a(·, ·) through G˚ arding’s inequality (5.2.2) provided that Re a is semidefinite. More precisely, we have the following lemma. Lemma 5.2.5. Let the sesquilinear form a(·, ·) satisfy G˚ arding’s inequality (5.2.2) and, in addition, have the property Re a(v, v) > 0 for all v ∈ H with v = 0 . Then a(·, ·) satisfies (5.2.4) and, consequently, is strongly H–elliptic. Proof: The proof rests on the well known weak compactness of the unit sphere in reflexive Banach spaces and in Hilbert spaces (Schechter [376, VIII Theorem 4.2]), i. e. every bounded sequence {vj }j∈IN ⊂ H with vj H ≤ M contains a subsequence vj  with a weak limit v0 ∈ H such that lim (g, vj  )H = (g, v0 )H for every g ∈ H .

j  →∞

We now prove the lemma by contradiction. If a were not strongly H– elliptic then there existed a sequence {vj }j∈IN ⊂ H with vj H = 1 and lim Re a(vj , vj ) = 0 .

j→∞

Then {vj } contained a subsequence {vj  } converging weakly to v0 ∈ H. G˚ arding’s inequality then implied   α0 vj  − v0 2H ≤ Re{a(vj  − v0 , vj  − v0 ) + C(vj  − v0 ), vj  − v0 } ≤

Re{a(vj  , vj  ) − a(v0 , vj  ) − a(vj  , v0 )

H

+ a(v0 , v0 ) + (Cvj  , vj  − v0 )H − (Cv0 , vj  − v0 )H } . Since C is compact, there existed a subsequence {vj  } ⊂ {vj  } ⊂ H such that Cvj  → w ∈ H for j  → ∞. Hence, due to the weak convergence vj   v0 we would have (Cvj  , vj  )H − (Cvj  , v0 )H   a(v0 , vj  ) = (jA)∗ v0 , vj  H

→ →

(w, v0 )H − (w, v0 )H = 0 ,   (jA)∗ v0 , v0 = a(v0 , v0 ) H

and corresponding convergence of the remaining terms on the right–hand side. This yielded

238

5. Variational Formulations

0 ≤ limj  →∞ α0 vj  − v0 2H ≤ −Re a(v0 , v0 ) and, consequently, with Re a(v0 , v0 ) ≥ 0, we could find lim vj  − v0 H = 0

j  →∞

together with Re a(v0 , v0 ) = 0 .

The latter implied v0 = 0 ; however, v0 H = lim vj  H = 1 , j →∞

which is a contradiction. Consequently, inf

vH =1

Re a(v, v) =: α0 > 0 ,

i. e., a is strongly H–elliptic.



In order to generalize the Lax–Milgram lemma to the case where the compact operator C is not zero, we need the following version of the classical Fredholm theorems in Hilbert spaces.

5.3 The Fredholm–Nikolski Theorems 5.3.1 Fredholm’s Alternative We begin with the Fredholm theorem for the standard equations of the second kind T u := (I − C)u = f in H , (5.3.1) where as before C is a compact linear operator on the Hilbert space H into itself. We adapt here the Hilbert space adjoint C ∗ : H → H of C which is defined by (Cu, v)H = (u, C ∗ v)H for all u, v ∈ H . In addition to (5.3.1), we also introduce the adjoint equation T ∗ v := (I − C ∗ )v = g in H .

(5.3.2)

The range of T will be denoted by (T ) and the nullspace of T by N (T ) = ker(T ). Theorem 5.3.1 (Fredholm’s alternative). For equation (5.3.1) and equation (5.3.2), respectively, the following alternative holds: Either i.) N (T ) = {0} and (T ) = H or ii.) 0 < dim N (T ) = dim N (T ∗ ) < ∞. In this case, the inhomogeneous equations (5.3.1) and (5.3.2) have solutions

5.3 The Fredholm–Nikolski Theorems

239

iff the corresponding right–hand sides satisfy the finitely many orthogonality conditions (f, v0 )H

=

0 for all v0 ∈ N (T ∗ )

(5.3.3)

(u0 , g)H

=

0 for all u0 ∈ N (T ) ,

(5.3.4)

and

respectively. If (5.3.3) is satisfied then the general solution of (5.3.1) is of the form k  c u0() u = u∗ + =1 ∗

where u is a particular solution of (5.3.1) depending continuously on f : u∗ H ≤ cf H with a suitable constant c . The u0() for  = 1, . . . , k are the linearly independent eigensolutions with N (T ) = span {u0() }k=1 . Similarly, if (5.3.4) is fulfilled then the solution of (5.3.2) has the representation v = v∗ +

k 

d v0()

=1

where v ∗ is a particular solution of (5.3.2) depending continuously on g; and v0() for  = 1, . . . , k are the linearly independent eigensolutions with N (T ∗ ) = span {v0() }k=1 . Our proof follows closely the presentation in Kantorowitsch and Akilow [222]. For ease of reading we collect the basic results in the following four lemmata and will give the proof at the end of this section. Lemma 5.3.2. The range (T ) is a closed subspace of H. Proof: Let {uk } ⊂ (T ) be a convergent sequence and suppose that u0 = limk→∞ uk . Our aim is to show that u0 ∈ (T ). First, note that by Projection Theorem 5.2.1, N (T ) is closed and we have the unique decomposition H = N (T ) ⊕ N (T )⊥ . Moreover, R(T ) = T (N (T )⊥ ). Since uk ∈ R(T ), there exists vk ∈ N (T )⊥ such that uk = T vk . Now we consider two cases. (1)We assume that the sequence vk H is bounded. By the compactness of the operator C, we take a subsequence {vk } for which Cvk → u when k  → ∞. We denote uk = T vk = vk − Cvk . This implies that

240

5. Variational Formulations

vk = uk + Cvk → v0 = u0 + u . Consequently, Cvk → Cv0 and u0 = v0 − Cv0 = T v0 which implies u0 ∈ R(T ), that is, R(T ) is closed. (2) Suppose that vk H is unbounded. Then there exists a subsequence vk  {vk } with vk H → ∞. Setting vk := , we have vk ∈ N (T )⊥ and vk H  vk H = 1. With uk = T vk we obtain uk  = T vk = vk − C vk → 0 since uk → u0 . vk H Now let { vk } be an other subsequence of vk such that C vk → u . This  implies the convergence vk → u . Hence, u = Cu with u H = 1 . However, vk ∈ N (T )⊥ and so is the limit u ∈ N (T )⊥ . Hence, u ∈ N (T )⊥ ∩ N (T ) = {0}. This contradicts the fact that u H = 1. Therefore, only case (1) is possible.  Lemma 5.3.3. The sequence of sets N (T j ) for j = 0, 1, 2, . . . with T 0 = I = identity is an increasing sequence, i.e. N (T j ) ⊆ N (T j+1 ). There exists a smallest index m ∈ IN0 such that ˙ (T j+1 ) for all j < m . N (T j ) = N (T m ) for all j ≥ m and N (T j )⊂N Proof: The assertion N (T j ) ⊆ N (T j+1 ) is clear. Observe that if there is an m ∈ IN with N (T m+1 ) = N (T m ) , then, by induction, one has N (T j ) = N (T m ) for all j ≥ m. We now prove by contradiction that there exists such an m. Suppose that it would not exist. Then for every j ∈ IN, the set N (T j ) was a proper subspace of N (T j+1 ). Hence, there existed an element vj+1 0 = vj+1 ∈ N (T j+1 ) ∩ N (T j )⊥ . Without loss of generality, we assume that vj+1 H = 1 . This would define a sequence {vj }j∈IN with vj ∈ N (T j ), and there existed a subsequence {vj  } for which Cvj  → u0 .

5.3 The Fredholm–Nikolski Theorems

241

Now, for m > n , consider the difference Cvm − Cvn = (vm − T vm ) − (vn − T vn ) = vm − zm n where 

zm n := vn + T vm − T vn ∈ N (T m −1 ) since 















T m −1 zm n = T m −n −1 (T n vn ) + T m vm − T m −n (T n vn ) = 0 . 



Because of T N (T m ) ⊂ N (T m −1 ) and n < m , this would imply that (zm n , vm )H = 0 . Consequently, vm − zm n 2H = vm 2H + zm n 2H ≥ 1 . On the other hand, we would have vm − zm n H = Cvm − Cvn H ≤ Cvm − u0 H + Cvn − u0 H , which implied that for m > n , vm − zm n H = Cvm − Cvn H → 0 as m , n → ∞ . Since vm − zm n H ≥ 1, we have a contradiction.



Lemma 5.3.4. The ranges j := (T j ) = T j H for j = 0, 1, 2, . . . with 0 = H, form a decreasing sequence of closed subspaces of H, i.e. j+1 ⊆ j . There exists a smallest index r ∈ IN0 such that ˙ j r = j for all j ≥ r and j+1 ⊂

for all j < r .

Proof: Lemma 5.3.2 guarantees that all j ’s are closed subspaces. The inclusion j+1 ⊆ j is clear from the definition. The existence of r will be proved by contradiction. Assume that for every j the space j+1 is a proper subspace of j . Then for every j there would exist an element vj ∈ j ∩ ⊥ j+1 with vj H = 1. Moreover, there existed a subsequence {vj  } such that Cvj  → u0 ∈ H for j  → ∞ , since C is compact. Choose m > n and consider the difference (Cvn − u0 ) − (Cvm − u0 ) = vn − zn m where

242

5. Variational Formulations

zn m := vm + T vn − T vm ∈ n +1 . 



Since with vm = T m um and vn = T n un and with T n ⊂ n +1 ⊆ m for m > n , there would hold 









zn m = T n +1 (T m −n −1 um − T m −n um + un ) whereas vn ∈ ⊥ n +1 . Then, in the same manner as in the previous lemma, we would have vn − zn m 2H = vn 2H + zn m 2H ≥ 1 but vn − zn m H ≤ Cvn − u0 H + Cvm − u0 H → 0 for m > n → ∞ ; 

a contradiction.

Lemma 5.3.5. Let m and r be the indices in the previous lemmata. Then we have: i. m = r. ii. H = N (T r ) ⊕ r , which means that every u ∈ H admits a unique decomposition u = u0 + u1 where u0 ∈ N (T r ) and u1 ∈ r . Proof: i.) We prove m = r in two steps: (a) Assume that m < r. Choose any w ∈ r−1 , hence w = T r−1 v for some v ∈ H. This yields that T w = T r v ∈ r = r+1 by definition of r. Then T w = T r+1 v for some v ∈ H , and, hence, T r (v − T v) = 0 . Now, since m < r by assumption, we have N (T r ) = N (T r−1 ) = · · · = N (T m ) and, thus, T r−1 (v − T v) = 0 implying r−1  w = T r−1 v = T r v ∈ r . Hence, we have the inclusions r−1 ⊆ r ⊆ r−1 and r−1 = r which contradicts the definition of r as to be the smallest index.

5.3 The Fredholm–Nikolski Theorems

243

(b) Assume that r < m. Set v ∈ N (T m ). Then, T m−1 v ∈ m−1 = · · · = r and m = m−1 = r , since m > r. Hence, T m−1 v = T m v for some v ∈ H . Moreover, T m+1 v = T m v = 0 , since v ∈ N (T m ) . The latter implies that v ∈ N (T m+1 ) = N (T m ) by the definition of m . Hence, T m−1 v = T m v = 0 which implies v ∈ N (T m−1 ). Consequently, we have the inclusions N (T m ) ⊆ N (T m−1 ) ⊆ N (T m ) and N (T m−1 ) = N (T m ) which contradicts the definition of m. The result m = r, i.e., i.) then follows from (a) and (b). ii.) For showing H = N (T r ) ⊕ r , let u ∈ H be given. Then T r u ∈ r = 2r and T r u = T 2r u  form some u  ∈ H. Let u1 be defined by u1 := T r u  ∈ r , and let u0 := u − u1 . For u0 , we find T r u0 = T r u − T r u1 = T 2r u  − T 2r u =0 and, hence, u0 ∈ N (T r ). This establishes the decomposition: u = u0 + u1 . For the uniqueness of the decomposition, let v ∈ N (T r ) ∩ r . Then v = T r w for some w ∈ H, and T r v = T 2r w = 0, since v ∈ N (T r ). Hence, w ∈ N (T 2r ) = N (T r ) which implies that v = T r w = 0 and N (T r ) ∩ r = {0} . 

This completes the proof.

Theorem 5.3.6. a)The operator T maps the subspace H := T r (H) = r bijectively onto itself and T : H → H is an isomorphism. b) The space H := N (T r ) is finite–dimensional and T maps H into itself. c) For the decomposition u = u0 + u1 =: P0 u + P1 u in Lemma 5.3.5, the projection operators P0 : H → H and P1 : H → H are both bounded.

244

5. Variational Formulations

d)The compact operator C admits a decomposition of the form C = C0 + C1 where C0 : H → H and C1 : H → H with C0 C1 = C1 C0 = 0 and both operators are compact. Moreover, T1 := I − C1 is an isomorphism on H. Proof: a) The definition of r gives T (r ) = r+1 = r , i. e. surjectivity. The injectivity of T on r follows from r = m since T u0 = 0 for u0 ∈ r implies u0 ∈ N (T ) ⊂ N (T r ) and Lemma 5.3.5 ii.) yields u0 = 0, i. e. injectivity. To show that T|H is an isomorphism on H , it remains to prove that (T|H )−1 on H is bounded. In fact, this follows immediately by using Banach’s closed graph theorem (Schechter [376, p. 65]) since T|H is continuous and bijective and H is a closed subspace of the Hilbert space H due to Lemma 5.3.5. However, we present an elementary contradiction argument by taking advantage of the special form T = I − C together with the compactness of C. Suppose (T|H )−1 is not bounded. Then there exists a sequence of elements vj ∈ H with vj H = 1 such that wj H → ∞ where wj = (T|H )−1 vj . For the corresponding sequence 7 uj := (wj wj H ) ∈ H with uj H = 1 one has T uj = uj − Cuj =

7 1 T wj = (vj wj H ) → 0 for j → ∞ . (5.3.5) wj H

Because of the compactness of the operator C, there exists a subsequence uj  ∈ H such that Cuj  → u0 in H . This implies from (5.3.5) that

7 uj  = Cuj  + vj  wj  H → u0 ,

hence, u0 ∈ H and, moreover, T u0 = u0 − Cu0 = 0 with u0 H = 1 . This contradicts the inectivity of T|H .

5.3 The Fredholm–Nikolski Theorems

b) Since

245

 T r = (I − C)r =: I − C

  the unit sphere {v ∈ H  v ≤ 1} is comwith a compact operator C, pact and, hence, H is finite–dimensional due to a well–known property of normal spaces, see e. g. Schechter [376, p.86]. c) For any u ∈ H define

Then

u1

:=

T1−r T r u =: P1 u ∈ r = H and

u0

:=

u − u1 = u − P1 u =: P0 u .

T r u0 = T r u − T r P1 u = T r u − T1r T1−r T r u = 0

and P0 u ∈ H . Clearly, P1 and P0 are bounded linear operators by definition. Moreover, the decomposition u = u0 + u1 = P0 u + P1 u

(5.3.6)

is unique from Lemma 5.3.5 and we have P0 P1 = P1 P0 = 0 . d) With (5.3.6) we define the compact operators C1 := CP1 and C0 := CP0 . Since C =I −T we have with a) that C|r = C|H = I|H − T|H : H → H . Hence, C1 : H → H . Similarly, for every v0 ∈ H we find that w0 := Cv0 satisfies T r w0 = T r (Cv0 ) = T r (I − T )v0 = T r v0 − T T r v0 = 0 , i. e.

C|H : H → H and C0 : H → H .

The equation C 1 C 0 = C0 C 1 = 0 follows now immediately. The invertibility of T1 = I − C1 follows from T1 u = T1 u1 + T1 u0 = T1 u1 + (I − C1 )u0 = T1 u1 + u0 = P0 f + P1 f with u = (T1|

H

)−1 P1 f + P0 f =: (I − C1 )−1 f

246

5. Variational Formulations

where all the operators on the right–hand side are continuous. Therefore,  I − C1 is an isomorphism on H. This completes the proof. Proof of Theorem 5.3.1: i.) a) If the solution of (5.3.1) is unique then the homogeneous equation T u0 = 0 admits only the trivial solution u0 = 0. Hence, r = m = 0, H = H and T (H) = H in Lemma 5.3.5. The latter implies f ∈ T (H) which yields the existence of u ∈ H with (5.3.1), i. e. solvability and that T : H → H is an isomorphism. b) Suppose that (5.3.1) admits a solution u ∈ H for every given f ∈ H. Then T (H) = H and Lemma 5.3.5 implies m = r = 0 and N (T ) = {0}, i. e. uniqueness. and T : H → H is an isomorphism. ii.) First let us show dim N (T ) = dim N (T ∗ ) < ∞. Since N (T ) ⊂ N (T r ) = H , it follows from Theorem 5.3.6 that dim N (T ) =: k < ∞. Since with C also C ∗ being a compact operator on H, the same arguments imply dim N (T ∗ ) =: k ∗ < ∞. Since a basis u0(1) , . . . , u0(k) is linearly independent, the well known Hahn–Banach theorem (Schechter [375]) implies that there exist ϕj ∈ H with (u0() , ϕj ) = δj for , j = 1, . . . , k . Similarly, there exist ψj ∈ H with (ψj , v0() ) = δj for j,  = 1, . . . , k ∗ . Now let us suppose k < k ∗ . Consider the new operator Tw := T w −

k 

(w, ϕ )ψ .

=1

If w0 ∈ N (T) then 0

=

(v0(j) , Tw0 ) = (T ∗ v0(j) , w0 ) −

k 

(w0 , ϕ )(v0(j) , ψ ) = −(w0 , ϕj )

=1

for j = 1, . . . , k . Hence, Tw0 = T w0 − 0 = 0 and w0 =

k 

α u0() ∈ N (T )

=1

where α are some constants for  = 1, . . . , k. However,

5.3 The Fredholm–Nikolski Theorems

0 = −(w0 , ϕj ) = −

k 

247

α (u0() , ϕj ) = −αj for j = 1, . . . , k

=1

and we find w0 = 0 . Hence, N (T) = {0} and T : H → H is an isomorphism due to i.). Then the equation k  ∗ ∗  (w∗ , ϕ )ψ = ψk+1 Tw = Tw − =1

has exactly one solution w∗ ∈ H for which we find 0 = (v0(k+1) , Tw∗ )

=

(v0(k+1) , T w∗ ) −

k 

(w∗ , ϕ )(v0(k+1) , ψ )

=1

=

(v0(k+1) , ψk+1 ) = 1 ,

an obvious contradiction. Hence, k ∗ ≤ k. In the same manner, by using T ∗∗ = T , we find k ≤ k ∗ which shows dim N (T ) = dim N (T ∗ ) = k . Finally, if Equation (5.3.1) has a solution u then (v0() , f ) = (v0() , T u) = (T ∗ v0() , u) = 0 for  = 1, . . . , k . Moreover, if u∗ is any particular solution of (5.3.1) then u0 := u−u∗ ∈ N (T ). Hence, k  α u0() . u = u∗ + u0 = u∗ + =1

Conversely, if (v0() , f ) = 0 is satisfied for  = 1, . . . , k then solve the equation Tu∗ = T u∗ −

k 

(u∗ , ϕ )ψ = f

=1 ∗

which admits exactly one solution u ∈ H due to i. and, furthermore, u∗ H ≤ T−1 H,H f H since T is an isomorphism. For this solution we see that (v0(j) , f ) = 0 =

(v0(j) , T ur −

k 

(u∗ , ϕ )ψ ) = −(u∗ , ϕj )

=1

for every j = 1, . . . , k . Hence, u∗ is a particular solution of (5.3.1). Clearly, any u = u∗ + u0 with u0 ∈ N (T ) solves (5.3.1). This completes the proof of Theorem 5.3.1.



248

5. Variational Formulations

5.3.2 The Riesz–Schauder and the Nikolski Theorems Fredholm’s alternative can be generalized to operator equations between different Hilbert and Banach spaces which dates back to Riesz, Schauder and, more recently, to Nikolski. Here, for simplicity, we confine ourselves to the Hilbert space setting. Let T : H 1 → H2 be a linear bounded operator between the two Hilbert spaces H1 and H2 . If T = TI − TC

(5.3.7)

where TI is an isomorphism while TC is compact from H1 into H2 , then for T and its adjoint T ∗ defined by the relation (T u, v)H2 = (u, T ∗ v)H1 for all u ∈ H1 , v ∈ H2 ,

(5.3.8)

Fredholm’s alternative remains valid in the following form. Theorem 5.3.7 (Riesz–Schauder Theorem). For the operator T of the form in (5.3.7) and its adjoint operator T ∗ defined by (5.3.8) the following alternative holds: Either i. N (T ) = {0} and (T ) = H2 or ii. 0 < dim N (T ) = dim N (T ∗ ) < ∞. In this case, the equations T u = f and T ∗ v = g have solutions u ∈ H1 or v ∈ H2 , respectively, iff the corresponding right– hand side satisfies the finitely many orthogonality conditions or

(f, v0 )H2 (g, u0 )H1

=0 =0

for all v0 ∈ N (T ∗ ) for all u0 ∈ N (T ) ,

respectively. If these conditions are fulfilled then the general solution is of the form u = u∗ + u 0 or v = v ∗ + v0 (5.3.9) with a particular solution u∗ or v ∗ depending continuously on f or g; and any u 0 ∈ N (T ) or v0 ∈ N (T ∗ ), respectively. Proof: In order to apply Theorem 5.3.1 we define the compact operator C := TI−1 TC in H1 . Since TI is an isomorphism, i. e. TI−1 H1 = H2 , one can show easily

5.3 The Fredholm–Nikolski Theorems

249

C ∗ = TC∗ (TI∗ )−1 from the definition of the adjoint operator, (Cu, w)H1 = (u, C ∗ w)H1 for all u, w ∈ H1 . Then Fredholm’s alternative, Theorem 5.3.1 is valid for the transformed equations TI−1 T u = (I − C)u = TI−1 f in H1

(5.3.10)

and (I − C ∗ )w = g in H1 where w = TI∗ v. Moreover, one verifies that the following relations hold: (T ) = TI (I − C) ∗

∗

(T ) = (I − C )

and

N (T ) = N (I − C) ; ∗

and N (T ) =

(TI∗ )−1 N (I

(5.3.11) ∗

− C ).

(5.3.12)

i. Equations (5.3.11) imply N (T ) = {0} = N (I − C) and H1 = (I − C) is equivalent to H2 = (T ) . ii. Because of (5.3.3), Equation (5.3.10) admits a solution iff (TI−1 f, w0 )H1 = 0 for all w0 ∈ N (I − C ∗ ) . The latter is equivalent to (f, (TI∗ )−1 w0 )H2 = (f, v0 )H2 = 0 for all v0 = (TI∗ )−1 w0 with w0 ∈ N (I − C ∗ ). Then it follows from (5.3.12) that v0 traces the whole eigenspace N (T ∗ ) = (TI∗ )−1 N (I − C ∗ ). This proves the claims for the equation T u = f . For the adjoint equation T ∗v = g we can proceed in the same manner by making use of both equations in (5.3.12) and the second relation in (5.3.11). The representation formulae (5.3.9) for the solutions can be derived by using the relations (5.3.11) and (5.3.12) again together with the corresponding representations in Theorem 5.3.1. This completes the proof.  For the Riesz–Schauder Theorem 5.3.7, the assumptions on the operator T in (5.3.7) are not only sufficient but also necessary as will be shown in the following theorem.

250

5. Variational Formulations

Theorem 5.3.8 (Nikolski’s Theorem). Let T : H1 → H2 be a bounded linear operator for which the Riesz–Schauder theorem 5.3.7 holds. Then T is of the form T = T I − TF (5.3.13) with some isomorphic operator TI ; H1 → H2 and a finite–dimensional operator TF . The latter means that there exist two finite–dimensional orthogonal sets {ϕj }kj=1 ∈ H1 and {ψj }kj=1 ∈ H2 such that TF u =

k 

(u, ϕj )H1 ψj .

(5.3.14)

j=1

Remark 5.3.1: Note that TF is a compact operator and, hence, the representation (5.3.13) is a special case of the operators in (5.3.7) considered in the Riesz–Schauder Theorem 5.3.7. Proof: Let u0() ∈ H1 and v0() ∈ H2 be orthogonal bases of the k– dimensional nullspaces N (T ) and N (T ∗ ) in the Riesz–Schauder theorem 5.3.7, respectively. Then define corresponding orthogonal systems ϕj ∈ H1 , ψj ∈ H2 satisfying (u0() , ϕj )H1 = δj and (ψj , v0() )H2 = δj for j,  = 1, . . . , k . With these elements we introduce the finite–dimensional operator TF as in (5.3.14) and define the bounded linear operator TI := T + TF . It remains now to show that TI : H1 → H2 . i. Injectivity: Suppose u ∈ H1 with TI u = 0 . Then u solves Tu = −

k 

(u, ϕj )H1 ψj .

j=1

Hence, the validity of the orthogonality conditions in Theorem 5.3.7 implies −

k 

(u, ϕj )H1 (ψj , v0() )H2 = −(u, ϕ )H1 = 0 for  = 1, . . . , k .

j=1

Hence, Tu = 0

(5.3.15)

5.3 The Fredholm–Nikolski Theorems

251

and, therefore, u=

k 

α u0()

=1

due to the assumed validity of the Riesz–Schauder theorem 5.3.7. However, equations (5.3.15) imply α = 0 for  = 1, . . . , k, hence, u ≡ 0. ii. Surjectivity: Given any f ∈ H2 . We show that there exists a solution w ∈ H1 of TI w = f . To this end consider f  := f −

k 

(f, v0(j) )H2 ψj

j=1

which satisfies

(f  , v0() )H2 = 0 for  = 1, . . . , k .

Hence, by the Riesz–Schauder theorem 5.3.7, there exists a family of solutions w ∈ H1 of the form  w = w∗ + αj u0(j) with αj ∈ C j=1

to

T w = f 

and

w∗ H1 ≤ cf  H2 ≤ c f H2 ,

the last inequality following from the construction of f  . In particular, for the special choice αj := (f, v0(j) )H2 − (w∗ , ϕj )H1 this yields TI w 

= =

T w  + TF w  k k   f + (w∗ + α u0() , ϕj )H1 ψj j=1

=

f−

k 

=1

(f, v0(j) )H2 ψj +

j=1

=

k  k  j=1 =1

f,

which completes the proof of the surjectivity. iii. Continuity of (TI )−1 : H2 → H1 :

(f, v0() )H2 (u0() , ϕj )H1 ψj

252

5. Variational Formulations

The definition of the αj implies with the continuous dependence of w∗ on f that |αj | ≤ c f H2 . Hence, we also have w H1 = (TI )−1 f H1 ≤ c f H2 

which is the desired continuity.

As an easy consequence of Theorems 5.3.7 and 5.3.8 and Remark 5.3.1 we find the following result. Corollary 5.3.9. Let T : H1 → H2 be a bounded linear operator. Then the following statements are equivalent: – T = TI − TC in (5.3.7). – Fredholm’s alternative in the form of the Riesz–Schauder theorem 5.3.7 is valid for T . – T = TI −TF in (5.3.13) where TI is an isomorphism (which can be different from TI ) and TF is a finite–dimensional operator. 5.3.3 Fredholm’s Alternative for Sesquilinear Forms In the context of variational problems one is lead to variational equations of the form (5.2.1) where the sesquilinear form a(·, ·) satisfies a G˚ arding inequality (5.2.2) and where, in general, the compact operator C does not vanish. Then Theorem 5.3.7 can be used to establish Fredholm’s alternative for the variational equation (5.2.1) which generalizes the Lax–Milgram theorem 5.2.3. For this purpose, let us also introduce the homogeneous variational problems for the sesquilinear form a and its adjoint: Find u0 ∈ H such that a(v, u0 )

=

0 for all v ∈ H

(5.3.16)

a∗ (u, v0 ) = a(v0 , u)

=

0 for all u ∈ H .

(5.3.17)

and v0 ∈ H such that

The latter is the adjoint problem to (5.3.16). Theorem 5.3.10. For the variational equation (5.2.1) in the Hilbert space H where the continuous sesquilinear form a(·, ·) satisfies G˚ arding’s inequality (5.2.2), there holds the alternative: Either i. (5.2.1) has exactly one solution u ∈ H for every given  ∈ H∗ or

5.3 The Fredholm–Nikolski Theorems

253

ii. The homogeneous problems (5.3.16) and (5.3.17) have finite–dimensional nullspaces of the same dimension k > 0. The nonhomogeneous problem (5.2.1) and its adjoint: Find v ∈ H such that a(v, w) = ∗ (w) for all w ∈ H have solutions iff the orthogonality conditions (v0(j) ) = 0 , respectively, ∗ (u0(j) ) = 0 for j = 1, . . . , k, are satisfied where {u0(j) }kj=1 spans the eigenspace of (5.3.16) and {v0(j) }kj=1 spans the eigenspace of (5.3.17), respectively. Proof: First let us rewrite the variational equation (5.2.1) in the form a(v, u) = (v, jAu)H = (v, f )H = (v) where the mappings A : H → H∗ and jA : H → H are linear and bounded and where f ∈ H represents  ∈ H∗ due to the Riesz representation theorem 5.2.2. Then we can identify the variational equation (5.2.1) with the operator equation (5.3.18) (TI − TC )u = jAu = f where TI := jA + C , TC = C with the compact operator C in G˚ arding’s inequality (5.2.2). Because of the Lax–Milgram theorem 5.2.3, the H–elliptic operator TI defines an isomorphism from H onto H. Similarly, the adjoint problem: Find v ∈ H such that (w, (jA)∗ )H = a(v, w) = ∗ (w) = (w, g)H for all w ∈ H , can be rewritten as to find the solution v ∈ H of the operator equation (TI∗ − TC∗ )v = (jA)∗ v = g

(5.3.19)

with TI∗ = (jA)∗ + C ∗ and TC∗ = C ∗ . The operators TI and TC satisfy all of the assumptions in the Riesz– Schauder theorem 5.3.7 with H1 = H2 = H. An application of that theorem to equations (5.3.18) and (5.3.19) provides all claims in Theorem 5.3.10.  Remark 5.3.2: In the case that the G˚ arding inequality (5.2.2) is replaced by the weaker condition |a(v, v) + (Cv, v)H | ≥ α0 v2H then Theorem 5.3.10 remains valid with the same proof.

(5.3.20)

254

5. Variational Formulations

5.3.4 Fredholm Operators We conclude this section by including the following theorems on Fredholm operators on Hilbert spaces. (For general Banach spaces see e.g. Mikhlin and Pr¨ ossdorf [305] where one may find detailed results and historical remarks concerning Fredholm operators.) Definition 5.3.1. Let H1 and H2 be two Hilbert spaces. The linear bounded operator A : H1 → H2 is called a Fredholm operator if the following conditions are satisfied: i The nullspace N (A) of A has finite dimension, ii the range (A) of A is a closed subspace of H2 , iii the range (A) has finite codimension. The number index(A) = dim N (A) − codim (A)

(5.3.21)

is called the Fredholm index of A. Theorem 5.3.11. The linear bounded operator A : H1 → H2 is a Fredholm operator if and only if there exist continuous linear operators Q1 , Q2 : H1 → H2 such that (5.3.22) Q1 A = I − C1 and AQ2 = I − C2 with compact linear operators C1 in H1 and C2 in H2 . If both B : H1 → H2 and A : H2 → H3 are Fredholm operators then A ◦ B also is a Fredholm operator from H1 to H3 and index(A ◦ B) = index(A) + index(B) .

(5.3.23)

For a Fredholm operator A and a compact operator C, the sum A + C is a Fredholm operator and index(A + C) = index(A) .

(5.3.24)

The set of Fredholm operators is an open subset in the space of bounded linear operators and the index is a continuous function. The adjoint operator A∗ of a Fredholm operator A is also a Fredholm operator and index(A∗ ) = −index(A) .

(5.3.25)

Au = f

(5.3.26)

The equation with f ∈ H2 is solvable if and only if (f, v0 )H2 = 0 for all v0 ∈ N (A∗ ) ⊂ H2 .

(5.3.27)

Moreover, dim N (A∗ ) = codim (A) and dim N (A) = codim (A∗ ) .

5.4 G˚ arding’s Inequality for Boundary Value Problems

255

5.4 G˚ arding’s Inequality for Boundary Value Problems In order to apply the abstract existence results for sesquilinear forms in Section 5.2 to the solution of the boundary value problems defined in Section 5.1, we need to verify the G˚ arding inequality for the sesquilinear form corresponding to the variational formulation of the boundary value problem. This is by no means a simple task. It depends on the particular underlying Hilbert space which depends on the boundary conditions together with the partial differential operators involved. In the most desirable situation, if the partial differential operators satisfy some strong, restrictive ellipticity conditions, the G˚ arding inequality can be established on the whole space and every regular boundary value problem with such a differential operator allows a variational treatment based on the validity of G˚ arding’s inequality. We shall present first the simple case of second order scalar equations and systems and then briefly present the cases of one 2m–th order equation. 5.4.1 G˚ arding’s Inequality for Second Order Strongly Elliptic Equations in Ω In this subsection we shall consider systems of the form (5.1.1) and begin with the scalar case p = 1. The case of one scalar second order equation For the G˚ arding inequality, let us confine to the case of one real equation, i.e. the ajk (x) are real–valued functions and Θ = 1 in (3.6.2). Then the uniform strong ellipticity condition takes the form n 

ajk (x)ξj ξk ≥ α0 |ξ|2 for all ξ ∈ IRn

(5.4.1)

j,k=1

with α0 > 0. Without loss of generality, we may also assume ajk = akj . In this case, we can easily show that G˚ arding’s inequality is satisfied on the whole space H 1 (Ω). Lemma 5.4.1. Let P in (5.1.1) be a scalar second order differential operator with real coefficients satisfying (5.4.1). Then aΩ (u, v) given by (5.1.3) arding is a continuous sesquilinear form on H = H 1 (Ω) and satisfies the G˚ inequality Re{aΩ (v, v) + (Cv, v)H 1 (Ω) } ≥ α0 v2H 1 (Ω) for all v ∈ H 1 (Ω) ,

(5.4.2)

where the compact operator C is defined by (Cu, v)H 1 (Ω) := −

  n Ω

j=1

bj (x)

 ∂u (x) + (c(x) − α0 )u(x) v(x)dx (5.4.3) ∂xj

256

5. Variational Formulations

via the Riesz representation theorem in H 1 (Ω). Moreover, for the scalar opaΩ , cf. (5.1.28) and a erator P , (5.4.2) remains valid if aΩ is replaced by   = C + additional lower order terms due to the obvious compact operator C modifications in (5.1.28). Proof: The continuity of aΩ (u, v) given by (5.1.3), in H 1 (Ω) × H 1 (Ω) follows directly from the Cauchy–Schwarz inequality and the boundedness of the coefficients of P . By definition, let bΩ (u, v) := aΩ (u, v) + (Cu, v)H 1 (Ω)    n  ∂u  ∂v ajk = dx + α0 u(x) v(x)dx ∂xk ∂xj Ω j,k=1

Ω

with C given by (5.4.3). As can be easily verified, bΩ (u, v) = bΩ (v, u) and, hence, bΩ (v, v) is real. Then, applying (5.4.1) to the first terms on the right for u = v yields the inequality   n      ∂v 2 2   dx + |v| dx = α0 v2H 1 (Ω) . Re bΩ (v, v) = bΩ (v, v) ≥ α0  ∂xj  j=1 Ω

Ω

What remains to be shown, is the compactness of the operator C. Since (u, C ∗ v)H 1 (Ω) = (Cu, v)H 1 (Ω) we find from (5.4.3) |(u, C ∗ v)H 1 (Ω) | ≤ c uH 1 (Ω) vL2 (Ω) where c = maxx∈Ω∧j=1,...,n {|bj (x)|, |C(x) − α0 |} which implies that C ∗ satisfies C ∗ vH 1 (Ω) ≤ c vL2 (Ω) . Hence, C ∗ maps L2 (Ω) into H 1 (Ω) continuously. Since strong Lipschitz domains enjoy the uniform cone property, Rellich’s lemma (4.1.31) implies compactness of C ∗ : H 1 (Ω) → H 1 (Ω). Therefore, C becomes compact, too. The G˚ arding inequality with  aΩ instead of aΩ follows from (5.4.2) since the two principal parts differ only by the skew–symmetric term (5.1.28) due to (5.1.26), which cancel each other for u = v, and lower order terms.  The case of second order systems In the scalar case, Lemma 5.4.1 shows that G˚ arding’s inequality is valid on the whole space. However, for the second order uniformly strongly elliptic system (5.1.1) with p > 1, G˚ arding’s inequality remains valid only on the arding’s theorem which we subspace H01 (Ω). The latter is the well–known G˚ state without proof.

5.4 G˚ arding’s Inequality for Boundary Value Problems

257

Theorem 5.4.2. (G˚ arding [137]) (see also Fichera [121, p.365], Neˇcas [326, Th´eor`eme 7.3. p.185], Wloka [439, Theorem 19.2]) For a bounded domain Ω ⊂ IRn and P having continuous coefficients in Ω, the uniform strong ellipticity (3.6.2) with Θ = I is necessary and sufficient for the validity of G˚ arding’s inequality on H01 (Ω): Re{aΩ (v, v) + (Cv, v)H01 (Ω) } ≥ α0 v2H 1 (Ω) for all v ∈ H01 (Ω) .

(5.4.4)

On the whole space H 1 (Ω), however, the validity of G˚ arding’s inequality is by no means trivial. We now collect some corresponding results. Definition 5.4.1. The system (5.1.1) of second order is called very strongly elliptic at the point x ∈ IRn , if there exists α0 (x) > 0 such that Re

n 

ζj ajk (x)ζk ≥ α0 (x)

j,k=1

n 

|ζ |2 for all ζ ∈ Cp

(5.4.5)

=1

(see Neˇcas [326, p.186] and for real equations Ladyˇzenskaya [257, p.296]). The system (5.1.1) is called uniformly very strongly elliptic in Ω if α0 (x) ≥ α0 > 0 for all x ∈ Ω with a constant α0 . Theorem 5.4.3. If the system (5.1.1) is uniformly very strongly elliptic in Ω then G˚ arding’s inequalities Re{aΩ (v, v) + (Cv, v)H 1 (Ω) } ≥ α0 v2H 1 (Ω)

(5.4.6)

holds for all v ∈ H 1 (Ω), i.e. on the whole space H 1 (Ω).  v)H 1 (Ω) } ≥ α0 v2 1 Re{ aΩ (v, v) + (Cv, H (Ω)

(5.4.7)

holds for all v ∈ H 1 (Ω) provided ajk = akj = a∗jk . The proof of (5.4.6) can be found in Neˇcas [326, p.186]. The very strong ellipticity is only a sufficient condition for the validity of (5.4.6). For instance, the important Lam´e system is strongly elliptic but not very strongly elliptic. Nevertheless, as will be shown, here we still have G˚ arding’s inequality on the whole space. This will lead to the formally positive elliptic systems which include the Lam´e system in an isotropic medium as a particular case. To illustrate the idea, we begin with the general linear system of elasticity in the form of the equations of equilibrium, (P u)j = −

n  ∂ σjk (x) = fj (x) for j = 1, · · · , n . ∂xk

(5.4.8)

k=1

Here, fj (x) denotes the components of the given body force field and σjk (x) is the stress tensor, related to the strain tensor

258

5. Variational Formulations

εjk (x) = via Hooke’s law σjk (x) =

1  ∂uj ∂uk  + 2 ∂xk ∂xj

n 

(5.4.9)

cjkm (x)εm (x) .

(5.4.10)

,m=1

The elasticity tensor cjkm has the symmetry properties cjkm = ckjm = cjkm = cmjk .

(5.4.11)

where cjkm are called elastic constants. If the elastic constants do not depend on the position in the medium, the medium is called homogeneous. Otherwise the medium is called inhomogeneous The stress and strain components and the elastic constants, related through Hook’s law, depend on the orientation of the coordinate axes. The mediumis said to be isotropic, if its elastic constants cjkm do not depend on the orientation of the coordinate axes or, in other words, if the elastic properties of the medium are the same in all directions. If the medium is not isotropic, it is called anisotropic. Therefore, in the case n = 3 or n = 2 there are only 21 or 6 different entries, respectively, the elasticities which define the symmetric positive definite 3(n − 1) × 3(n − 1) Sommerfeld matrix ⎞ ⎛ c1111 c1122 c1133 c1123 c1131 c1112 ⎜ · c2222 c2233 c2223 c2231 c2212 ⎟ ⎟ ⎜ ⎜ · · c3333 c3323 c3331 c3312 ⎟  ⎟ for n = 3 , ⎜ C(x) = ⎜ · · c2323 c2331 c2312 ⎟ ⎟ ⎜ · ⎝ · · · · c3131 c3112 ⎠ · · · · · c1212 ⎛

and

c1111  C(x) =⎝ · ·

c1122 c2222 ·

⎞ c1112 c2212 ⎠ c1212

for n = 2 .

For the isotropic material, the elasticities are explicitly given by c1111 c2323 c1122 cjkm

= = = =

c2222 c3131 c1133 0

= = =

c3333 c1212 c2233 for all remaining indices .

= = =

2μ + λ , μ, λ and

Following Leis [263], we introduce the matrix G of differential operators generalizing the Nabla operator: ⎛

∂ ∂x1

G := ⎝ 0 0

0 ∂ ∂x2

0

0 0 ∂ ∂x3

0 ∂ ∂x3 ∂ ∂x2

∂ ∂x3

0

∂ ∂x1

∂ ⎞ ∂x2 ∂ ⎠ ∂x1

0

for n = 3 ,

5.4 G˚ arding’s Inequality for Boundary Value Problems

whereas for n = 2, delete the columns containing For convenience, we rewrite G in the form G=

n 

Gk

k=1

∂ ∂x3

259

and the last row.

∂ ∂xk

where, for each fixed k = 1, . . . , n, Gk = ((Gmk ))=1...,3(n−1) is a constant m=1,··· ,n

3(n − 1) × n matrix of the same form as G. In terms of the Sommerfeld matrix  and of G, the equations of equilibrium (5.4.8) can be rewritten in the form C Pu = −

n  j,k=1



∂ ∂xj

ajk

∂u ∂xk



 := −G  CGu =−

n  j,k=1

∂  ∂ Gj CGk u=f. ∂xj ∂xk (5.4.12)

Then (5.1.2) reads 





(P u) v¯dx = aΩ (u, v) − Ω

(∂ν u) v¯ds

Γ

where the sesquilinear form aΩ is given by aΩ (u, v) =

  n Ω

∂u     ∂¯ v  Gk dx C ∂xj ∂xk n

Gj

j=1

(5.4.13)

k=1

and the boundary traction is the corresponding conormal derivative, ∂ν u =

n 

 k nj Gj CG

j,k=1

  ∂u = nk σjk . ∂xk j=1,··· ,n n

(5.4.14)

k=1

 In this sesquilinear form, the elasticities C(x) form a symmetric 3(n − 1) × 3(n − 1) matrix, which is assumed to be uniformly positive definite. Consequently, we have  aΩ (u, u) ≥ Ω

 n  ∂u 2  α0 (x) Gk  dx ≥ α0 |Gu|2IR3(n−1) dx ∂xk k=1

Ω

with a positive constant α0 , and where | · |IR3(n−1) denotes the 3(n − 1)– Euclidean norm. A simple computation shows  2 n  n  n    ∂uj 2   ∂uj ∂uk    +  |Gu|2IR3(n−1) = + ≥ ε2jk .  ∂xj   ∂xk  ∂x j j,k=1 j=1 j,k=1

j=k

Hence,

260

5. Variational Formulations

 aΩ (u, u) ≥ α0

n 

εjk (x)2 dx ,

(5.4.15)

Ω j,k=1

and G˚ arding’s inequality in the form (5.4.6) on the whole space H 1 (Ω) follows from the celebrated Korn’s inequality given in the Lemma below, in combination with the compactness of the imbedding H 1 (Ω) → L2 (Ω). Lemma 5.4.4. (Korn’s inequality) For a bounded strong Lipschitz domain Ω there exists a constant c > 0 such that the inequality   n

εjk (x)2 dx ≥ cu2H 1 (Ω) − u2L2 (Ω)

(5.4.16)

Ω j,k=1

holds for all u ∈ H 1 (Ω). The proof of Korn’s inequality (5.4.16) is by no means trivial. A direct proof of (5.4.16) is given in Fichera [121], see also Ciarlet and Ciarlet [69], Kondratiev and Oleinik [236] and Nitsche [338]. However, as we mentioned previously, the elasticity system belongs to the class of systems called formally positive elliptic systems. For this class, under additional assumptions, one can establish G˚ arding’s inequality on the whole space H 1 (Ω) by a different approach which then also implies Korn’s inequality. Definition 5.4.2. (Neˇcas [326, Section 3.7.4], Schechter [375], Vishik [426]) The system (5.1.1) is called formally positive elliptic if the coefficients of the principal part can be written as ajk (x) =

t 

¯rj (x) rk (x) for j, k = 1, · · · , n

(5.4.17)

r=1

  with t ≥ p complex vector–valued functions rk = rk1 (x), · · · , rkp (x) which satisfy the condition t  n     rj (x)ξj  = 0 for all 0 = ξ = (ξ1 , · · · , ξn ) ∈ IRn  r=1

(5.4.18)

j=1

which is referred to as the collective ellipticity Schechter [375]. Note that ajk = a∗kj

(5.4.19)

We now state the following theorem. Theorem 5.4.5. (Neˇcas [326, Th´eor`eme 3.7.7]) Let Ω be a bounded, strong Lipschitz domain and let the system (5.1.1) be formally positive elliptic with coefficients rj (x) in (5.4.17) satisfying the additional rank condition:

5.4 G˚ arding’s Inequality for Boundary Value Problems

rank

n 

 rj ξj

j=1

r=1,··· ,t

261

= p for every ξ = (ξ1 , · · · , ξn ) ∈ Cn \ {0} .

(5.4.20) aΩ in (5.1.28) satisfy Then the sesquilinear form aΩ in (5.1.3) as well as  G˚ arding’s inequalities (5.4.6) and (5.4.7), respectively, on the whole space H 1 (Ω). We note that the rank condition (5.4.20) always implies the collective ellipticity condition (5.4.18); the converse, however, is not true in general. The G˚ arding inequality for  aΩ follows from that for aΩ with the same arguments as before in Theorem 5.4.3. The proof of Theorem 5.4.5 relies on the following crucial a–priori estimate: u2H 1 (Ω)

≤

t    c u2L2 (Ω) + Lr u2L2 (Ω)

(5.4.21)

r=1

where Lr u

=

n 

 rk

k=1

∂u , ∂xk

which holds under the assumptions of Theorem 5.4.5 for the collectively elliptic system Lr . For its derivation see Neˇcas [326, Th´eor`eme 3.7.6]. For the elasticity system (5.4.12) we now show that it satisfies all the above assumptions of Theorem 5.4.5. We note that the Sommerfeld matrix  C(x) admits a unique square root which is again a 3(n − 1) × 3(n − 1) positive definite symmetric matrix which will be denoted by 1

( )  12 (x) = (( crq2 ))3(n−1)×3(n−1) . C

In particular, for isotropic materials we have explicitly ⎛ ⎞ ζ +η ζ +η ζ +η 0 0 0 0 ⎜ · ζ −η ζ −η 0 0 0 0 ⎟ ⎜ ⎟ ⎜ · ζ −η 0 0 0 0 ⎟  12 = 1 ⎜ · ⎟ (5.4.22) √ C · · 3 μ 0 0 0 ⎟ 3⎜ ⎜ · ⎟ √ ⎝ · 0 ⎠ · · · · 3 μ √ · · · · · 0 3 μ 8 8 for n = 3 with ζ = 2μ + nλ and η = 2μ , and where, for n = 2 the rows and columns with r, q = 3, 5 and 6 are to be  12 , the system (5.4.12) takes the form deleted. In terms of C −

n  j,k=1

∂ ∂xj

 ajk (x)

∂u ∂xk

 =−

n  j,k=1

 ∂  1  12 Gk ) ∂u = f . (C 2 Gj ) (C ∂xj ∂xk

262

5. Variational Formulations

Then the product of the two matrices can also be written as 

3(n−1)

 12 (x)Gj ) (C  12 (x)Gk ) = (C

rj (x) rk (x) = ajk (x)

(5.4.23)

r=1

with the real valued column vector fields rj (x) given by  2 (x) = (1j (x), · · · , 3(n−1),j (x)) for j = 1, · · · , n . Gj C 1

In order to apply Theorem 5.4.5 to the elasticity system it remains to verify the rank condition (5.4.20) which will be shown by contradiction. Then (5.4.20) implies condition (5.4.18). Suppose that the rank is less than p = n. Then there existed some complex n vector ξ = (ξ1 , · · · , ξn ) = 0 and j=1 rj ξj = 0 for all r = 1, · · · , 3(n − 1). Then, clearly, 3(n−1) n   ¯ (rj (x) rk (x))ξj ξk = 0 r=1

j,k=1

which would imply with (5.4.23) that the n × n matrix equation n 

¯  (Gj ξj ) C(x)(G k ξk ) = 0

j,k=1

holds and, hence, the positive ⎧ ⎛ ⎪ ξ¯1 ⎪ ⎪ ⎪ ⎝0 ⎪ n ⎨  0 0= Gk ξ¯k = ⎪  ⎪ k=1 ⎪ ¯ ⎪ ξ1 ⎪ ⎩ 0

 definiteness of C(x) would imply 0 ξ¯2 0 0 ξ¯2

0 0 0 ξ¯3 ¯ ξ3 ξ¯2  ξ¯2 ξ¯1

ξ¯3 0 ξ¯1

⎞ ξ¯2 ξ¯1 ⎠ 0

for n = 3 ,

for n = 2 .

The latter yields ξ = (ξ1 , · · · , ξn ) = 0 which contradicts our supposition ξ = 0. Hence, the rank equals p = n. This completes the verification of all the assumptions required for the elasticity system to provide G˚ arding’s inequality on H 1 (Ω) for the associated sesquilinear form (5.4.13). 5.4.2 The Stokes System The Stokes system (2.3.1) does not belong to the class of second order systems, elliptic in the sense of Petrovskii but nevertheless, in Section 2.3, Tables 2.3.3 and 2.3.4, we formulated boundary integral equations of various kinds for the hydrodynamic Dirichlet and traction problems. Here we formulate corresponding bilinear variational problems in the domain Ω and appropriate G˚ arding inequalities based on the first Green’s identity for the Stokes system

5.4 G˚ arding’s Inequality for Boundary Value Problems

263

(see Dautray and Lions [95], Galdi [136], Kohr and Pop [233], Ladyˇzenskaya [256], Temam [417], Varnhorn [424]). Find (u, p) ∈ H 1 (Ω) × L2 (Ω) such that   aΩ (u, v) − b(p, v) + b (u, q) = f · vdx + T (u) · vds (5.4.24) Ω

Γ

is satisfied for (v, q) ∈ H 1 (Ω) × L2 (Ω). Here the bilinear forms in (5.4.24) are defined as n  

n   ∂uj ∂vk aΩ (u, v) := μ ∇uj · ∇vj dx + μ dx , ∂x k ∂xj j=1 Ω j,k=1 Ω   b(p, v) := p∇ · vdx , b (u, q) := (∇ · u)qdx , Ω

(5.4.25)

Ω

and μ is the fluid’s dynamic viscosity. Clearly, aΩ is continuous on H 1 (Ω) × H 1 (Ω) and b on L2 (Ω) × H 1 (Ω). In addition to (5.4.24) one has to append boundary conditions on Γ such as, e.g., the Dirichlet or traction conditions, and to specify the trial and test fields accordingly. Since the bilinear form aΩ can also be written as aΩ (u, v) = 2μ

n  

εjk (u)εjk (v)dx

(5.4.26)

j,k=1 Ω

with the strain tensor εjk given by (5.4.9), the Korn inequality (5.4.16) yields the G˚ arding inequality: aΩ (v, v) ≥ cv2H 1 (Ω) − (Cv, v)H 1 (Ω) for all v ∈ H 1 (Ω)

(5.4.27)

where (Cv, u)H 1 (Ω) = μ(v, u)L2 (Ω) and C is compact in H 1 (Ω) due to the Rellich Lemma (4.1.31). Hence, (5.4.24) is to be treated as a typical saddle point problem (Brezzi and Fortin [45, II Theorem11]). In order to formulate the boundary condition with the weak formulation (5.4.24) properly we again need an extension of the mapping (2.3.6), (u, p) → T (u) to an appropriate subspace of H 1 (Ω) × L2 (Ω) as in Lemma 5.1.1, and therefore define  −1 (Ω)} H 1 (Ω, Pst ) := {(u, p) ∈ H 1 (Ω) × L2 (Ω) | ∇p − μΔp ∈ H 0

(5.4.28)

equipped with the graph norm (u, p)H 1 (Ω,Pst ) := uH 1 (Ω) + pL2 (Ω) + ∇p − μΔuH −1 (Ω) . (5.4.29)

264

5. Variational Formulations

Lemma 5.4.6. For given fixed (u, p) ∈ H 1 (Ω, Pst ), the linear mapping   v → v · T (u)ds := aΩ (Zv, u) − b (Zv, p) − Zv · (∇p − Δu)dx (5.4.30) Γ

Ω 1

1

for v ∈ H 2 (Γ ) defines a continuous linear functional T (u) ∈ H − 2 (Γ ) where Z is a right inverse to the trace operator γ0 in (4.2.41). The linear mapping 1 (u, p) → T (u) from H 1 (Ω, Pst ) into H − 2 (Γ ) is continuous, and there holds   (∇p − Δu) · vdx + T (u) · vds = aΩ (u, v) − b(p, v) (5.4.31) Ω Γ for (u, p) ∈ H 1 (Ω, Pst ) and v ∈ H 1 (Ω) . Hence, T (u) is an extension of (2.3.6) to H 1 (Ω, Pst ). The Interior Dirichlet Problem for the Stokes System  −1 (Ω) and ϕ ∈ H 12 (Γ ) are given where ϕ satisfies the necessary Here f ∈ H 0 compatibility conditions  . ϕ · n ds = 0 ,  = 1, . . . , L (5.4.32) Γ

Then we solve for the desired solenoidal field 1 (Ω) := {w ∈ H 1 (Ω) | divw = 0 in Ω} u ∈ Hdiv

(5.4.33)

1 the variational problem: Find u ∈ Hdiv (Ω) with u|Γ = ϕ as the solution of  1 1 aΩ (u, v) = f · vdx for all v ∈ H0,div (Ω) := {v ∈ Hdiv (Ω) | v|Γ = 0} . Ω

(5.4.34) 1 ˙ 01 (Ω)⊂H ˙ 1 (Ω) is a closed subspace of H 1 (Ω) and aΩ Since H0,div (Ω)⊂H satisfies the G˚ arding inequality (5.4.27), for (5.4.34) Theorem 5.3.10 can be applied. Moreover, (5.4.34) is equivalent to the following variational problem with the Dirichlet bilinear form.

Lemma 5.4.7. The variational problem (5.4.34) is equivalent to 1 (Ω) with u|Γ = ϕ of Find u ∈ Hdiv μ

n   j=1 Ω

 ∇uj · ∇vj dx =

1 f · vdx for all v ∈ H0,div (Ω) . Ω

(5.4.35)

5.4 G˚ arding’s Inequality for Boundary Value Problems

265

The latter with ϕ = 0 only has the trivial solution. Hence, (5.4.35) as well as (5.4.34) have a unique solution, respectively. Proof of Lemma 5.4.7: If u ∈ C 2 (Ω) ∩ C 1 (Ω) with ∇ · u = 0 and 1 v ∈ H0,div (Ω) then integration by parts yields aΩ (u, v) = μ

n  

∇uj · ∇vj dx

j=1 Ω

+μ

   n j,k=1

Ω

=μ

n  

  ∂  ∂uj    ∂ vk − ∇ · u vk dx ∂xj ∂xk ∂xk n

k=1

 ∇uj · ∇vj dx + μ

j=1 Ω

=μ

n  

∂uj vk nk ds ∂xk

Γ

∇uj · ∇vj dx

j=1 Ω

since v|Γ = 0. Since the bilinear forms on the left and the right hand side 1 are continuous on Hdiv (Ω), by completion we obtain the proposed equality 1  also for u ∈ Hdiv (Ω). 1 (Ω) available we obtain T (u) from (5.4.31), i.e., With u ∈ Hdiv   1 T (u) · vds = f · vds + aΩ (u, v) for all v ∈ Hdiv (Ω) (5.4.36) Γ

Ω − 12

defines T (u) ∈ H (Γ ). Then the pressure p can be obtained from (2.3.12). The corresponding mapping properties of the potentials in (2.3.12) will be shown later on. Clearly, for the solution we obtain the a priori estimate uH 1 (Ω) ≤ c{ϕ

1

H 2 (Γ )

+ f H −1 (Ω) } .

(5.4.37)

The Interior Neumann Problem for the Stokes System  −1 (Ω) and ψ = T (u) ∈ H − 12 (Γ ) are given For the Neumann problem f ∈ H 0 satisfying the necessary compatibility conditions   f · mk dx + ψ · mk ds = 0 , k = 1, 3(n − 1) (5.4.38) Ω

Γ

where the mk are the rigid motion basis in Ω. The desired solenoidal field 1 u ∈ Hdiv (Ω) will be obtained as the solution of the variational problem: 1 (Ω) satisfying Find u ∈ Hdiv

266

5. Variational Formulations



 f · vdx +

aΩ (u, v) = Ω

1 ψ · vds for all v ∈ Hdiv (Ω) .

(5.4.39)

Γ

Since  = span {mk } is the null–space of aΩ (u, v) we modify (5.4.38) to get the stabilized uniquely solvable bilinear equation: 

3(n−1)

aΩ (up , v) + 

 f · vdx +

= Ω

(up , mk )L2 (Ω) mk

k=1

(5.4.40)

ψ · vds for all v ∈

1 Hdiv (Ω) .

Γ

The particular solution up of (5.4.40) is uniquely determined whereas the general solution then has the form 

3(n−1)

u = up +

αk mk for n = 2 or 3 ,

k=1

with arbitrary αk ∈ R. For up we have the a priori estimate up H 1 (Ω) ≤ c{ϕ

1

H − 2 (Γ )

+ f H −1 (Ω) } .

(5.4.41)

With u and ψ = T (u) available, the pressure p also here can be calculated from (2.3.12). 5.4.3 G˚ arding’s Inequality for Exterior Second Order Problems Analogously to the interior problems, one may establish all properties for the sesquilinear form aΩ c (u, v) on HE × HE for later application of existence theory. First we need the following inequalities between the exterior Dirichlet integrals and the HE (Ω c )–norm for the operator P . Here we require the conditions (5.4.42) bj (x) = 0 for all x ∈ Ω c in addition to the assumptions (5.1.36). Lemma 5.4.8. Let P in (5.1.1) be uniformly strongly elliptic in Ω c with coefficients satisfying the conditions (5.1.36), (5.4.42). Then the inequality ∇v2L2 (Ω c ) ≤ cv2HE,0

(5.4.43)

holds for all v ∈ HE,0 (Ω c ). In the scalar case, i.e., p = 1 then also ∇v2L2 (Ω c ) + γc0 v2L2 (Γ ) ≤ cv2HE,0

(5.4.44)

holds for all v ∈ HE (Ω c ). For a second order system, i.e., for p > 1, (5.4.44) holds for all v ∈ HE (Ω c ) provided P is very strongly elliptic or formally positive elliptic.

5.4 G˚ arding’s Inequality for Boundary Value Problems

267

Proof: For the proof of (5.4.43) note that the functions in HE,0 vanish on Γ = ∂Ω c and can be extended by zero into Ω. The inequality then follows by Fourier transformation in IRn with the help of Parseval’s equality and (3.6.2). To establish inequality (5.4.44), we proceed as in Costabel et al [90] and introduce an auxiliary boundary ΓR : {x ∈ IRn with |x| = R} with R sufficiently large so that |y| < R for all y ∈ Ω. Let η ∈ C ∞ (IRn ) be a cut–off function such that 0 ≤ η ≤ 1 , η(x) = 1 for |x| ≤ R and η(x) = 0 for |x| ≥ 2R. Define v := ηv for v ∈ HE . Then, according to the corresponding assumptions on P , the application of Lemma 5.4.1 or Theorem 5.4.3 or c Theorem 5.4.5 to the annular domains ΩjR := {x ∈ Ω c | |x| ≤ jR}, j = 1, 2, imply with the help of Lemma 5.2.5 the following inequalities: There exist constants α01 , α02 > 0 such that α0j {∇v2L2 (Ω c

jR )

2 c (v, v) + γc0 v 2 + γc0 v2L2 (Γ ) } ≤ aΩjR L (Γ )

(5.4.45)

c for all v ∈ H 1 (ΩjR ) and for j = 1 and 2. Clearly, (5.4.45) yields for v

α02 {∇ v 2L2 (Ω c ) + γc0 v2L2 (Γ ) } ≤  v 2HE .

(5.4.46)

v ∗ ∈ L2 (Ω c ), and Moreover, for v∗ = (1 − η)v = (v − v) ∈ H01 (Ω c ) we have ∇ the G˚ arding inequality in the exterior is valid on the subspace HE,0 ,i.e., v ∗ 2L2 (Ω c ) ≤ aΩ c ( v ∗ , v∗ ) . α03 ∇

(5.4.47)

From (5.4.46) and (5.4.47) we obtain α04 {∇v2L2 (Ω c ) + γc0 v2L2 (Γ ) } n     ∂   ∂ ajk (ηv) (η¯ v) ≤ γc0 v2L2 (Γ ) + ∂xk ∂xj j,k=1Ω c

≤

v2HE + cv2H 1 (Ω c

  ∂   ∂ (1 − η)v (1 − η)¯ v dx + ajk ∂xk ∂xj

c 2R \ΩR )

,

(5.4.48)

where α04 = min{α02 , α03 } > 0. To complete the proof of the inequality (5.4.44), we suppose the contrary. This implies the existence of a sequence vj ∈ HE (Ω c ) with ∇vj 2L2 (Ω c ) + γc0 vj 2L2 (Γ ) = 1 and vj HE → 0

(5.4.49)

1 1 (Ω c ), i.e. there exists v0 ∈ Hloc (Ω c ) and which converges weakly in Hloc 1 2 1 c vj  v0 in Hloc (Ω ). With the compact imbedding H 2 (Γ ) → L (Γ ), this yields γc0 vj L2 (Γ ) → 0 = γc0 v0 L2 (Γ )

due to (5.4.49). Then (5.4.45) for j = 2 yields with (5.4.49)

268

5. Variational Formulations

α02 {∇vj 2L2 (Ω c

2R )

+ γc0 vj 2L2 (Γ ) } ≤ vj 2HE → 0

c \Ω c ) → 0 together with vj H → 0. Hence, which implies that vj H 1 (Ω2R E R from (5.4.48) we find

∇vj 2L2 (Ω c ) + γ0 vj 2L2 (Γ ) → 0 

which contradicts (5.4.49). Thus, (5.4.44) must hold.

Remark 5.4.1: From the proof we see that the result remains valid for the symmetrized sesquilinear form (5.1.62) since P is to be replaced by the self adjoint operator 12 (P + P ∗ ). For the latter, the additional conditions (5.4.42) are not required. Lemma 5.4.9. Under the assumptions of Lemma 5.4.8, the sesquilinear form aΩ c (u, v) is continuous on HE and satisfies G˚ arding’s inequality Re{aΩ c (v, v) + (Cv, v)HE } ≥ v2HE for all v ∈ HE (Ω c )

(5.4.50)

where C is defined by 

u vds −

(Cu, v)HE :=

n   

bj (x)

j=1Ω c

Γ

∂u  v(x)dx ∂xj

(5.4.51)

via the Riesz representation theorem in HE (Ω c ) and is therefore a compact operator. Proof: i) Continuity: By definition (5.1.3) where Ω is replaced by Ω c , assumptions (5.1.36) and (5.4.42) and the Schwarz inequality we have the estimate |aΩ c (u, v)| ≤ c∇uL2 (Ω c ) {∇vL2 (Ω c ) + vL2 (ΩRc ) } 0

 c where ΩR = {x ∈ Ω c |x| < R0 }. The application of the following version of 0 the Poincar´e–Friedrichs inequality (4.1.40) vL2 (ΩRc

0

)

c ≤ c(ΩR ){vL2 (Γ ) + ∇vL2 (Ω c ) } , 0

together with (5.4.44) yields the desired estimate |aΩ c (u, v)| ≤ cuHE vHE . ii) G˚ arding’s inequality: The inequality (5.4.50) follows from the definition of C in (5.4.51) and the definition (5.1.60) of (v, v)HE . It only remains to show that the mapping C : HE → HE is compact. To this end we consider the adjoint mapping C ∗ in the form

5.4 G˚ arding’s Inequality for Boundary Value Problems

269

(Cu, v)HE = (u, C ∗ v)HE = (u, C1∗ v)HE + (u, C2∗ v)HE where



(u, C1∗ v)HE

(γc0 u) γc0 vds ,

:= Γ

(u, C2∗ v)HE

:=

−

n   

bj (x)

j=1Ω c R

∂u  v(x)dx . ∂xj

0

The mapping C1∗ can be written as the composition of the mappings γc0

1

∗ C

ic

1 HE (Ω c ) HE (Ω c ) −→ H 2 (Γ ) → L2 (Γ ) −→

ic  ∗ is where → denotes the compact imbedding ic of the trace spaces and C 1 defined by the Riesz representation theorem from  1∗ w)H := (γc0 u) wds (u, C E Γ

with

1∗ w)|H ≤ γc0 uL2 (Γ ) wL2 (Γ ) ≤ uH wL2 (Γ ) . |(u, C E E

 ∗ : L2 (Γ ) → HE (Ω c ). As a This implies the continuity of the mapping C 1 composition of continuous linear mappings with the compact mapping ic , the mapping C1∗ is compact, and so is C1 . The mapping C2∗ can be written as the composition of the mappings v|Ω c

R

∗ C

ic

2 c c HE (Ω c ) −→0 H 1 (ΩR ) → L2 (ΩR ) −→ HE (Ω c ) 0 0

c where the restriction to ΩR ⊂ Ω c is continuous due to the equivalence of 0 c 1 c ) because of the boundedness of the domain the spaces HE (ΩR0 ) and H (ΩR 0 ic

c ΩR (see Triebel [421, Satz 28.5]). For the same reason, the imbedding → is 0  ∗ : L2 (Ω c ) → HE (Ω c ) is given by compact. The mapping C 2 R0

 ∗ w)H := − (u, C 2 E

n   

bj (x)

j=1Ω c R

and satisfies

∂u  wdx ∂xj

0

 ∗ w)H | ≤ cuH wL2 (Ω c ) ; |(u, C 2 E E R 0

which implies its continuity. Thus, C2∗ is compact due to the compactness of ic , and so is C2 . 

270

5. Variational Formulations

Alternatively, one also may use the weighted Sobolev spaces H 1 (Ω c ; 0) and H01 (Ω c ; 0) (see (4.4.1) – (4.4.3)) instead of HE and HE,0 , respectively, by following Giroire [148] and Nedelec [328]; see also Dautray and Lions [97]. However, the following shows that the formulation (5.1.65) and Lemmata 5.4.8 and 5.4.9 both remain valid if the function spaces HE and HE,0 are replaced by H 1 (Ω c ; 0) and H01 (Ω c ; 0), respectively. Lemma 5.4.10. The weighted Sobolev space H 1 (Ω c ; 0) (and H01 (Ω c ; 0)) with the scalar product (4.4.2) is equivalent to the energy space HE (and HE,0 ) associated with ajk = δjk and also to the energy space HE (and HE,0 ) given with the norm (5.1.60) (and by (5.1.63)) for general ajk belonging to a formally positive elliptic system of second order. Proof: For the equivalence we have to show that there exist two positive constants c1 and c2 such that c1 vH 1 (Ω c ;0) ≤ vHE ≤ c2 vH 1 (Ω c ;0) for all v ∈ H 1 (Ω c ; 0) .

(5.4.52)

Since the coefficients ajk in (5.1.60) are supposed to be uniformly bounded and also to satisfy the uniform strong ellipticity condition (3.6.2) with Θ = 1, it suffices to prove the lemma for ajk = δjk . i) We show the right inequality in (5.4.52). v2HE

:=

v2L2 (Γ ) + ∇v2L2 (Ω c )

=

v2L2 (Γ ) + ∇vL2 (ΩRc

0

)

+ ∇v2L2 (|x|≥R0 ) 1

Using (4.1.40) which states that {vL2 (Γ ) + ∇vL2 (ΩRc ) } 2 and vH 1 (ΩRc ) 0 0 c are equivalent in the bounded domain ΩR , we find with some positive con0 stant cR0 v2HE ≤ cR0 v2H 1 (Ω c ) + ∇v2L2 (|x|≥R0 ) . R0

Since for the weight functions in (4.4.1) we have (|x|)0 (|x|) ≤ (R0 )0 (R0 ) for all |x| ≤ R0 , we find v2HE ≤ cR0 2 (R0 )20 (R0 )v2H 1 (Ω c

R0 ;0)

+ v2H 1 (Ω c ;0) ,

which implies the right inequality in (5.4.52)). ii) For the second inequality we use the Friedrichs–Poincar´e inequality (4.4.12) for weighted Sobolev spaces in exterior domains, v2H 1 (Ω c ;0) ≤ c1 ∇v2L2 (Ω c ) + c2R0 v2H 1 (Ω c

R0 )

Using (4.1.40) and (4.1.41), again we find

.

5.5 Existence of Solutions to Boundary Value Problems

271

v2H 1 (Ω c ;0) ≤ c1 ∇v2L2 (Ω c ) + c2 {v2L2 (Γ ) + ∇u2L2 (Ω c ) } R0

which gives the left inequality in (5.4.52). Since H 1 (Ω c ; 0) and HE are complete as Hilbert spaces and both contain the same dense subset C0∞ (IRn ) ⊕ Cp , it follows from inequality (5.4.52) that both spaces are equivalent.  5.4.4 G˚ arding’s Inequality for Second Order Transmission Problems For the transmission problems 1.1, 1.2 and 2.1–2.4 (i.e., (5.1.79), (5.1.80) and (5.1.84)–(5.1.87)) the corresponding sesquilinear form is given by aΩ c (u, v)  a(u, v) =:  aΩ (u, v) + 

(5.4.53)

on u, v ∈ H 1 (Ω)×HE (Ω c ). By making use of the G˚ arding inequalities (5.4.6), (5.4.7) in the bounded domain Ω as well as (5.4.50) for the exterior Ω c , we obtain G˚ arding’s inequality for these transmission problems. Corollary 5.4.11. Let P given in (5.1.1) be either scalar strongly elliptic or very strongly elliptic or positive elliptic satisfying in Ω c the additional assumptions in Lemma 5.4.8 (i.e., (5.1.36) and (5.4.42)). Then the sesquilinear form  a(u, v) of the transmission problems 1.1, 1.2, 2.1–2.4 is continuous on H 1 (Ω) × HE (Ω c ) and satisfies G˚ arding’s inequality    v  ds Re  a(v, v) + (C1 v, v)H 1 (Ω) + ((C2 v, v)HE (Ω c ) + v¯ (5.4.54) Γ ≥ α0 {v2H 1 (Ω) + v2HE (Ω c ) } for all v ∈ H 1 (Ω) × HE (Ω c ). As for the remaining transmission problems 2.5, 2.6 the G˚ arding inequalities for the interior and exterior problems both are already available from the previous sections. For 2.5 and 2.6 they are needed for solving the interior and exterior problems consecutively. c = Ω c ∩ BR with BR := If we replace Ω c by a bounded domain ΩR n {x ∈ IR | |x| < R} and R sufficiently large such that Ω ⊂ BR then we c by replacing may consider the previous transmission problems in Ω and ΩR the radiation conditions by appropriate boundary conditions on ΓR = ∂BR , arding’s e.g., homogeneous Dirichlet conditions Bγ0 u = 0 on ΓR . Then G˚ c ). inequality (5.4.54) remains valid if HE (Ω c ) is replaced by H 1 (ΩR

5.5 Existence of Solutions to Strongly Elliptic Boundary Value Problems We are now in the position to summarize some existence results for the boundary value and transmission problems in Section 5.1. These results are based on

272

5. Variational Formulations

the corresponding variational formulations together with G˚ arding inequalities which have been established in Section 5.4. The latter implies the validity of Fredholm’s alternative, Theorem 5.3.10. Hence, uniqueness implies existence of solutions in appropriate energy spaces. In many cases, uniqueness needs to be assumed. In particular, the transmission problems play a fundamental role for establishing the mapping properties of the boundary integral operators in Section 5.6. These mapping properties will be needed to derive G˚ arding inequalities for the boundary integral operators which are a consequence of the G˚ arding inequalities for the sesquilinear forms associated with the transmission problems. 5.5.1 Interior Boundary Value Problems We begin with the existence theorem for the general Dirichlet problem (5.1.19) in variational form.  −1 (Ω) and ϕ ∈ H 12 (Γ ), there exists a unique Theorem 5.5.1. Given f ∈ H 0 1 solution u ∈ H (Ω) with B00 γ0 u|Γ = ϕ such that aΩ (u, v) = f, vΩ for all v ∈ H = H01 (Ω) ,

(5.5.1)

provided the homogeneous adjoint equation aΩ (u0 , v) = 0 for all v ∈ H = H01 (Ω)

(5.5.2)

has only the trivial solution u0 ≡ 0. Proof: The proof is an obvious consequence of Fredholm’s alternative, Theorem 5.3.10 together with G˚ arding’s inequality (5.4.6) from Theorem 5.4.3, where u = Φ + w with some Φ ∈ H 1 (Ω) satisfying B00 γ0 Φ = ϕ on Γ and with w ∈ H; thus a(v, w) := aΩ (w, v) and (v) := f, vΩ − aΩ (Φ, v) .  In the case of those Dirichlet problems where uniqueness does not hold, as a consequence of Theorem 5.3.10 (the second part of Fredholm’s alternative) we have the following existence theorem. Theorem 5.5.2. Under the same assumptions as for Theorem 5.5.1, there exists a solution u ∈ H 1 (Ω) of (5.1.19) in the form u = up +

k  j=1

αj u0(j)

(5.5.3)

5.5 Existence of Solutions to Boundary Value Problems

273

where up is a particular solution of (5.5.1) and u0(j) ∈ H01 (Ω) are linearly independent eigensolutions of aΩ (u0(j) , v) = 0 for all v ∈ H01 (Ω) provided the orthogonality conditions (v0(j) ) = f, v0(j) Ω − aΩ (Φ, v0(j) ) = 0

(5.5.4)

for all eigensolutions v0(j) ∈ H01 (Ω) of aΩ (w, v0(j) ) = 0 for all w ∈ H01 (Ω) ,

(5.5.5)

are satisfied; j = 1, . . . , k < ∞. In order to characterize these eigensolutions, we need the adjoint differential operator and a generalized second Green’s formula corresponding to (5.1.1). The standard second Green’s formula in Sobolev spaces reads as 



∗





{u (P v) − (P u) v}dx = Ω

{(∂ν u) v − u (∂ν v)}ds − 



Γ

  n Γ

(nj bj u) vds ,

j=1

provided u, v ∈ H 2 (Ω). The formal adjoint P ∗ is defined by P ∗v = −

n  j,k=1

(5.5.6)

n ∂  ∗ ∂v   ∂ ∗ ajk − (bj v) + c∗ v ∂xj ∂xk ∂x j j=1

whose conormal derivative is ∂ν v =

n  k,j=1

nj a∗jk

∂v . ∂xk

(5.5.7)

The generalized second Green’s formula can be derived in terms of the mapping τ and its adjoint. More precisely, we have the following lemma. Lemma 5.5.3 (Generalized second Green’s formula). For every pair u, v ∈ H 1 (Ω, P ) there holds  {u (P ∗ v) − (P u) v}dx (5.5.8) Ω = τ u, γ0 vΓ − γ0 u, τvΓ − (n · b )γ0 u, γ0 vΓ ,

1 where τv = ∂ν v for v ∈ C ∞ ; its extension τ to H − 2 (Γ ) is defined by completion in the same manner as for τ .

274

5. Variational Formulations

Proof: The proof follows from Lemma 5.1.1 with the operator P and P ∗ , subtraction of the corresponding equations (5.1.8), and from additional ap plication of the divergence theorem in H 1 (Ω). Clearly, the functions u0(j) in (5.5.3) are nontrivial solutions of the homogeneous Dirichlet problem. It is also easy to see that the eigensolutions v0(j) of (5.5.5) are distributional solutions of the adjoint homogeneous boundary value problem P ∗ v0(j) = 0 in Ω satisfying B00 γ0 v0(j) = 0 on Γ .

(5.5.9)

Moreover, the orthogonality conditions (5.5.4) can be rewritten in the form     −1 ∗ (5.5.10) f  v 0(j) dx − ϕ (B00 ) τv0(j) ds = 0 . Ω

Γ

For the general interior Neumann problem (5.1.29) we also have the following existence theorem: Theorem 5.5.4. Let Γ ∈ C 1,1 and let the differential operator P belong to one of the following cases: P is a scalar strongly elliptic operator viz. (5.4.1) or a very strongly elliptic system of operators viz. (5.4.5) or a formally positive elliptic system of operators viz. (5.4.17), (5.4.18). Then, given  −1 (Ω) and ψ ∈ H − 12 (Γ ), there exists a unique solution u ∈ H 1 (Ω) f ∈ H 0 satisfying (5.1.29), i. e.  aΩ (u, v) = f, v¯Ω + ψ, SγvΓ

for all v ∈ H 1 (Ω) ,

(5.5.11)

with the bilinear form  aΩ given by (5.1.28) associated with the operator P and S = B, provided the homogeneous equation  aΩ (u0 , v) = 0 for all v ∈ H 1 (Ω) admits only the trivial solution u0 ≡ 0. This theorem is again a direct consequence of Fredholm’s alternative, Theorem 5.3.10, and G˚ arding’s inequality (5.4.2), (5.4.7) on H = H 1 (Ω) (Lemma 5.4.1 and Theorems 5.4.3 and 5.4.5). It is worth mentioning that the variational solution u is also in H 1 (Ω, P ). 1  γ0 u : Hence the mappings τ : H 1 (Ω, P )  u → τ u ∈ H − 2 (Γ ) as well as N 1 1 −2  H (Ω, P )  u → N γ0 u ∈ H (Γ ) via (5.1.27) are continuous. Therefore 1 (5.1.23) implies that Rγu = N γu ∈ H − 2 (Γ ) is well defined and (5.5.11) yields N γu − ψ, BγvΓ = 0 for all v ∈ H 1 (Ω). Hence, the variational solution of (5.5.11) satisfies the general Neumann condition in the form

5.5 Existence of Solutions to Boundary Value Problems 1

N γu = ψ in H − 2 (Γ ) .

275

(5.5.12)

In the case of non–uniqueness, we have a theorem, analogous to Theorem 5.5.2. Theorem 5.5.5. Under the same assumptions as for Theorem 5.5.4, there exists a solution u ∈ H 1 (Ω) of (5.5.11) in the form u = up +

k 

αj u0(j)

(5.5.13)

j=1

where up is a particular solution of (5.5.11) and u0(j) ∈ H 1 (Ω) are linearly independent eigensolutions of  aΩ (u0(j) , v) = 0 for all v ∈ H 1 (Ω)

(5.5.14)

provided the necessary orthogonality conditions for f and ψ (v0(j) ) = v0(j) , f Ω + ψ, Bγv0(j) Γ = 0

(5.5.15)

with all eigensolutions v0(j) ∈ H 1 (Ω) of  aΩ (w, v0(j) ) = 0 for all w ∈ H 1 (Ω)

(5.5.16)

are satisfied; j = 1, . . . , k < ∞. In order to gain more insight into the eigensolutions u0(j) , we recall the variational equation (5.1.29), i. e.  aΩ (u, v) = P u, v¯Ω + N γu, BγvΓ

for all v ∈ H 1 (Ω) .

(5.5.17)

Hence, u0(j) must be a nontrivial distributional solution of the homogeneous Neumann problem, P u0(j) = 0 in Ω , N γu0(j) = 0 on Γ .

(5.5.18)

To characterize the eigensolutions v0(j) of the adjoint equation (5.5.16), we apply the generalized second Green’s formula to (5.5.17) and obtain  aΩ (w, v0(j) )

=

P w, v0(j) Ω + N γw, Bγv0(j) Γ

=

w, P ∗ v0(j) Ω + N γw, Bγv0(j) Γ

− τ w, γ0 v0(j) Γ + γ0 w, τv0(j) Γ + (b · n)γ0 w, γ0 v0(j) Γ . We note that the relation (5.1.23),  γ0 w, γ0 v0(j) Γ , τ w, γ0 v0(j) Γ = N γw, Bγv0(j) Γ + N implies that (5.5.19) takes the form

(5.5.19)

276

5. Variational Formulations

 aΩ (w, v0(j) )

=

w, P ∗ v0(j) Ω  γ0 w, γ0 v0 Γ + γ0 w, τv0(j) + (b · n)γ0 v0(j) Γ + N

=

 ∗ γ0 + τ + (b · n)γ0 )v0(j) Γ . w, P ∗ v0(j) Ω + γ0 w, (N

Hence, it follows from (5.5.16) that the eigensolutions v0(j) are the nontrivial distributional solutions of the adjoint general Neumann problem P ∗ v0(j) = 0 in Ω and  ∗ γ0 v0(j) + (b · n)γ0 v0(j) = 0 on Γ . τv0(j) + N

(5.5.20)

 in (5.1.25) via integration by parts  ∗ is derived from R The operator N and is of the form  ∗ γ0 v0(j) =− N

n  n−1   j,k=1

 ∂ ∂yk −1∗ ∗ (γ μ nj a∗jk v0(j) )+N00 N01 nj a∗jk nk v0(j) . ∂σ ∂σμ |Γ μ,=1

5.5.2 Exterior Boundary Value Problems 1

−1 Theorem 5.5.6. Given f ∈ Hcomp (Ω c ) and ϕ1 ∈ πH 2 (Γ ) and − 12  there exists a ψ2 ∈ (I − π)H (Γ ) together with d ∈ C ,  = 1, . . . , L, unique solution u = u0 + p + E  (•; 0)q with u0 + p ∈ HE (Ω c ) , q ∈ Cp such  L =1

κ q ∈ HEπB subject to the side conditions (5.1.73) and the constraint (5.1.74), provided the homogeneous equation that Equation (5.1.72) holds for all v = v0 +

 aΩ c (u0 + p, v) = 0 for all v ∈ HEπB has only the trivial solution u ≡ 0. If the homogeneous equation has nontrivial solutions, then the second part of Fredholm’s alternative holds accordingly. The proof follows with the same arguments as in the proof of Theorem 5.5.4 based on G˚ arding’s inequality (5.4.50) and Fredholm’s alternative. Details are omitted. 5.5.3 Transmission Problems From the weak formulation (5.1.81) of the transmission problems and the 0 corresponding G˚ arding’s inequality (5.4.53) on all of H 1 (Ω) × HE (Ω c ) it follows that we have the following existence result:  −1 (Ω) × H −1 (Ω c ) and any set of boundary Theorem 5.5.7. Given f ∈ H comp 0 conditions in Table 5.1.1 with (5.1.78) or in Table 5.1.2 with (5.1.83) then

5.6 Solution of Integral Equations via Boundary Value Problems

277

0 there exists a unique solution u ∈ H 1 (Ω)×HE (Ω c ) provided the corresponding homogeneous problem has only the trivial solution. If the corresponding homogeneous problem has nontrivial solutions then the second part of Fredholm’s alternative holds accordingly.

We may also consider the above transmission problems in a bounded c annular domain where Ω c is replaced by ΩR := Ω c ∩ BR with R > 0 large enough so that the ball BR contains Ω in its interior. In this case, no radiation conditions are needed and on the outer boundary ∂BR = {x ∈ IRn | |x| = R}, we may describe any of the regular boundary conditions and Theorem 5.5.6  −1 (Ω) × H  −1 (Ω c ) and u ∈ H 1 (Ω) × H 1 (Ω c ). remains valid with f ∈ H 0 0 0 R R

5.6 Solutions of Certain Boundary Integral Equations and Associated Boundary Value Problems In this section we consider the weak formulation of the representation formulae for the solutions of boundary value problems, in particular the problems of Section 5.1. In order to establish existence and uniqueness results for the boundary integral equations we need to establish coerciveness properties of the integral operators on the boundary, which follow from the coerciveness of the corresponding interior, exterior and transmission problems in variational form as formulated in the previous section. 5.6.1 The Generalized Representation Formula for Second Order Systems The representation formula can be derived from the generalized second Green’s formula (5.1.8) in Sobolev spaces as in the classical approach by applying the generalized second formula (5.1.9) to the variational solution in   Ωε = Ω \ {y  |y − x| ≤ ε} and Ωεc = Ω c \ {y  |y − x| ≤ ε} with v(y) replaced by E  (x, y) and then taking the limit ε → 0 in connection with the explicit growth conditions of E(x, y) and its derivatives when y → x. Since we so far have not yet discussed the growth conditions of E at y = x, we therefore do not present this standard procedure here. Instead, we begin with the representation formula (3.6.9) and formula (3.6.18) for a classical solution with Γ ∈ C 1,1 , f ∈ C0∞ (IRn ) and for M (x; u) = 0,    u(x) = E (x, y)f (y)dy + E  (x, y) {∂ν u − (b · n)γ0 u} (y)dsy Ω



− Γ

Γ

(∂νy E(x, y)) γ0 u(y)dsy for x ∈ Ω 

(5.6.1)

278

5. Variational Formulations

and







E  (x, y) {∂cν u − (b · n)γc0 u} dsy

E (x, y)f (y)dy −

0= Ωc



+

Γ

(∂cνy E(x, y)) γc0 u(y)dsy for x ∈ Ω

(5.6.2)



Γ

whereas we have    u(x) = E (x, y)f (y)dy − E  (x, y) {∂cν u − (b · n)γc0 u} dsy Ωc



+

Γ

(∂cνy E(x, y)) γc0 u(y)dsy for x ∈ Ω c

(5.6.3)



Γ

together with  0= Ω −

E  (x, y)f (y)dy + 

∂νy E(x, y)





E  (x, y){∂v u − (b · n)γ0 u}

Γ

γ0 u(y)dsy for x ∈ Ω c .

Γ

By adding (5.6.1) and (5.6.2), we obtain   u(x) = E  (x, y)f (y)dy − E  (x, y)[∂ν u − (b · n)γ0 u]dsy IRn

+

Γ

(∂νy E(x, y)) [γ0 u]dsy for x ∈ Ω ,

(5.6.4)

Γ

where [v] := vc − v denotes the jump of v across Γ . The same formula also holds for x ∈ Ω c ; hence, (5.6.4) is valid for all x ∈ IRn \ Γ . By density and completion arguments, one obtains the corresponding formulation in Sobolev spaces and for Lipschitz domains. Theorem 5.6.1 (Generalized representation formula). Let Ω be a  −1 (Ω) ⊗ H −1 (Ω c ) with supp(f )  IRn bounded Lipschitz domain. Let f ∈ H comp 0 and let u be a variational solution of P u = f in IRn \ Γ , i.e., of (5.1.81) with 1 (Ω c , P ) satisfying the radiation condition u|Ω ∈ H 1 (Ω, P ) and u|Ω c ∈ Hloc M (x; u) = 0 (cf. (3.6.15)). Then u(x) admits the representation u(x) = E(x, ·), f IRn − [τ u − (b · n)γ0 u], E(x, ·)Γ + (∂ν E(x, ·)), [γ0 u]Γ for almost every x ∈ IRn \ Γ .

(5.6.5)

5.6 Solution of Integral Equations via Boundary Value Problems

279

Remark 5.6.1: In case f = 0, formula (5.6.5) holds for every x ∈ IRn \ Γ . 1 (Ω c , P ) there holds M (x; u) = 0. Note that for u|Ω c ∈ Hcomp

Proof of Theorem 5.6.1: For the proof we shall use the following estimates:   E(x, ·), P uIRn \Γ H 1 (ΩRc ) ≤ cx,R uH 1 (Ω,P ) + uH 1 (ΩRc ,P ) (, 5.6.6) |[τ u − (b · n)γ0 u], E(x, ·)Γ | + |(∂ν E(x, ·)), [γ0 u]Γ |   for x ∈ Γ . ≤ cx,R uH 1 (Ω,P ) + uH 1 (ΩRc ,P )

(5.6.7)

 c In these estimates ΩR = Ω c ∩ {y ∈ IRn  |y| < R} for R sufficiently large such c . The estimate (5.6.6) is a standard a–priori estimate for that supp(f ) ⊂ ΩR Newton potentials (see e.g. (10.0.19) in Chapter 10). Since E(x, ·) for fixed x ∈ Γ is a smooth function on the boundary Γ , the estimates (5.6.7) follow from Lemma 5.1.1 and Theorem 5.5.4. With these two estimates, the representation (5.6.5) can be established by the usual completing procedure. More precisely, for smooth Γ and if u is a variational solution with the required properties we can approximate u by a sequence uk with uk |Ω ∈ C ∞ (Ω) and uk |Ω c ∈ C ∞ (Ω c ) satisfying (5.6.4) so that 1 (Ω c ,P ) → 0 as k → ∞ . u − uk H 1 (Ω,P ) + u − uk Hloc

Then, because of (5.6.7), for any x ∈ Γ the two boundary potentials in (5.6.4) generated by uk will converge   to the corresponding boundary potentials (u − F )(x) where F (x) = E (x, y)P u(y)dy in (5.6.5). From (5.6.6), howIRn

ever, we only have E(x, y), P uk IRn \Γ → E(x, ·), P uIRn \Γ = F (x) for almost every x ∈ IRn \ Γ . If Ω is a strong Lipschitz domain with a strong Lipschitz boundary, then for fixed x ∈ Γ , the generalized Green formulae can still be applied to the annular domains Ωε for any sufficiently small ε > 0. For the domain Bε (x) =  {y  |y − x| < ε}, the previous arguments can be used since Bε (x) has a smooth boundary. Note that the sum of the boundary integrals over ∂Bε will be canceled. This completes the proof.  From the representation formula (5.6.5) we may represent the solution of (5.1.19) or of (5.1.20) for x ∈ Ω by setting u|Ωc ≡ 0 which coincides with our previous representation formulae in Ω. Similarly, if one is interested in the exterior problem we may set u|Ω ≡ 0 and obtain a corresponding representation of solutions in Ω c . In fact, the representation formula (5.6.5) can be

280

5. Variational Formulations

applied to the more general transmission problems which will be discussed below. Based on the representation formula (5.6.5) we now consider associated boundary value problems. 5.6.2 Continuity of Some Boundary Integral Operators Similar to the classical approach, the generalized Green’s formulae lead us to the boundary integral equations in Sobolev spaces. For this purpose we need the mapping properties of boundary potentials including jump relations which can be derived from the theory of pseudo–differential operators as we shall discuss later on. For a restricted class of boundary value problems, however, these properties can be obtained by using solvability and regularity of associated transmission problems as in Section 5.1.3. To illustrate the idea, let us consider the generalized representation formula (5.6.5) for the variational solution of the transmission problem (5.1.81) for second order equations with f = 0 and the radiation condition M = 0; namely u(x) = −V [τ u − (b · n)γ0 u](x) + W [γ0 u](x) for x ∈ IRn \ Γ ,

(5.6.8)

where V σ(x) is the simple layer potential and W ϕ(x) is the double layer potential corresponding to (5.6.5), namely V σ(x) = E(x, ·), σΓ , W ϕ(x) = ∂ν E(x, ·), ϕΓ .

(5.6.9)

From this formula (5.6.8) we now establish the following mapping properties provided that Theorem 5.6.1 is valid. Otherwise, the proof presented below must be modified although the results remain valid (see Chapters 8 and 9). Theorem 5.6.2. The following operators are continuous: 1

V : H − 2 (Γ )

→

1 H 1 (Ω, P ) × Hloc (Ω c , P ) ,

(Γ )

→

H 2 (Γ ) ,

τ V and τc V : H − 2 (Γ )

1

→

H − 2 (Γ ) ,

1

→

1 H 1 (Ω, P ) × Hloc (Ω c , P ) ,

1 2

→

H 2 (Γ ) ,

1

→

H − 2 (Γ ) .

γ0 V and γc0 V : H

− 12

W : H 2 (Γ ) γ0 W and γc0 W : H (Γ ) τ W and τc W : H 2 (Γ )

1

1

1

1

Proof: For V we consider the transmission problem (5.1.76)–(5.1.78) with π = I and ϕ1 = 0, ψ1 = σ: 0 (Ω c , P ) satisfying the differential equation Find u ∈ H 1 (Ω, P ) and u ∈ HE P u = 0 in Ω and in Ω c ,

5.6 Solution of Integral Equations via Boundary Value Problems

281

together with the transmission conditions [γ0 u]Γ = 0 and [∂ν u − (b · n)γ0 u]Γ = σ 1

with given σ ∈ H − 2 (Γ ) and some radiation condition. This is a special case of Theorem 5.6.1 which yields the existence of u. Moreover, u depends on σ continuously. The latter implies that the mappings 1 0 σ → u from H − 2 (Γ ) to H 1 (Ω, P ) and to HE (Ω c , P ), respectively, are continuous. By the representation formula (5.6.8), it follows that the mapping σ → u = V σ is continuous in the corresponding spaces. This together with the continuity of the trace operators γ0 : H 1 (Ω, P ) as well as of τ : H 1 (Ω, P )

1

→ H 2 (Γ ) 1

→ H − 2 (Γ )

1

1 and γc0 : Hloc (Ω c , P )

→ H 2 (Γ )

1 and τc : Hloc (Ω c , P )

→ H − 2 (Γ )

1

implies the continuity properties of the operators associated with V . Similarly, the solution of the transmission problem (5.1.76)–(5.1.78) with π = I and ψ1 = 0, ϕ1 = ϕ: 0 (Ω c , P ) satisfying Find u ∈ H 1 (Ω, P ) and u ∈ HE P u = 0 in Ω and in Ω c with

1

[γ0 u]|Γ = ϕ ∈ H 2 (Γ ) and [∂ν u − (b · n)γ0 u]Γ = 0;

together with the representation formula (5.6.8), u(x) = W ϕ(x) for x ∈ Γ provides the desired continuity properties for the mappings associated with W.  As a consequence of the mapping properties, we have the following jump relations. 1

1

0, ϕ,

[τ V σ]|Γ [τ W ϕ]|Γ

Lemma 5.6.3. Given (σ, ϕ) ∈ H − 2 (Γ ) × H 2 (Γ ), then the following jump relations hold: [γ0 V σ]|Γ [γ0 W ϕ]|Γ

= =

= =

−σ ; 0.

Proof: We see from the representation formula (5.6.8) that u(x) = −V σ(x) for x ∈ IRn \ Γ is the solution of the transmission problem (5.1.76)–(5.1.78) with π = I and ϕ1 = 0 , ψ1 = σ with the transmission conditions [γ0 u]Γ = 0 and − σ = [τ (u)]Γ . Inserting u = −V σ gives the jump relations involving V .

282

5. Variational Formulations

Likewise, u(x) = W ϕ(x) is the solution of the transmission problem (5.1.76)–(5.1.78) with π = I and ϕ1 = ϕ , ψ1 = 0 satisfying [γ0 u]Γ = ϕ and [τ (u)]Γ = 0 which gives the desired jump relations involving W .



In accordance with the classical formulation (1.2.3)–(1.2.6), we now in1 troduce the boundary integral operators on Γ for given (σ, ϕ) ∈ H − 2 (Γ ) × 1 H 2 (Γ ) defined by: Vσ

:=

γ0 V σ ,

Kϕ

:=

K σ

:=

1 τV σ − σ , 2

Dϕ

:=

1 γ0 W ϕ + ϕ 2 −τ W ϕ .

(5.6.10)

Clearly, Theorem 5.6.2 provides us the continuity of these boundary integral operators. The corresponding Calder´ on projectors can now be defined in a weak sense by  1 I −K , V 2 CΩ := and CΩ c := I − CΩ , (5.6.11) D , 12 I + K   1 1 which are continuous mappings on H 2 (Γ ) × H − 2 (Γ ) . 5.6.3 Continuity Based on Finite Regions In deriving the continuity properties of the boundary integral operators we rely on the properties of the solutions to transmission problems in IRn \ Γ . As we have seen, this complicates the analysis because of the solution’s behaviour at infinity. Consequently, we needed restrictions on the coefficients of the differential operators. Alternatively, as we shall see, if one is only interested in the local properties of the solution, a similar analysis can be achieved by replacing the transmission problem in IRn \ Γ by one in a bounded domain BR \ Γ . To be more precise, for illustration we consider the supplementary transmission problem: c Find u ∈ H 1 (Ω, P ) × H 1 (ΩR , P ) satisfying c P u = f1 in Ω and P u = f2 in ΩR

(5.6.12)

together with [γ0 u]|Γ = ϕ and [∂ν u − (b · n) γ0 u]|Γ = σ and γ0 u||x|=R = 0 (5.6.13)  −1 (Ω) , f2 ∈ H  −1 (Ω c ) ∩ H −1 (BR ) , ϕ ∈ H 21 (Γ ) , σ ∈ with given f1 ∈ H 0 0 1 H − 2 (Γ ).

5.6 Solution of Integral Equations via Boundary Value Problems

283

The corresponding weak formulation then reads: c , P ) with [γ0 u]Γ = ϕ and γ0 u||x|=R = 0 such Find u ∈ H 1 (Ω, P ) × H 1 (ΩR that aΩRc (u, v)  a(u, v) =  aΩ (u, v) +     (5.6.14) v ds + f1 v¯dx + f2 v¯dx = − σ¯ Γ

c ΩR

Ω

for all c , P ) with [γ0 v]Γ = 0 and γ0 v||x|=R = 0} . v ∈ H := {v ∈ H 1 (Ω, P ) × H 1 (ΩR Since  a(u, v) satisfies the G˚ arding inequality on H, the Fredholm alternative is valid for the problem (5.6.14). For convenience let us assume that the homogeneous problem for (5.6.14) has only the trivial solution. Then the unique solution u of (5.6.14) satisfies an estimate  uH 1 (Ω) + uH 1 (ΩRc ) ≤ c f1 H −1 (Ω) + f2 H −1 (Ω c )∩H −1 (BR )  (5.6.15) + σ − 12 + ϕ 12 H

(Γ )

H (Γ )

and can be represented in the form      u(x) = E (x, y)f1 (y)dy + E (x, y)f2 (y)dy − E  (x, y)σ(y)dsy Ω



+



∂νy E(x, y)



c ΩR

Γ

ϕ(y)dsy for x ∈ BR \ Γ .

Γ

(5.6.16) Now let us reexamine the continuity properties of the simple layer potential operator V again. Let σ ∈ C ∞ (Γ ) be given. Then consider  u(x) := V σ(x) = E(x, y) σ(y)dsy for x ∈ Γ . Γ

Also, let η ∈ C0∞ (IRn ) be a cut–off function with η(x) = 1 for |x| ≤ R1 < R and supp η ⊂ BR

(5.6.17)

where R1 > 0 is chosen sufficiently large such that Ω ⊂ BR1 . Now we define u  := η(x)u(x) . Then u (x) is a solution of the transmission problem (5.6.12), (5.6.13) where ϕ = 0 , f1 = 0 and f2 := (P η − cη)V σ −

n  j,k=1

ajk

 ∂η ∂V σ ∂η ∂V σ  + . ∂xk ∂xj ∂xj ∂xk

(5.6.18)

284

5. Variational Formulations

Hence, supp f2  BR \ BR1 and, since the coefficients of P are assumed to be C ∞ (IRn ), we have f2 ∈ C0∞ (BR \ BR1 ) since dist (Γ, BR \ BR1 ) ≥ d0 > 0. (Note that this smoothness may be reduced depending on the smoothness of the coefficients of P .) So, the estimate (5.6.15) implies for u  V σH 1 (Ω) + ηV σH 1 (ΩRc ) ≤ c{f2 H −1 (BR \BR1 ) + σ

1

H − 2 (Γ )

}.

Since f2 ∈ C0∞ (BR \ BR1 ) and because of |x − y| ≥ d0 > 0 for x ∈ (BR \ BR1 ) and y ∈ Γ from (5.6.18) it is clear that f2 H −1 (BR \BR1 ) ≤ cf2 L2 (BR \BR1 ) ≤ c σ

1

H − 2 (Γ )

.

Consequently, with Theorem 4.2.1, the trace theorem, we find V σ

1

H 2 (Γ )

≤ c1 {V σH 1 (Ω) + V σH 1 (ΩRc ) } ≤ c2 σ 1

1

H − 2 (Γ )

,

the first two continuity properties for V in Lemma 5.6.3, by the standard 1 completion argument approximating σ ∈ H − 2 (Γ ) by C ∞ –functions.  ∈ Applying Lemma 5.1.1 to u  ∈ H 1 (Ω, P ) and Lemma 5.1.2 to u 1 c H (ΩR , P ) we obtain the mapping properties of τ V and τc V in Theorem 5.6.2. For the properties of W ϕ in Theorem 5.6.2 we may proceed in the same manner as for V ϕ where ψ1 = 0 and ϕ1 = ϕ = 0 in the transmission problem (5.1.78). 5.6.4 Continuity of Hydrodynamic Potentials Although the Stokes system (2.3.1) is not a second order Petrovskii elliptic system, its close relation with the Lam´e system can be exploited to show continuity properties of the hydrodynamic potentials V, W and D in (2.3.15), (2.3.16), (2.3.31) (see Kohr et al [234]). Lemma 5.6.4. The mappings given by the hydrodynamic potentials define the following linear operators:

γ0 V

γ0 W

1

1 1 → Hdiv (Ω, Δ) × Hdiv,loc (Ω c , Δ) ,

1

→ H 2 (Γ ) ,

V

: H − 2 (Γ )

and γc0 V

: H − 2 (Γ ) 1

1 1 → Hdiv (Ω, Δ) × Hdiv,loc (Ω c , Δ) ,

1

→ H 2 (Γ ) ,

1

→ H − 2 (Γ ) .

W

: H 2 (Γ )

and γc0 W

: H 2 (Γ )

D

: H 2 (Γ )

Here we denote

1

1

1

5.6 Solution of Integral Equations via Boundary Value Problems 1 Hdiv (Ω, Δ) 1 Hdiv,loc (Ω c , Δ)

:= {v ∈ H 1 (Ω, Δ)

| div v

= 0 in Ω} ,

1 := {v ∈ Hloc (Ω, Δ)

| div v

= 0 in Ω c } .

285

Proof: From (2.3.38) we see that for V = Vst we have   2c Vst = (1 + c) Ve − VΔ I μ which yields with Theorem 5.6.2 for Ve with any c > 0 and for VΔ the desired properties of Vst since the Stokes simple layer potential is solenoidal in Ω and in Ω c . Similarly, from (2.3.42) and (2.3.43) we obtain Wst = (1 + c)We − c(WΔ + L1 ) and L1 =

 1  (1 + c1 )We,c1 − (1 + c2 )We,c2 − WΔ c1 − c2

with any 0 < c2 < c1 . Hence, the desired mapping properties again follow with Theorem 5.6.2 from those of We,c1 , We,c2 and WΔ . For Dst we use (2.3.48) and (2.3.49), Dst = De −

c L2,0 1+c

where L2,0 does not depend on c. Hence, L2,0 =

(1 + c1 )(1 + c2 ) {De,c1 − De,c2 } . c1 − c2

Theorem 5.6.2 for De,c1 and for De,c2 where 0 < c2 < c1 yields that L2,0 : 1 1 H 2 (Γ ) → H − 2 (Γ ) is continuous which then implies the desired continuity  for Dst as well. For the same reasons, the jump relations for the hydrodynamic potentials follow also from those of the Lam´e system and the Laplacian. 1

1

Lemma 5.6.5. Given (τ , ϕ) ∈ H − 2 (Γ ) × H 2 (Γ ), then the hydrodynamic potentials satisfy the following jump relations:  [γ0 V τ ]|Γ = 0 , T c (V τ ) − T (V τ ) |Γ = −τ  c [γ0 W ϕ]|Γ = ϕ , T (W ϕ) − T (W ϕ) |Γ = 0 . Proof: The relation [γ0 V τ ]|Γ = 0 follows from (2.3.38) and Lemma 5.6.3 applied to Ve and VΔ . For smooth ϕ and τ with (2.3.26) and (2.3.27) we obtain

286

5. Variational Formulations

2ϕ = ϕ + [γ0 W ϕ]|Γ − [γ0 V τ ]|Γ

on Γ ,

hence, ϕ = [γ0 W ϕ]|Γ − 0 . With the continuity properties of γ0 W and γc0 W , the proposition follows by completion arguments. From Lemma 2.3.1 we obtain with (2.3.30):  c  T (W ϕ) − T (W ϕ) Γ = − lim [De ϕ]|Γ = − lim [De ϕ]|Γ = 0 . c→0

λ→+∞

For T (V τ ) we invoke the second row of the Calder´on projections (2.3.27) and (2.3.28) to obtain  2τ = −[Dϕ]|Γ + τ + T (V τ ) − T c (V τ ) |Γ which is, because of [Dϕ]|Γ = 0, the desired relation; first for smooth ϕ and 1  τ but then with Lemma 5.6.4 by completion in H 2 (Γ ). For the final computation of the pressure p via (2.3.12) we need the mapping properties of the pressure operators Φ (2.3.15), Π (2.3.16) and of Q(x, y)f (y)dy. Ω

Lemma 5.6.6. The pressure operators define the continuous mappings



1

Φ

: H − 2 (Γ )

Π

: H 2 (Γ )

→ L2 (Ω) ,

 −1 (Ω) : H 0

→ L (Ω) .

Q(x, y)f (y)dy

1

→ L2 (Ω) , (5.6.19)

2

Ω

Proof: With Q(x, y) given in (2.3.10) and γn in (2.2.3) as well as the simple layer potential VΔ of the Laplacian in (1.2.1) we have Φτ = −divVΔ τ . 1

Since Theorem 5.6.2 ensures the continuity VΔ : H − 2 (Γ ) → H 1 (Ω) differentiation yields the first of the assertions. Similarly, Πϕ = −2μdivWΔ ϕ with WΔ for the Laplacian given in (1.2.2). Theorem 5.6.2 shows continuity 1 of WΔ : h 2 (Γ ) → H 1 (Ω) and we get (5.6.19) for Π. For the volume potential we have   Q(x, y)f (y(dy = −div EΔ (x, y)f (y)dy Ω

Ω

5.6 Solution of Integral Equations via Boundary Value Problems

287

where the Newton potential of the Laplacian is a pseudodifferential operator  −1 (Ω) can also be seen of order −2 in IRn of the type in (7.2.21). Since f ∈ H 0 n −1 as f ∈ H (IR ) if extended by zero we get   EΔ (·, y)f (y)dy = EΔ (·, y)f (y)dy ∈ H 1 (Ω) . Ω

IRn



So, differentiation implies the last continuous mapping in (5.6.19). 5.6.5 The Equivalence Between Boundary Value Problems and Integral Equations

For the boundary value problems given in Section 5.5, from the corresponding representation formulae, together with the continuity properties of the boundary potentials given in Theorem 5.6.2 and the jump relations in Lemma 5.6.3, we arrive at boundary integral equations of the form Aλ = Bμ on Γ

(5.6.20)

where μ is given and λ is unknown. The boundary integral operators A and B depend on the boundary value problem to be solved. In contrast to boundary integral equations in classical function spaces on Γ , here (5.6.20) should be understood as an operator equation in the corresponding Sobolev spaces 1 1 on Γ ; here H − 2 (Γ ) or H 2 (Γ ). For the boundary value problems (5.1.19), (5.1.52), (5.1.29) and (5.1.70), the specific forms of A, B and the meaning of λ and μ together with the Sobolev spaces on Γ are listed in Table 5.6.3. Theorem 5.6.7. For given μ in the appropriate function space on Γ , the boundary value problems and the associated boundary integral equations (5.6.20) on Γ as listed in Table 5.6.3 are equivalent in the following sense: 1 i. Every variational solution u ∈ H 1 (Ω, P ) (or u ∈ Hloc (Ω c , P ), respectively) of the boundary value problem defines the boundary data μ and λ where λ is a solution of the associated boundary integral equations (5.6.20) with A and B given in Table 5.6.3. ii. Conversely, every pair μ, λ where λ is a solution of the boundary integral equations (5.6.20) defines, via the representation formula, a variational solution of the corresponding boundary value problem. Every variational solution of the boundary value problem can be obtained in this manner.

Remark 5.6.2: Theorem 5.6.7 assures that every solution of the boundary integral equations (5.6.20) indeed generates solutions of the corresponding boundary value problem and that every solution of the boundary value problem can be obtained in this way. However, the theorem does not imply

288

5. Variational Formulations

that the number of solutions of both problems coincides. It is possible that nontrivial solutions of the homogeneous boundary integral equations via the representation formula are mapped onto u ≡ 0. Proof: Since the arguments of the proof are the same in each of the listed cases, we here consider only the first case to illustrate the main ideas: i.) Let u ∈ H 1 (Ω, P ) be the variational solution of the interior Dirichlet 1 boundary value problem (5.1.19) with given μ = ϕ = γ0 u|Γ = Rγu ∈ H 2 (Γ ) 1 and f = 0. Then τ u|Γ ∈ H − 2 (Γ ) is well defined. Hence, λ := Sγu = τ u|Γ − 1 b · nϕ is also well defined and λ ∈ H − 2 (Γ ). As was shown in Section 5.6.1, the solution of (5.1.19) can be represented by u(x) = V λ(x) − W μ(x) for x ∈ Ω . Taking the trace γ0 |Γ on both sides we obtain with Lemma 5.6.3 and with the operators A = V and B = 12 I + K, as given in (5.6.10), the desired equation (5.6.20) between μ and λ on Γ . 1 1 ii.) Now suppose that μ ∈ H 2 (Γ ) and λ ∈ H − 2 (Γ ) satisfy the boundary integral equation (5.6.20) in the form  V λ = 12 I + K μ on Γ . Then the function u(x) := V λ(x) − W μ(x) for x ∈ Ω satisfies the differential equation P u = P (V λ) − P (W μ) = 0 in Ω since V λ and W μ are boundary potentials defined by the fundamental solution E(x, y) of P . Moreover, both potentials are in H 1 (Ω, P ) because of Theorem 5.6.2. Therefore, u admits the trace γ0 u|Γ . Taking the trace on both sides of the representation formula we have with (5.6.10) 1 γ0 u = γ0 V λ − γ0 W μ = V λ − Kμ + μ on Γ . 2 Now replacing V λ by using the boundary integral equation (5.6.20), one obtains 1 1 γ0 u = Kμ + μ − Kμ + μ = μ = ϕ on Γ . 2 2 Clearly, the remaining cases can be treated in the same manner. We omit the details. 

5.6 Solution of Integral Equations via Boundary Value Problems

289

5.6.6 Variational Formulation of Direct Boundary Integral Equations Because of the equivalence Theorem 5.6.7, solutions of the boundary value problems can now be constructed from the solutions of the boundary integral equations (5.6.20). It will be seen that the existence of their solutions can be established by applying Fredholm’s alternative Theorem 5.3.10 to (5.6.20) in corresponding Sobolev spaces. Similar to the elliptic partial differential equations, we consider the weak, bilinear formulation of (5.6.20). For simplicity, let us consider the case that A is a boundary integral operator of the first kind with the mapping property A : H α (Γ ) → H −α (Γ ) continuously .

(5.6.21)

In the examples of Table 5.6.3, we have A = V where the Sobolev order is α = − 12 or A = D + {± 12 I − K  } ◦ (b · n) where α = 21 . As for partial differential operators, the number 2α is called the order of A. The variational formulation for the boundary integral equations of the first kind (5.6.20) now reads: Find λ ∈ H α (Γ ) such that aΓ (χ, λ) := χ, AλΓ = χ, f Γ for all χ ∈ H α (Γ ) ,

(5.6.22)

where f = Bμ ∈ H −α (Γ ) is given. Now we shall show that the sesquilinear form aΓ (χ, λ) in (5.6.22) is conarding’s intinuous on H α (Γ ), the energy space for A, and also satisfies G˚ equality (5.4.1) under suitable assumptions on the corresponding boundary value problems. Theorem 5.6.8. To the integral operators A of the first kind in Table 5.6.3 there exist compact operators CA : H α (Γ ) → H −α (Γ ) and positive constants γA such that   Re aΓ (λ, λ) + λ, CA λΓ ≥ γA λ2H α (Γ ) for all λ ∈ H α (Γ ) . (5.6.23)

μ = τc u|Γ = ψ

μ ∈ H − 2 (Γ ),

1

μ = τ u|Γ = ψ

μ ∈ H − 2 (Γ ),

1

μ = γc0 u|Γ = ϕ

μ ∈ H 2 (Γ ),

1

μ = γ0 u|Γ = ϕ

μ ∈ H 2 (Γ ),

1

Given data

1 I 2

λ = γc0 u|Γ

1

λ ∈ H 2 (Γ ),

λ = γ0 u|Γ

1

λ ∈ H 2 (Γ ),

λ = τc u|Γ − (b · n)|Γ ϕ

1

λ ∈ H − 2 (Γ ),

λ = τ u|Γ − (b · n)|Γ ϕ

+ K

+ K + V ◦ (b · n)

1 I 2

− K − V ◦ (b · n)

D + ( 12 I − K  ) ◦ (b · n)

1 I 2

D − ( 12 I + K  ) ◦ (b · n)

1 I 2

V

− K

V

λ ∈ H − 2 (Γ ),

1

A

Unknown data 1 I 2

+K

B

V

− K

−V

−( 12 I + K  )

1 I 2

−D − (b · n)I

− 12 I + K

D − (b · n)I

Table 5.6.3. Boundary value problems and their associated boundary integral equations

ENP (5.1.70)

INP (5.1.29)

EDP (5.1.52)

IDP (5.1.19)

Boundary value problem

2nd

1st

2nd

1st

2nd

1st

2nd

1st

Type of BIE

(5.6.3)

(5.6.1)

(5.6.3)

(5.6.1)

Representation

290 5. Variational Formulations

5.6 Solution of Integral Equations via Boundary Value Problems

291

Proof: 1 i) Let A = V , choose λ ∈ H 2 (Γ ) and define u(x) := −V λ(x) for x ∈ IRn . 1 (Ω c , P ), respectively. Moreover, Lemma Then u ∈ H 1 (Ω, P ) and u ∈ Hloc 5.6.3 yields [γ0 u]|Γ = 0 and [τ u]|Γ = λ .

By adding the generalized first Green formulae (5.1.7) and (5.1.13) we obtain    λV λds + (P ηV λ) (ηV λ)dx Re Γ



Ωc



= Re −



[τ u]γ0 uds +

 (P ηu) ηudx = Re {aΩ (u, u) + aΩ c (ηu, ηu)}

Ωc

Γ

where η ∈ C0∞ (IRn ) with η|Ω ≡ 1 and dist({x|η(x) = 1}, Ω) =: d0 > 0 is any fixed chosen cut–off function since the first Green’s formula for Ω c is valid for v = ηu having compact support. On the other hand, from Theorem 5.6.2, Lemma 5.6.3 and Lemma 5.1.1, 1 1 1 the mappings τ : H 1 (Ω, P ) → H − 2 (Γ ) and τc : Hloc (Ω c , P ) → H − 2 (Γ ) are continuous providing the inequality λ2

1

H − 2 (Γ )

= [τ u]2

1

H − 2 (Γ )

≤ ≤

2τ u2 − 1 + 2τc u2 − 1 H 2 (Γ ) H 2 (Γ )   c u2H 1 (Ω) + ηu2H 1 (Ω  ) + P ηu2L2 (Ω  )

arding’s inequality (5.4.2) for aΩ in the where Ω  := Ω c ∩ supp η. Then G˚ domain Ω and for aΩ  in Ω  implies   α0 u2H 1 (Ω) + ηu2H 1 (Ω  )   ≤ Re aΩ (u, u) + aΩ  (ηu, ηu) + (Cu, u)H 1 (Ω) + (C  ηu, ηu)H 1 (Ω  ) with a positive constant α0 and compact operators C and C  depending also on Ω and Ω  . Collecting the inequalities, we get α λ2

1

≤ Re{aΓ (λ, λ)

H − 2 (Γ ) +c1 P ηu2L2 (Ω  )

+ c2 (Cu, u)H 1 (Ω) + c3 (C  ηu, ηu)H 1 (Ω  ) + (P ηu, ηu)L2 (Ω  ) } . 1

1

Accordingly, we define the operator CA : H − 2 (Γ ) → H 2 (Γ ) by the sesquilinear form χ, CA λΓ = c(χ, λ) := c2 (V χ, CV λ)H 1 (Ω) + c3 (ηV χ, C  ηV λ)H 1 (Ω  ) +c1 (P ηV χ, P ηV λ)L2 (Ω  ) + (P ηV χ, V λ)L2 (Ω  ) .

μ = τc u|Γ = ψ

μ ∈ H − 2 (Γ ),

1

μ = τ u|Γ = ψ

μ ∈ H − 2 (Γ ),

1

μ = γc0 u|Γ = ϕ

μ ∈ H 2 (Γ ),

1

μ = γ0 u|Γ = ϕ

μ ∈ H 2 (Γ ),

1

Given data

1 I 2

λ = γc0 u|Γ

1

λ ∈ H 2 (Γ ),

λ = γ0 u|Γ

1

λ ∈ H 2 (Γ ),

λ = τc u|Γ − (b · n)|Γ ϕ

1

λ ∈ H − 2 (Γ ),

λ = τ u|Γ − (b · n)|Γ ϕ

+ K

+ K + V ◦ (b · n)

1 I 2

− K − V ◦ (b · n)

D + ( 12 I − K  ) ◦ (b · n)

1 I 2

D − ( 12 I + K  ) ◦ (b · n)

1 I 2

V

− K

V

λ ∈ H − 2 (Γ ),

1

A

Unknown data 1 I 2

+K

B

V

− K

−V

−( 12 I + K  )

1 I 2

−D − (b · n)I

− 12 I + K

D − (b · n)I

Table 5.6.4. Boundary value problems and their associated boundary integral equations

ENP (5.1.70)

INP (5.1.29)

EDP (5.1.52)

IDP (5.1.19)

Boundary value problem

2nd

1st

2nd

1st

2nd

1st

2nd

1st

Type of BIE

(5.6.3)

(5.6.1)

(5.6.3)

(5.6.1)

Representation

292 5. Variational Formulations

5.6 Solution of Integral Equations via Boundary Value Problems

293

We note that the operator CA is well defined since each term on the 1 right–hand side is a bounded sesquilinear form on χ, λ ∈ H − 2 (Γ ). Hence, by the Riesz representation theorem 5.2.2 there exists a linear mapping jCA : 1 1 H − 2 → H − 2 (Γ ) such that c(χ, λ) = (χ, jCA λ)

1

H − 2 (Γ )

.

 ∗ 1 1 1 Since j −1 : H − 2 (Γ ) → H − 2 (Γ ) = H 2 (Γ ), this representation can be written in the desired form 1

χ, CA λΓ = c(χ, λ) for all χ, λ ∈ H − 2 (Γ ) where CA = j −1 jCA . It remains to show that CA is compact. Let us begin with the first sesquilinear form on the right–hand side, χ, C1 λΓ := c2 (V χ, CV λ)H 1 (Ω) where C : H 1 (Ω) → H 1 (Ω) 1

1

is compact. Since V : H − 2 (Γ ) → H 1 (Ω) is continuous, CV : H − 2 (Γ ) → H 1 (Ω) is compact. Moreover, we have the inequality C1 λ

1

H 2 (Γ )

≤

sup χ

−1 H 2 (Γ )

≤1

|χ, C1 λΓ | ≤ cCV λH 1 (Γ ) · V 

1

H − 2 (Γ ),H 1 (Ω)

.

1

Hence, any bounded sequence {λj } in H − 2 (Γ ) provides a convergent subsequence {CV λj  } in H 1 (Ω), and the corresponding subsequence {C1 λj  } con1 1 verges in H 2 (Γ ) due to the above inequality. This shows that C1 : H − 2 (Γ ) → 1 H 2 (Γ ) is a compact linear operator. By the same arguments with Ω replaced by Ω  , the operator C2 corresponding to χ, C2 λΓ = c3 (ηV χ, C  ηV λ)H 1 (Ω  ) is also compact. In the last two terms we observe that Ω  := supp(P ηu) ∩ Ω  ⊂ Ω  and dist (Ω  , Γ ) =: d0 > 0 . Now consider the mapping P ηV λ(x) = Px η(x)E(x, ·), λΓ and observe that the support of this function is in Ω  . Hence, this function 1 is in C ∞ (Ω  ) for λ ∈ H − 2 (Γ ). This implies that 1

P ηV : H − 2 (Γ ) → H 1 (Ω  ) is continuous and because of the compact imbedding H 1 (Ω  ) → L2 (Ω  ),

294

5. Variational Formulations 1

P ηV : H − 2 (Γ ) → L2 (Ω  ) is compact . As a consequence, the operator C3 defined by χ, C3 λΓ = c1 (P ηV χ, P ηV λ)L2 (Ω  ) = c1 (P ηV χ, P ηV λ)L2 (Ω  ) is compact since C3 λ

1

H 2 (Γ )

≤ c1 P ηV λL2 (Ω  ) · P ηV 

1

H − 2 (Γ ),L2 (Ω  )

.

The last term can be written in the form χ, C4 λΓ = C4∗ χ, λΓ = (P ηXχ, V λ)L2 (Ω  ) and C4∗ is compact which follows from the estimate C4∗ χ

1

H 2 (Γ )

≤ P ηV χL2 (Ω  ) V 

1

H − 2 (Γ ),L2 (Ω  )

and the compactness of P ηV as previously shown. This completes the proof for A = V . ii) Now we consider A = D + {± 12 I − K  } ◦ (b · n) =: D + Dc 1

where α = 12 . For any λ ∈ H 2 (Γ ) define u(x) := V ◦ (b · n)λ(x) + W λ(x) for x ∈ IRn \ Γ

(5.6.24)

according to (5.6.5) and u (x) := η(x)u(x) for x ∈ Ω c with η given as in case i.). Then Theorem 5.6.2 implies that u ∈ H 1 (Ω, P ) for x ∈ Ω and u  ∈ H 1 (Ω  , P ) for x ∈ Ω  . Moreover, Lemma 5.6.3 yields the jump relations [γ0 u]Γ = λ , [τ u]Γ = −(b · n)λ and the one sided jump relations from Ω, γ0 u

=

τu

=

1 V ◦ (b · n)λ + (− I + K)λ and 2 1 ( I + K  ) ◦ (b · n)λ − Dλ on Γ . 2

As before, adding the generalized first Green’s formulae (5.1.8) and (5.1.13), we obtain λ, DλΓ

=

aΩ (u, u) + aΩ c (ηu, ηu)  − (ηu) P ηudx − (b · n)λ, V ◦ (b · n)λΓ . Ω 

5.6 Solution of Integral Equations via Boundary Value Problems

295

Hence, ≥

ReaΓ (λ, λ)

  α0 u2H 1 (Ω) + ηu2H 1 (Ω  )





(ηu) P ηudx

−Re{(u, Cu)H 1 (Ω) + (Cηu, C ηu)H 1 (Ω  ) + Ω 

+(b · n)λ, V ◦ (b · n)λΓ − λ, Dc λΓ } . From the definition of λ we see that λ2

1 2

H (Γ )

= [γ0 u]2

1 2

H (Γ )

  ≤ c u2H 1 (Ω) + ηu2H 1 (Ω  )

following from the trace theorem in the form (4.2.40). This implies (5.6.23) with α = 12 and χ, CA λΓ = (v, Cu)H 1 (Ω) + (ηv, C  ηu)H 1 (Ω  )   + (ηv) P ηudx + χ, (b · n)V ◦ (b · n)λ L2 (Γ ) Ω 

− χ, {± 12 I − K  } ◦ (b · n)λΓ where v(x) u(x)

= =

V ◦ (b · n)χ(x) + W χ(x) and V ◦ (b · n)λ(x) + W λ(x) for x ∈ IRn \ Γ .

The first three sesquilinear forms are defined by compact operators which can be analyzed as in case i.). For the last two terms we have the compact 1 1 imbeddings ic1 : H 2 (Γ ) → L2 (Γ ) and ic2 : L2 (Γ ) → H − 2 (Γ ). Then, with 1 1 the continuity properties of V : H − 2 (Γ ) → H 2 (Γ ) and (b · n) : L2 (Γ ) → L2 (Γ ) (note that b · n ∈ L∞ (Γ )), the following composition of mappings 1

H 2 (Γ )

ic2

1

ic2

1

1

→ L2 (Γ )

ic1

(b·n)

→ H − 2 (Γ ) → H 2 (Γ )

ic1

(b·n)

→ H − 2 (Γ )

→ L2 (Γ )

→ L2 (Γ ) → L2 (Γ )

V

gives the compact mapping 1

1

ic2 ◦ (b · n) ◦ ic1 ◦ V ◦ ic2 ◦ (b · n) ◦ ic1 : H 2 (Γ ) → H − 2 (Γ ) . Similarly, we see that the mapping 1

1

{± 12 I − K  } ◦ ic2 ◦ (b · n) ◦ ic1 : H 2 (Γ ) → H − 2 (Γ ) is compact. Collecting these results completes the proof.



296

5. Variational Formulations

Integral equations of the second kind Next, let us consider the cases that A is one of the boundary integral operators of the second kind in Table 5.6.3 with the mapping properties A : H α (Γ ) → H α (Γ ) continuously

(5.6.25)

where α = 12 or − 12 , respectively. The order of A now is equal to zero. Therefore, the variational formulation of (5.6.20) here requires the use of the H α (Γ )–scalar product instead of the L2 –duality: Find λ ∈ H α (Γ ) such that aΓ (χ, λ) := (χ, Aλ)H α (Γ ) = (χ, f )H α (Γ ) for all χ ∈ H α (Γ )

(5.6.26)

where f = Bμ ∈ H α (Γ ) is given. From Theorem 5.6.8 we have G˚ arding’s inequalities available for the 1 1 operators V and D on H − 2 (Γ ) and H 2 (Γ ), respectively. Moreover, we have 1 1 1 1 the mapping properties V : H − 2 (Γ ) → H 2 (Γ ) and D : H 2 (Γ ) → H − 2 (Γ ). This allows us to modify the variational formulation (5.6.26) by replacing the H α (Γ )–scalar products in the following way: Variational formulation of the boundary integral equations of the second kind i.) For α = − 12 and A = ( 12 I ± K  ) in the IDP and EDP: 1 Find λ ∈ H − 2 (Γ ) such that 1

aΓ (χ, λ) := V χ, { 12 I ± K  }λΓ = V χ, f Γ for all χ ∈ H − 2 (Γ ) . (5.6.27) ii.) For α = 12 and A = { 12 I ± K ± V ◦ (b · n)} in the INP and ENP: 1 Find λ ∈ H 2 (Γ ) such that aΓ (χ, λ) := Dχ, { 12 I ± K ± V ◦ (b · n)}λΓ = Dχ, f Γ

(5.6.28)

1

for all χ ∈ H 2 (Γ ). In the following, let us first consider G˚ arding’s inequalities for the boundary sesquilinear forms (5.6.26) and (5.6.27), (5.6.28). As will be seen, the boundary integral equations of the second kind are also intimately related to the domain variational formulations. Whereas for the first kind boundary integral equations, the transmission problems played the decisive role, here, the boundary integral operators of the second kind associated with interior boundary value problems are related to the variational formulation of the differential equation in the exterior domain and, likewise, the boundary integral operator of the second kind associated with the exterior boundary value problems are related to the interior variational problems.

5.6 Solution of Integral Equations via Boundary Value Problems

297

Theorem 5.6.9. To the boundary sesquilinear forms (5.6.27) and (5.6.28) there exist compact operators CA : H α (Γ ) → H −α (Γ ) and positive constants γA such that Re{aΓ (λ, λ) + cΓ (λ, λ)} ≥ γA λ2H α (Γ ) for all λ ∈ H α (Γ ) ,

(5.6.29)

where cΓ (·, ·) is a compact sesquilinear form with cΓ (χ, λ) = χ, CA λΓ . Proof: i.) We begin with the IDP (5.1.19), and the equation in Table 5.6.3 where (5.6.20) reads Aλ = ( 12 I − K  )λ = f := {D − (b · n) ◦ I}μ = Bμ 1

1

1

for λ ∈ H − 2 (Γ ) and given μ ∈ H 2 (Γ ) and f ∈ H 2 (Γ ). Define u(x) := V λ in IRn and take u  = φu in Ω c as in the proof of Theorem 5.6.8. Then we obtain γc0 u  = V λ and τc u  = −( 12 I − K  )λ on Γ . Hence, with the generalized Green’s formula (5.1.13) we find  u, u ) − Re (P u ) u dx . ReV λ, AλΓ = −Reγc0 u, τc uΓ = ReaΩ c ( Ω

Since aΩ c satisfies G˚ arding’s inequality (5.4.50) for u  having compact support in Ω  , we obtain  u2H 1 (Ω  ) − Re(C  u , u )H 1 (Ω  ) − Re (P ηu) ηudx ReV λ, AλΓ ≥ c0  Ω

and with the trace theorem, Theorem 4.2.1, (4.2.29), ReV λ, AλΓ ≥ c0 V λ2

1

H 2 (Γ )

− Re{(C  ηu, ηu)H 1 (Ω  ) +



(P ηu) ηudx} .

Ω 

(5.6.30) For V we already have established G˚ arding’s inequality in Theorem 5.6.8, hence ReV λ, λΓ + ReCV λ, λΓ ≥ c0 λ2 − 1 H

− 12

2

(Γ )

1 2

with c0 > 0 and CV : H (Γ ) → H (Γ ) compactly. This inequality together with (4.2.22) yields the estimate 1

1

Here and in the sequel, c0 > 0 denotes a generic constant whose value may change from step to step.

298

5. Variational Formulations

c0 λ

1

H − 2 (Γ )

≤ V λ

1

H 2 (Γ )

+ CV λ

1

H 2 (Γ )

and, consequently, c20 λ2H − 1 (Γ ) ≤ (V λ, V λ) 2

1

H 2 (Γ )

+ (CV λ, CV λ)

Inserting this inequality into (5.6.30) we obtain   Re V λ, AλΓ + cΓ (λ, λ) ≥ c20 λ2

1

H 2 (Γ )

.

1

H − 2 (Γ )

where cΓ (χ, λ) := (ηV χ, C  ηV λ)H 1 (Ω  ) +



(ηV χ) (P ηV λ)dx + (CV χ, CV λ)

1

H 2 (Γ )

.

Ω 

As in the proof of Theorem 5.6.8, all three bilinear forms defining cΓ (·, ·) are compact; and cΓ can be represented in the proposed form. ii.) For the EDP with the corresponding equation in Table 5.6.3 we have Aλ = ( 12 I + K  )λ = f := −{D + (b · n) ◦ I}μ = Bμ and define now u := V λ in IRn with γ0 u = V λ and τ u = ( 12 I + K  )λ on Γ . With the generalized first Green’s formula (5.1.8) we now have ReV λ, AλΓ = Reγ0 u, τ uΓ = ReaΩ (u, u) . The remaining arguments of the proof are the same as in i.) but with the interior domain Ω instead of the exterior domain, Ω  ⊂ Ω c . iii) For the INP with the corresponding equations in Table 5.6.3 the operators in (5.6.20) now read 1

Aλ = { 12 I + K + V ◦ (b · n)}λ = f := V μ = Bμ for λ ∈ H 2 (Γ ) 1

1

with given μ ∈ H − 2 (Γ ) and f ∈ H 2 (Γ ). Now we define  := ηu in Ω c . u := W λ in IRn \ Γ and u Similarly, we obtain , γc0 u Γ + ReDλ, V ◦ (b · n)λΓ ReDλ, AλΓ = −Reτc u  = Re aΩ c ( u, u ) − Re (P ηu) ηudx + ReDλ, V ◦ (b · n)λΓ . Ω

5.6 Solution of Integral Equations via Boundary Value Problems

299

G˚ arding’s inequality (5.4.50) implies that u2H 1 (Ω  ) − Re(C  u , u )H 1 (Ω  ) − Re ReDλ, AλΓ ≥ c0 



(P ηu) ηudx .

Ω 

Now we need the continuity of the generalized conormal derivative τc u  from Lemma 5.1.2 which yields τc u2

1

H − 2 [Γ )

= Dλ2

1

H − 2 (Γ )

≤ c u2H 1 (Ω  ) + c(P u , P u )L2 (Ω  )

providing ReDλ, AλΓ ≥

c0 Dλ2 − 1 H 2 (Γ )





− Re{(C ηu, ηu)H 1 (Ω  ) +  +c

(P ηu) ηudx

Ω 

(P ηu) P ηudx − Dλ, V ◦ (b · n)λΓ } .

Ω 

G˚ arding’s inequality for D as shown in Theorem 5.6.8 gives the estimate Dλ2

1

H − 2 (Γ )

1

≥ c0 λ2

1

H 2 (Γ )

− c(CD λ, CD λ)

1

H − 2 (Γ )

1

where CD : H 2 (Γ ) → H − 2 (Γ ) is compact. Collecting the above estimates finally yields   Re Dλ, AλΓ + cΓ (λ, λ) ≥ c0 λ2 1 H 2 (Γ )

where + c(ηW χ, C  ηW λ)H 1 (Ω  ) cΓ (χ, λ) = c(CD χ, CD λ) − 12 H (Γ )   +c (ηW χ) P ηW λdx + c (P ηW χ) (P ηW λ)dx − Dχ, V ◦ (b · n)λΓ . Ω 

Ω 

The compactness of all the sesquilinear forms on the right–hand side except the last one has already been shown in the proof of Theorem 5.6.8. The compactness of the last one follows again from the compact composition of the mappings 1

1

D ◦ V ◦ ic2 ◦ (b · n) ◦ ic1 : H 2 (Γ ) → H − 2 (Γ ) . iv) For the ENP (5.1.70), the operator A in Table 5.6.3 is of the form 1

Aλ = { 12 I − K − V ◦ (b · n)}λ = f := −V μ = Bμ for λ ∈ H 2 (Γ ) 1

1

with given μ ∈ H − 2 (Γ ) and f ∈ H 2 (Γ ). As in case iii) take

300

5. Variational Formulations

u = W λ in Ω with γ0 u = −( 12 I − K)λ and τ u = −Dλ on Γ . Then ReDλ, AλΓ

=

Reτ u, γ0 uΓ − ReDλ, V ◦ (b · n)λΓ

=

aΩ (u, u) − ReDλ, V ◦ (b · n)λΓ .

The rest of the proof follows in the same manner as in case iii) with Ω c replaced by the interior domain Ω. This completes the proof of Theorem 5.6.9.  Remark 5.6.3: For the bilinear form (5.6.26), G˚ arding’s inequality Re{(λ, Aλ)H α (Γ ) + cΓ (λ, λ)} ≥ γA λ2H α (Γ )

(5.6.31)

with the operators A of the second kind in Table 5.6.3 follows immedi1 1 ately provided K : H 2 (Γ ) → H 2 (Γ ) and, consequently, the dual operator  − 12 − 12 K : H (Γ ) → H (Γ ), are compact. Here, the corresponding compact and bilinear forms are defined by cΓ (χ, λ) = ±(χ, {K + V ◦ (b · n)}λ) 12 H (Γ )

±(χ, K  λ) − 12 , respectively. However, the compactness of K or K  can H (Γ ) only be established for a rather limited class of boundary value problems excluding the important problems in elasticity where K and K  are Cauchy singular integral operators, but including the classical double layer potential operators of the Laplacian, of the Helmholtz equation, of the Stokes problem and the like, provided Γ is at least C 1,α with α > 0 excluding C 0,1 –Lipschitz boundaries which are considered here. (See also Remark 1.2.1 and Chapter 8.) Theorem 5.6.9 is valid without these limitations since it is based only on continuity, the trace theorems on Lipschitz domains and G˚ arding’s inequality for the domain sesquilinear forms. 5.6.7 Positivity and Contraction of Boundary Integral Operators Here we follow the presentation in Steinbach et al [401]. In [84], Costabel showed that these relations are rather general and have a long history. As in Carl Neumann’s classical method for the solution of the Dirichlet or Neumann problem in potential theory, we now show that for a certain class of boundary integral equations of the second kind Carl Neumann’s method for solving these equations with the Neumann series can still be carried out in the corresponding Sobolev spaces. For this purpose we consider the special class of boundary value problems leading to the boundary integral operators of the first kind with V and D which are positive in the sense that they satisfy G˚ arding inequalities and in addition

5.6 Solution of Integral Equations via Boundary Value Problems 1

V λ, λ ≥ cV1 λ2

1

H − 2 (Γ )

2 Dμ, μ ≥ cD 1 μ

for all λ ∈ H − 2 (Γ ) 1 2

1

H 2 (Γ )

for all μ ∈ H (Γ )

301

(5.6.32) (5.6.33)

where 1

1

H2 (Γ )

:=

{μ ∈ H 2 (Γ ) | μ, m = 0 for all m ∈ }



:=

{μ0 ∈ H 2 (Γ ) | Dμ0 = 0} ,

(5.6.34)

and 1

(5.6.35)

1 with constants 0 < cV1 ≤ 12 , 0 < cD 1 ≤ 2. Since D satisfies a G˚ arding inequality the dimension of  is finite. Moreover, these inequalities are satisfied if in the previous sections the corresponding energy forms of the underlying partial differential operators are finite (Costabel [84]). With the Calder´on projectors (5.6.11) we have also the relations (1.2.24)– 1 1 (1.2.28) in the Sobolev spaces H 2 (Γ ) and H − 2 (Γ ), respectively, as in Theorem 5.6.2 for the four basic boundary integral operators V, K, K  , D in (5.6.10). In order to show the contraction properties of the operators 12 I ± K and 1  2 I ± K , we now introduce the norms 1

λV := V λ, λ 2

1

and μV −1 := V −1 μ, μ 2

(5.6.36)

and μ 12 , respectively. More prewhich are equivalent to λ − 12 H (Γ ) H (Γ ) cisely, the following estimates are valid. Lemma 5.6.10. With the constants cV1 and cD 1 in (5.6.32) and (5.6.33) we have cV1 μ2V −1 ≤ μ2

1

1 2

H (Γ )

2 cD 1 μ

1

H 2 (Γ )

and Here

≤ Sμ, μ 1

for all μ ∈ H 2 (Γ ) , 1

for all μ ∈ H2 (Γ ) 1

V − 2 μL2 (Γ ) = μV −1 = V −1 μ, μ 2 . S = D + ( 12 I + K  )V −1 ( 12 I + K) = V −1 ( 12 I + K)

(5.6.37) (5.6.38)

(5.6.39) (5.6.40)

denotes the Steklov–Poincar´e operator associated with Ω. As we can see, for the solution u of the homogeneous equation (5.1.1) 1 with f = 0 and given Dirichlet data μ ∈ H 2 (Γ ) on Γ , the Steklov–Poincar´e 1 operator in (5.6.40) maps μ into the Neumann data ∂ν u = λ ∈ H − 2 (Γ ) on Γ . Therefore S is often also called the Dirichlet to Neumann map.

302

5. Variational Formulations

Proof: By definition, V λ

1

H 2 (Γ )

=

sup 1 0=τ ∈H − 2

(Γ )

V λ, τ  τ  − 12 H

≥

|V λ, λ| ≥ cV1 λ − 12 H (Γ ) λ − 12 H

(Γ )

(Γ )

due to (5.6.32). On the other hand, with (4.1.27) and μ = V λ we have V −1 μ

1 H− 2

(Γ )

=

sup 1

0=χ∈H 2 (Γ )

=

|V −1 μ, χ| χ 12 H (Γ )

sup 1

0=V λ∈H 2 (Γ )

|μ, λ| V λ 12

≤

H (Γ )

1 μ 12 H (Γ ) cV1

in view of the previous estimate. So, again with (4.1.27), V −1 μ, μ = μ2V −1 ≤ V −1 μ

1

H − 2 (Γ )

μ

1

H 2 (Γ )

≤

1 μ2 1 H 2 (Γ ) cV1

which establishes (5.6.37). The second inequality follows from the symmetric form of the Steklov–Poincar´e operator, i.e.,  Sμ, μ =  D + ( 12 I + K  )V −1 ( 12 I + K) μ, μ 1

2 = Dμ, μ + ( 12 I + K)μ2V −1 ≥ cD 1 μ

for μ ∈ H2 (Γ )

1 2

H (Γ )

which is the desired inequality (5.6.38). The relation(5.6.39) is a simple consequence of the positivity and selfad1 1 jointness of V which defines an isomorphic mapping H − 2 (Γ ) → H 2 (Γ ), 1 1 hence V −1 : H − 2 (Γ ) → H 2 (Γ ) is also positive and the square root 1 1 V − 2 : H − 2 (Γ ) → L2 (Γ ) is well defined (Rudin [367, Theorem 12.33]).  We are now in the position to formulate the contraction properties of the operators ( 12 I ± K) and ( 12 I ± K  ). Theorem 5.6.11. The operators 12 I + K and the corresponding energy spaces and

1 2I

+ K  are contractions on

 ( 12 I

± K)μ|V −1 ≤ cK μV −1

for all μ ∈

−1

 ( 21 I



± K )λ|V ≤ cK λV

for all λ ∈ 9

where cK =

1 2

+

1 4

1

H 2 (Γ )

− cV1 cD 1 0 and λ0 > 0 are constants. Since H 2 (Ω) → L2 (Ω) is compactly imbedded (see (4.1.31)),  v 2 dx =: (Cv, v)H 2 (Ω) Ω

where C is a compact linear operator in H 2 (Ω). Clearly, this lemma is also c with fixed R > 0 sufficiently large. valid for aΩRc (u, v) and ΩR Since a in (5.7.5) satisfies a G˚ arding inequality as (5.7.6) on H, the Fredholm alternative for sesquilinear forms, Theorem 5.3.10 is valid. The homogeneous problem for (5.7.5) where f1 = 0 , f2 = 0 , ϕ = 0 , ψ = 0 has only a trivial solution. Therefore (5.7.5) always has a unique solution u. Moreover, it satisfies the a priori estimate uH 2 (Ω) + uH 1 (ΩRc ) ≤ c{f1 H −2 (Ω) + f2 H −2 (Ω c )∩H −2 (BR ) + ϕ

3

1

H 2 (Γ )×H 2 (Γ )

+ ψ

3

(5.7.7) 1

H − 2 (Γ )×H − 2 (Γ )

},

and can be represented in the form   E(x, y)f2 (y)dy . (5.7.8) u(x) = W ϕ(x) − V ψ(x) + E(x, y)f1 (y)dy + Ω

c ΩR

310

5. Variational Formulations

For proving the continuity properties of the boundary integral operators in the Calder´ on projector (2.4.23) we again exploit the fact that V in (2.4.26) as well as D in (2.4.30) define potentials which are solutions of the transmission problem (5.7.4) with f1 = 0 and f2 = 0 together with the transmission conditions ϕ = 0 for V and ψ = 0 for W , respectively. Theorem 5.7.3. The following operators are continuous: 1

3

2 V : H − 2 (Γ ) × H − 2 (Γ ) → H 2 (Ω, Δ2 ) × Hloc (Ω c , Δ2 ) ,  ∂  ∂  ∂nc V = ∂n V =: V : H − 12 (Γ ) × H − 32 (Γ ) → H 12 (Γ ) × H 32 (Γ ) , γc0 γ0     3 3 1 1 Mc M V : H − 2 (Γ ) × H − 2 (Γ ) → H − 2 (Γ ) × H − 2 (Γ ) , V and N Nc 1

3

2 W : H 2 (Γ ) × H 2 (Γ ) → H 2 (Ω, Δ2 ) × Hloc (Ω c , Δ2 ) ,  ∂  ∂  ∂nc W and ∂n W : H 12 (Γ ) × H 32 (Γ ) → H 12 (Γ ) × H 32 (Γ ) , γc0 γ0     1 3 1 3 Mc M W =: D : H 2 (Γ ) × H 2 (Γ ) → H − 2 (Γ ) × H − 2 (Γ ) . W = N Nc

In the theorem, the corresponding operators are given by:   ∂u   Mu → u , W : ∂n → u V : Nu u 

and V=

V23 V13

V24 V14



 , D=

D32 D42

D31 D41

 (5.7.9)

defined in (2.4.26), (2.4.30), respectively. Proof: In fact, the proof can be carried out in the same manner as for Theorem 5.6.2 for second order systems. Therefore, here we only present the main ideas. First we consider the continuity properties of V . To this end, we consider the related variational transmission problem (5.7.2), (5.7.3) with ϕ = 0 which c , Δ2 ) satisfying (5.7.7) which has a unique solution u in H 2 (Ω, Δ2 ) × H 2 (ΩR can be represented in the form (5.7.8) with ϕ = 0. In particular, we may consider f1 = 0 , f2 = 0 which leads to the first three continuity properties in Theorem 5.7.3 by making use of the trace theorem 4.2.1. Next, for the mapping properties of W , we again consider the transmission 3 1 problem (5.7.2), (5.7.3) but now with ψ = 0 and ϕ ∈ H 2 (Γ ) × H 2 (Γ ) , f1 = 0 , f2 = 0. With representation formula (5.7.8) of the corresponding solution c , Δ2 ), i.e., with u = W ϕ and the trace theorem u in H 2 (Ω, Δ2 ) × H 2 (ΩR together with the definition of the operators M and N , i.e., of τ based on

5.7 Partial Differential Equations of Higher Order

311

the generalized Green’s formula in Lemma 5.7.1, we obtain the last three continuity properties in Theorem 5.7.3.  As a consequence of the mapping properties in Theorem 5.7.3 we find the jump relations as in Lemma 5.6.3. 1

3

1

3

Lemma 5.7.4. Given ϕ ∈ H 2 (Γ ) × H 2 (Γ ) and ψ ∈ H − 2 (Γ ) × H − 2 (Γ ), the following jump relations hold:  ! ∂ " [Vψ]|Γ = 0 , [(M, N ) V ψ]|Γ = −ψ , ∂n , γ0 W ϕ |Γ = ϕ , [Dϕ]|Γ = 0 (5.7.10) We omit the proof since the arguments are the same as for Lemma 5.6.3. In order to consider the boundary integral equations collected in Table 2.4.5 and Table 2.4.6 in the above Sobolev spaces and in variational form we now present the G˚ arding inequalities for the matrix operators V and D in (5.7.9). Theorem 5.7.5. With the real Sobolev spaces, the boundary integral matrix operators V and D satisfy G˚ arding’s inequalities as ψ, Vψ + ψC V ψ ≥ cV ψ2

1

3

H − 2 (Γ )×H − 2 (Γ )

Dϕ, ϕ + C D ϕ, ϕ ≥ cD ϕ2

1

,

3

H 2 ×H 2 (Γ )

(5.7.11) (5.7.12)

where cV and cD are positive constants and the linear operators C V and 1 3 1 3 C D are compact operators on H − 2 (Γ ) × H − 2 (Γ ) and H 2 (Γ ) × H 2 (Γ ), respectively. Proof: Since, again, the proof can be carried out in the same manner as for Theorem 5.6.8, we just outline the main ideas in this case without the details. For (5.7.11) consider the function u(x) := −η(x)V ψ(x) for x ∈ IR2 \ Γ 1

3

with any ψ ∈ H − 2 (Γ ) × H − 2 (Γ ) given , where η ∈ C0∞ (IR2 ) is a cut–off function as in (5.6.17). Then u is a solution of the transmission problem c (5.7.2), (5.7.3) with ϕ = 0 and f2 = Δ2 (ηV ψ) ∈ C0∞ (ΩR ). As in Section 5.6.6, the generalized first Green’s formula yields with the G˚ arding inequality (5.7.6):   ∂u ψ, Vψ = ∂n ψ1 + uψ2 ds Γ



= aΩ (u, u) + aΩRc (u, u) −

uΔ2 (ηV ψ)dy R1 ≤|y|≤R

≥

γ0 {u2H 2 (Ω)

+

u2H 2 (Ω c ) R

− c1 ψ2H −1 (Γ )×H −2 (Γ ) }

312

5. Variational Formulations

Next, use Lemma 5.7.1 which yields ψ2

1

3

H − 2 (Γ )×H − 2 (Γ )

≤ c2 {u2H 2 (Ω) + u2H 2 (Ω c ) + Δ2 u2H −2 (Ω) } R

So, ψ, Vψ ≥

γ0 2 c2 ψH − 12 (Γ )×H − 32 (Γ )

− c3 ψ2H −1 (Γ )×H −2 (Γ )

since Δ2 u2H −2 (Ω) ≤ cψH −1 (Γ )×H −2 (Γ ) . Then with the compact imbed1

3

ding H − 2 (Γ )×H − 2 (Γ ) → H −1 (Γ )×H −2 (Γ ) the G˚ arding inequality (5.7.11) follows. For (5.7.12) we use the function u(x) := η(x)W ϕ(x) for x ∈ IR2 \ Γ . Then

 Dϕ, ϕ =

(ϕ1 M u + ϕ2 N u)ds Γ



= aΩ (u, u) + aΩRc (u, u) −

uΔ2 (ηW ϕ)dy . R1 ≤|y|≤R2

G˚ arding inequalities (5.7.6) and the trace theorem lead to the desired estimate (5.7.12).  Remark 5.7.1: For the integral operator equations of the second kind as in Tables 2.4.5 and 2.4.6, corresponding boundary bilinear forms are related to the domain bilinear forms (2.4.3) and are given by ) aΩ c (W ϕ, W ϕ = ( 12 I ± K11 )ϕ1 ∓ V12 ϕ2 , D41 ϕ 1 + D42 ϕ 2  ) aΩ (W ϕ, W ϕ 1 1 + D32 ϕ 2  +∓D21 ϕ1 + ( 2 I ∓ K22 )ϕ2 , D31 ϕ (5.7.13) and  aΩ c (V ψ, V ψ)  aΩ (V ψ, V ψ)

= ( 12 I ± K33 )ψ1 ∓ V34 ψ2 , V13 ψ1 + V14 ψ2  +∓D34 ψ1 + ( 1 I ∓ K44 )ϕ2 , V23 ψ1 + V24 ψ2  2

(5.7.14) Therefore, these bilinear forms also satisfy G˚ arding inequalities in the same manner as in Theorem 5.6.9 for second order partial differential equations. As we can see, the test functions are images under the operators of the first kind which serve as preconditioners.

5.8 Remarks

313

With G˚ arding inequalities available, Fredholm’s alternative, Theorem 5.3.10 can be applied to all the integral equations in Tables 2.4.5 and 2.4.6 in the corresponding Sobolev spaces. In fact, as formulations in Table 2.4.6 imply uniqueness, for these versions the existence of solutions in the Sobolev spaces is ensured. Remark 5.7.2: As was pointed out by Costabel in [84] the positivity of the boundary integral operators V and D on appropriate Sobolev spaces as in Section 5.6.7 for the second order equations, the convergence of corresponding successive approximation for integral equations of the second kind, can be established. The details will not be pursued here. Some further works on the biharmonic equation and boundary integral equations can be found in Costabel et al [87, 91], Giroire [148], Hartmann and Zotemantel [174], [208] and Kn¨ opke [230].

5.8 Remarks 5.8.1 Assumptions on Γ In this chapter we have for the most part, not been very specific concerning the assumptions on Γ which should at least be a strong Liptschitz boundary. For strongly elliptic second order elliptic equations and systems including the equations of linearized elasticity and also for the Stokes system, a strong Lipschitz boundary is sufficient as long as Dirichlet or Neumann conditions or boundary and transmission conditions of the form (5.1.21) with B00 = I and N00 = 0 are considered (see Costabel [80], Fabes et al [114, 115], Neˇcas [326]). For the general Neumann–Robin conditions (5.1.21), however, one needs at least a C 1,1 boundary. For higher order equations of order 2m , one needs as least Γ ∈ C ,κ with  + κ = m as C 1,1 for the biharmonic equation. D. Mitrea, M. Mitrea and M. Taylor develop in Mitrea et al [309] the boundary integral equations for Lipschitz boundaries and for scalar strongly elliptic second order equations, in particular the Hodge Laplacian on Riemannian manifolds. 5.8.2 Higher Regularity of Solutions Let us collect some facts regarding higher regularity of solutions without proofs. Higher regularity is often needed in applications and for the numerical approximation and corresponding error analysis (see e.g. [204], Sauter and Schwab [371], Steinbach [398]). Without higher requirement for Γ , Theorem 5.6.2, Lemma 5.6.4 and Lemma 5.1.1 can be extended as follows due to the results by Fabes, Kenig, Verchota in [114, 115] and Costabel [80].

314

5. Variational Formulations

Theorem 5.8.1. For |σ| ≤ 12 , the operators in Theorem 5.6.2 and in Lemma 5.6.4 can be extended to the following continuous mappings:

γ0 V τW

γ0 W τW

1

1+σ → H 1+σ (Ω) × Hloc (Ω c ) ,

1

→ H 2 +σ (Γ ) ,

1

→ H − 2 +σ (Γ ) ,

V

: H − 2 +σ (Γ )

and γc0 V

: H − 2 +σ (Γ )

and τc V

: H − 2 +σ (Γ )

1

1

1

1+σ → H 1+σ (Ω) × Hloc (Ω c ) ,

1

→ H 2 +σ (Γ ) ,

1

→ H − 2 +σ (Γ ) .

W

: H 2 +σ (Γ )

and γc0 W

: H 2 +σ (Γ )

and τc W

: H 2 +σ (Γ )

1

1

For 0 ≤ σ < 12 , the generalized conormal derivative satisfies τ u ∈ 1 H − 2 +σ (Γ ) for u ∈ H 1+σ (Ω). (For the Stokes system see Kohr et al [234].) Moreover, the variational solutions in H 1 (Ω) gain slightly more regularity and satisfy corresponding a priori estimates of the form uH 1+σ (Ω) ≤ {ϕ

1

H 2 +σ (Γ )

+ f H −1+σ (Ω) }

(5.8.1)

1  −1+σ (Ω) or for the Dirichlet problem if (ϕ, f ) ∈ H 2 +σ (Γ ) × H 0

uH 1+σ (Ω) ≤ c{ψ

1

H − 2 +σ

+ f H −1+σ (Ω) }

(5.8.2)

1  −1+σ (Ω) and 0 < σ ≤ 1 . for the Neumann problem if (ψ, f ) ∈ H − 2 +σ (Γ )× H 0 2 1 Correspondingly, if for IDP and EDP in Table 5.1.2, μ ∈ H 2 +σ (Γ ) is 1 1 given, then λ ∈ H − 2 +σ (Γ ). If for INP and ENP μ ∈ H − 2 +σ (Γ ) is given 1 then λ ∈ H 2 +σ (Γ ). If Γ is much smoother then one may obtain even higher regularity as will be seen in Chapters 8–11.

5.8.3 Mixed Boundary Conditions and Crack Problem We return to the example of the mixed boundary value problem (2.5.1) for  −1 (Ω) , ϕ ∈ H 12 (ΓD ) and the Lam´e system; but now with given f ∈ H 0 1 1 1 ψ ∈ H − 2 (ΓN ). Again, we write with ϕ ∈ H 2 (Γ ) and ψ ∈ H − 2 (Γ ) according to (2.5.2),  where ϕ  ∈H  12 (ΓN ) and ψ  − 12 (ΓD )  , Tu = ψ + ψ  ∈H γ0 u = ϕ + ϕ (5.8.3) are the yet unknown Cauchy data. The boundary integral equations (2.5.6) can now be written in variational form:  ϕ  − 12 (Γ ) satisfying ) ∈ H Find (ψ,

5.8 Remarks

315

  ϕ  τ − K ϕ  χ  ) , (  := V ψ,  , τΓ + K  ψ,  + Dϕ  , χ  Γ τ , χ) aΓ (ψ,  − 12 (Γ ) × H  12 (Γ ) (5.8.4)  for all (  ∈H = ( τ , χ) τ , χ) where the linear functional  is given as  = ( 12 I + K)ϕ, τ − V ψ, τ − N f , τ ( τ , χ)

(5.8.5)

 − Dϕ, χ − Tx N f , χ  . + ( 12 I − K  )ψ, χ arding inequality Since aΓ is continuous and satisfies the G˚   ϕ  ϕ  ψ  + Dϕ  ) , (ψ,  ) = V ψ, , ϕ  aΓ (ψ, 2  ≥ γ0 {ψ +  ϕ 1 } 1 H − 2 (Γ )

(5.8.6)

H 2 (Γ )

 ϕ  12 (ΓN ) ⊂ H − 12 (Γ ) × H 12 (Γ ) as in (5.6.32) and  − 12 (ΓD ) × H ) ∈ H for all (ψ,  − 12 (ΓD ) × H  12 (ΓN )–elliptic and (5.8.4) is uniquely solvable. (5.6.33), aΓ is H In fact, this formulation underlies the efficient and fast boundary element methods in Of et al [339, 340], Steinbach [398]. For the Stokes system, mixed boundary value problems are treated in [234] and for the biharmonic equation in Cakoni et al [58]. 1 For the classical insertion problem (2.5.14), now ϕ ∈ H 2 (Γ ) and ϕ± ∈ 1 1  2 (Γc ) are given, and the variational form of (2.5.16) H 2 (Γc ) with [ϕ]|Γc ∈ H 1  − 12 (Γc ) reads: for the desired Cauchy data ψ ∈ H − 2 (Γ ) and [ψ]|Γc ∈ H 1  − 12 (Γc ) satisfying Find (ψ, [ψ]|Γc ) ∈ H − 2 (Γ ) × H  (5.8.7) aΓ (ψ, [ϕ]|Γc ) , (τ , τ) := VΓ ψ, τ Γ − VΓc [ϕ]|Γc , τ Γ − VΓ ψ, τΓc + VΓc [ϕ]|Γc , τΓc  − 2 (Γc ) = (τ , τ) for all (τ , τ) ∈ H − 2 (Γ ) × H 1

1

where the linear functional is given via (2.5.16). The bilinear form aΓ is continuous and satisfies a G˚ arding inequality  +  τ 2 − 1 aΓ (ψ, τ) , (ψ, τ) ≥ γ0 {ψ2 − 1 H 2 (Γ ) H 2 (Γc ) (5.8.8) 2 2 c(ψH −1 (Γ ) +  τ H −1 (Γc ) } Since, in addition, the homogeneous problem with ϕ = 0 , ψ = 0 admits only the trivial solution, (5.8.7) is uniquely solvable. 1  − 12 (Γc ) For the classical crack problem, ψ ± ∈ H − 2 (Γc ) with [ψ]|Γc ∈ H 1 and, in our example, ϕ ∈ H 2 (Γ ) in (2.5.17) are given. Then the variational form of (2.5.18) reads: 1  12 (Γc ) satisfying Find (ψ, [ϕ]|Γc ) ∈ H − 2 (Γ ) × H   := VΓ ψ, τ Γ + KΓc [ϕ]|Γc , τ Γ aΓ (ψ, [ϕ]|Γc ) , (τ , χ) (5.8.9)   Γc − KΓ ψ, χ  Γc + DΓc [ϕ]|Γc , χ  2 (Γc ) , = (τ , τ) for all (τ , τ) ∈ H − 2 (Γ ) × H 1

1

316

5. Variational Formulations

where the linear functional is given via (2.5.18). Here aΓ satisfies the G˚ arding inequality   , (ψ, χ)  ≥ γ0 {ψ2 − 1 +χ  2 1 −c(ψ2H −1 (Γ ) +χ  2L2 (Γ )} . aΓ (ψ, χ) H

2

(Γ )

H 2 (Γ )

(5.8.10) Also here the homogeneous problem (5.8.9) with ψ ± |Γc = 0 and ϕ|Γ = 0, i.e.,  = 0, admits only the trivial solution, hence (5.8.9) has a unique solution. For more detailed analysis of these problems we refer to Costabel and Dauge [85], Duduchava et al [107], [209, 210], Natroshvili et al [324, 106], Stephan [403].

6. Electromagnetic Fields

6.1 Introduction In this chapter we are concerned with the application of boundary integral methods to boundary value problems for time harmonic Maxwell equations in electromagnetic theory. In what follows in this chapter we assume that the results stated are formulated in classical function spaces if not specified otherwise. In particular, Γ = ∂Ω is assumed to be at least C 1+α , i.e. Lyaponov. However, most of the analysis can be generalized to Lipschitz continuous boundaries. The determination of the state of an electromagnetic system requires the knowledge of several physical quantities. We list these below: E, the electric field, H, the magnetic field, D, electric displacement field or flux, B, the magnetic induction or flux, J , the conduction current density, , the charge density. These are five vector fields and one scalar field depending on space and time. There are four basic laws which connect these quantities: – Gauss’ law for electric fields . This states that for any subregion Ω ⊂ R3 with boundary ∂Ω, the electrical flux across Ω is equal to the total charge in Ω:   D · ds = d. (6.1.1) ∂Ω

Ω

– Gauss’ law for magnetic fields. The total magnetic flux across any closed surface ∂Ω is zero:  B · ds = 0. ∂Ω

© Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_6

(6.1.2)

318

6. Electromagnetic Fields

– Faraday’s law. The line integral of the electric field about any closed regular curve ∂S in the field is equal to the negative time rate of change of the magnetic flux through any surface S with the boundary ∂S; i.e.  : ∂B · ds. (6.1.3) E · d = − ∂t S

∂S

– Ampere–Maxwell law. The rate of change with time of the electrical flux and the electrical current through a surface S with boundary ∂S produces a circulating magnetic field around the closed path ∂S; i.e.,    ∂ D · ds + J · ds = H · d. (6.1.4) ∂t S

S

∂S

Equations (6.1.1) - (6.1.4) are the Maxwell equations in integral form [286] which are ingenious.

6.2 Maxwell Equations Since the Maxwell equations in integral form are valid for arbitrary domains Ω and arbitrary surfaces S, by using the divergence theorem and the Stokes theorem, we obtain the Maxwell differential equations in Ω × [0, T ]: ∂B , ∂t ∂D +J , curl H = ∂t div D =  , div B = 0. curl E = −

(6.2.1) (6.2.2) (6.2.3) (6.2.4)

These equations are not linearly independent since Equations (6.2.1) imply (6.2.4) provided (divB)(x, 0) = 0. (6.2.5) Also one may notice that from Equations (6.2.2) and (6.2.3) the charge density  and the current density J satisfy the equation ∂ + div J = 0. ∂t

(6.2.6)

This equation expresses the fundamental law that the electrical charge is conserved — any flow of charge must come from supply. Conversely, (6.2.6), (6.2.1) and (6.2.2) imply (6.2.3) provided

6.3 Constitutive Equations

(D − )(x, 0) = 0.

319

(6.2.7)

This means, (6.2.1)–(6.2.3) are linearly independent, and under condition (div B)(x, 0) = 0, Equation (6.2.4) is a consequence. Alternatively, (6.2.1), (6.2.2) and (6.2.6) are linearly independent and under conditions (6.2.5) and (6.2.7) the equations (6.2.3) and (6.2.4) are the consequences.

6.3 Constitutive Equations We have 16 unknowns but only 7 independent equations. Therefore, to characterize the electromagnetic fields completely, one needs 9 additional equations. These are constitutive relations taking into account the properties of media which must play a role in the complete system of equations. Experiments show the relations    D(x, t) = F E(x, t) , B(x, t) = G H(x, t) , J (x, t) = H J (x, t) with appropriate functions F, G and H. We are going to consider linear isotropic homogeneous media. In this case, we adopt the following constitutive equations: D = εE , B = μH , J = σE + J0 , (6.3.1) where ε, μ and σ are constant; ε ≥ 0 is the electric permittivity, μ > 0 the magnetic permeability, σ ≥ 0 the conductivity and J0 is a prescribed current density. An energy estimate For the Cauchy problem in Ω × (0, T ] with Γ = ∂Ω consisting of ∂B 1 + curl D = 0 , ∂t ε ∂D 1 σ − curl B + D = 0 , ∂t μ ε div D = 0 , div B = 0 , in terms of the constitutive equations for  = 0 and J0 = 0, let us consider the following energy estimate:     ε ∂ 2 |B| dx = − (curl D) · Bdx = n · (D × B)ds − D · (curl B)dx, 2 ∂t Ω

μ 2

 Ω

Ω

∂ |D|2 dx = ∂t

 Ω

σ (curl B) · Ddx − ε

Γ

Ω

(6.3.2)

 |D|2 dx. Ω

320

6. Electromagnetic Fields

Adding the equations in (6.3.2) yields the differential equation   d σ E=− |D|2 dx + n · (B × D)ds dt ε Ω

Γ

for the electromagnetic energy E(t) =

1 2





 ε|B|2 + μ|D|2 dx.

Ω

Lemma 6.3.1. Under the condition n · (B × D)|Γ = 0, the homogeneous Cauchy problem only has the trivial solution B = D = 0. As a consequence of Lemma 6.3.1, it is suggested that the Cauchy problem with boundary conditions (n × D)|Γ = Φ or (n × B)|Γ = Ψ.

(6.3.3)

has at most one solution Proof of of Lemma 6.3.1: Since E(0) = 0    σ 1 2 μ|D| dx ≤ E(t) − E(0) = − |D|2 dxdt ≤ 0 0≤ 2 ε 0 Ω

Ω

it holds

1 2

 μ|D|2 dx = 0, Ω

hence D ≡ 0. This implies ∂B = 0 and B(x, t) = B(x, 0) = 0 ∂t as proposed.



Lemma 6.3.2. Let 0 ∈ L2 (R3 ) have compact support and J0 = 0. Then for σ > 0 , and ε > 0 σt D(x, t) = e− ε B0 for t ≥ 0, where B denotes the Bogovskii operator in R3 . Proof: Apply div to (6.2.2) and use (6.2.3) and (6.3.1) together with (6.2.3) to obtain ∂ σ +  = 0. ∂t ε Hence, with (6.2.3) , σt divD =  = e− ε 0 ,

6.3 Constitutive Equations

321

and we obtain the proposed relation in terms of B, the Bogovskii operator  as a right inverse to div which is a ψdO−1 . Remark 6.3.1: For positive constant ε and σ we have perfect electric and magnetic media Dautray and Lions [95]. Some special media are characterized in terms of the media parameters. So, substances for which σ is not negligibly small are called conductors. • The perfectly conducting medium is given for σ → ∞. Under the assumption that curl H remains bounded, Equation (6.3.2) yields E(x) = 0 and then (6.3.2) implies H(x) = 0. That means, no electromagnetic fields exist in a perfectly conducting medium. • If σ = 0 or negligibly small the medium is called dielectric or insulator. For σ = 0 we have perfect dielectrics. Then  = 0 and J = 0. For a general conducting medium, σ = 0, if the conductivity increases with temperature over a wide range, the medium is called semiconductor. In free −9 −7 space, ε = ε0 = 10 Henry/meter. 36π Faraday/meter and μ = μ0 = 4π10 2 Moreover, ε0 μ0 c = 1 with c the speed of light. Remark 6.3.2: The relation (6.2.6) of the conservation of charge with the last of the equations (6.3.1) yields ∂ σ +  = 0 and (x, 0) = 0 (x) ∂t ε

(6.3.4)

σ

in view of (6.2.2) and (6.2.3). Hence,  = e− ε t 0 where 0 (x) is given. Remark 6.3.3: The generalized current concept: It was Maxwell who first noted that Ampere’s law for the statics: curl H = J was incomplete for time-varying fields. He amended the law to include an electric displacement current ∂D/∂t in addition to the conduction current. In dielectrics, part of the term ∂D/∂t is a motion of bound particles and is thus a current in the true sense of the word. However, it is convenient to consider the entire ∂D/∂t term as a current. In view of the symmetry of Maxwell’s equations, it also is convenient to consider the term ∂B/∂t as a magnetic displacement current. To represent sources, we amend the field equations to include impressed currents, electric and magnetic. These are the currents we view as the cause of the field. The impressed currents represent energy sources. The symbols J and M will be used to denote electric and

322

6. Electromagnetic Fields

magnetic currents in general, with superscripts indicating the type of current. We define total currents: ∂D + J c + Ji ∂t ∂B + Mi Mt = ∂t Jt =

where the superscripts t, c, and i denote total, conduction, and impressed currents. The symbols i and k will be used to denote net electric and magnetic currents, and the same superscripts will indicate the type. Thus the circuit form corresponding to the above equations is ∂ψ e + ic + i i ∂t ∂ψ m + ki kt = ∂t it =

The i and k are related to the J and M by   i = J · nds, k = M · nds. Γ

Γ

where these apply to any of the previous types of current. Remark 6.3.4: The terminology of the electromagnetic quantities is by no means universal and one may find the following alternative terminologies for the corresponding electromagnetic quantities: E, the electric field is also called the electric field density; B, the magnetic induction is also called the magnetic field, magnetic field density or magnetic flux density; D, the electric displacement field is also called electric induction or electric flux density; H, the magnetic field is also called auxiliary magnetic field, magnetic field density, magnetizing field; J , the current density is also called the total current density consisting of free and bound current; , the charge density is also called total charge density consisting of free and bound charge; ε, the permittivity is also called the dielectric constant; μ, the permeability is also called the magnetic constant.

6.4 Time Harmonic Fields

323

6.4 Time Harmonic Fields We consider only the monochromatic theory of electromagnetic waves, that is, not every thing is independent of time but rather that things vary periodically in time with a fixed frequency ω. Then with i2 = −1     E(x, t) = Re E(x)e−iωt , D(x, t) = Re D(x)e−iωt ,     H(x, t) = Re H(x)e−iωt , B(x, t) = Re B(x)e−iωt ,     J (x, t) = Re J(x)e−iωt , (x, t) = Re (x)e−iωt . By using these formulae the derivation with respect to time t transforms into multiplication by −iω. The current density J commonly consists of two terms: one, J0 associated with external sources of electromagnetic disturbances and the other, Jc associated with conduction currents produced as a result of the electric field governed by Ohm’s law. The Maxwell equations (6.2.1)–(6.2.4) together with the constitutive equations (6.3.1) in a bounded domain Ω read now curl E(x) = iωμH(x), curl H(x) = (−iωε + σ)E(x) + J0 , div (ε E)(x) = , div (μH)(x) = 0.

(6.4.1) (6.4.2) (6.4.3) (6.4.4)

Equation (6.2.6) then reduces to (x) = div J0 (x)/(iω − σ/ε).

(6.4.5)

We remark that for time–harmonic fields  and J0 are given source distributions satisfying the equation of conntinuity (6.4.5). In the homogeneous isotropic medium, from the Maxwell equations we can eliminate either E or H and obtain the inhomogeneous Helmholtz equations ΔE + k 2 E = grad /ε − iωμJ0

and div E = /ε

(6.4.6)

and div H = 0,

(6.4.7)

or ΔH + k 2 H = −curl J0 with k 2 = ω 2 με + iωμσ,

(6.4.8)

where we have used the vector identity Δ u = grad div u − curl curl u for a vector field u.

(6.4.9)

324

6. Electromagnetic Fields

In what follows, we introduce the complex wave number k ∈ C on the branch of the square root of k 2 in (6.4.8) with Re k ≥ 0 and Im k ≥ 0. Lossless and lossy media: A medium is generally referred to as a lossless medium, if the conductivity σ = 0 in the medium, and it is referred to as a lossy medium if σ = 0 in the medium. In the lossy medium, if the permittivity possesses an imaginary part, then the medium exhibits loss, and waves attenuate as they propagate through lossy media. Loss is incorporated into the permittivity via the conductivity σ which relates Jc to E by Ohm’s law: Jc = σE. The Maxwell–Ampere’s law in a lossy medium now reads: curl H = (σ − iω%)E = −iω %ef f E where %ef f := % (1 + iσ/ω%) is called the effective permittivity and the quantity σ/ω% is called the loss tangent of the medium and is a measure of how successfully the medium conducts (see, e.g. Staelin, Morgenthaler and Kong [397, 237]). Mathematically, the conductor is considered good if σ  ω% and poor if the reverse is true. 6.4.1 Plane waves In homogeneous, lossless and source free-media, plane waves E(x) = a exp{ik · x},

H(x) = b exp{ik · x}

(6.4.10)

are solutions of source–free Maxwell’s equations, where a and b are wave amplitudes, k is called the propagation vector. They are related constant vectors to be specified. For the plane wave solution, Maxwell’s equations become: k×E = k×H = k · μH = k · %E =

ω μH, −ω %E, 0, 0.

* where k = ω √μ% is the The propagation vector can be written as k = k k, ˆ is the magnitude of k, the wave number and is real in a lossless medium; k unit vector, which we may assume also real. The vector a is perpendicular ˆ and the vector b is related to a as to k, ; k * μ −1 * b= (k × a) = Z (k × a), Z = is the impedance . ωμ % ˆ and k ˆ form a right–hand orthonormal ˆ, b Geometrically, the unit vectors a system (see Kong [237]). When the medium is lossy, k 2 = (iωμ)(σ − iω%) = ω 2 μ%c is complex with %c = (% + iσ/ω). The wave number k is also complex. In this case the wave

6.5 Electromagnetic potentials

325

propagating in a certain direction also decays exponentially in that direction. √ The imaginary part of the wave number k = ω μ%c determines how fast the wave attenuates inside the medium. We define a penetration depth to be 1/Im(k). When the medium is very conductive so that %c ≈ iσ/ω, we may approximate the wave number k ≈ (1 + i)/δ. Here ; 2 1 δ= = Im(k) ωμσ which is called the skin depth of the conductor. More precisely, the distance the wave must travel in a lossy medium to reduce its value to e−1 = 0.368 of its original value is known as the penetration or skin depth δ ( see, e.g. Balanis [21], Barkeshl [24] ).

6.5 Electromagnetic potentials It is a common practice in the analysis of electromagnetic boundary value problems to use auxiliary vector potentials as aids in obtaining solutions for the electric and magnetics fields. Two most commonly used electromagnetic vector potentials are the vector potential A and vector potential Q. The vector potential A is used in the absence of the magnetic current density, to express the magnetic field H, which is also called magnetic Hertz potential, while the vector potential Q is used to express the electric field E in the absence of the electric current density. The latter is called the electric Hertz potential. To illustrate this idea, the magnetic flux density B = μH is divergence free and, hence, can be expressed as B = curl A = μH where A is called the magnetic vector potential. In the homogeneous isotropic medium from (6.4.1), Faraday’s law reads curl (E − iωA) = 0 which implies E = iωA − grad ϕ where the scalar function ϕ is called the electric scalar potential. Then from the Ampere–Maxwell equation (6.4.2) we see that curl H = which implies

1 curl curl A μ

326

6. Electromagnetic Fields

curl curl A = (σ − iωε)μE + μJ0 , = (σ − iωε)[iωμA − μ grad ϕ] + μJ0 , = k 2 A − (σ − iωε)μ grad ϕ + μJ0 . Hence, −(Δ + k 2 )A = −grad div A − (σ − iωε)μ grad ϕ + μJ0 ,  = −grad div A + (σ − iωε)μϕ + μJ0 . In order to determine A uniquely, we choose div A = −(σ − iωε)μϕ (see the Remark 6.5.1 below concerning gauge conditions). This implies that −(Δ + k 2 )A = μJ0 . Hence,

 Gk (x − y)J0 (y)dy,

A=μ Ω ik|x−y|

1 e where Gk (x − y) = 4π |x−y| is the fundamental solution of the Helmholtz equation. Consequently we see that  H(x) = curl Gk (x − y)J0 (y)dy. Ω

Now, for the electric field, we obtain E = iωA − grad ϕ 1 1 = iωA + grad div A μ (σ − iωε)  1 2 = (k I + grad div) Gk (x − y)J0 (y)dy. σ − iωε Ω

Moreover, from (6.4.3) we get div E =

 ε

which implies that −Δϕ + iω divA =

 . ε

Thus  , ε  − (Δ + k 2 )ϕ = . ε

−Δϕ − (iωμ)(σ − iωε)ϕ = or

6.5 Electromagnetic potentials



We have ϕ(x) =

327

 Gk (x − y) (y)dy , x ∈ Ω. ε

Ω

Note:

(x) −1 = divΓ J0 (x) ε (σ − iωε)

from (6.4.5). Similarly, for the electrical vector potential Q and the magnetic scalar potential ψ, the following Maxwell system in terms of a magnetic current density M0 and a magnetic charge density ζ has the form curl E = iωμH + M0

(6.5.1)

curl H = (σ − iωε)E, div (εE) = 0, div (μH) = ζ.

(6.5.2) (6.5.3) (6.5.4)

We obtain

1 1 curl Q , H = (σ − iωε)Q − grad ψ ε ε where Q satisfies the inhomogeneous Helmholtz equation E=

−(Δ + k 2 )Q = εM0 and ψ the equation −(Δψ + k 2 )ψ =

ζ . μ

By using the fundamental solution Gk we obtain  Gk (x − y)M0 (y)dy, E(x) = curl Ω

1 H(x) = {k 2 I + grad div} iωμ

(6.5.5)

 Gk (x − y)M0 (y)dy.

(6.5.6)

Ω

Remark 6.5.1: Gauge condition. A fundamental theorem due to Helmholtz states that a vector field can be uniquely determined only by specifying both its curl and its divergence. Hence from equation curl A = B to specify the curl of A exactly by the magnetic flux density B is not enough to determine A uniquely without specifying the divergence of A. Different choices for the divergence of A are referred to as choice of gauge. A physical problem may thus be described by a vector potential with different

328

6. Electromagnetic Fields

choice of gauge, which are said to be equivalent to within a gauge transformation, and each particular choice of ∇ · A is called a gauge condition. In the previous formulation, we have chosen divA = −(σ − iωε)μϕ which is called Lorentz’s gauge. A best–known gauge condition used in electromagnetics is extremely simple, e. g. ∇·A = 0, which is called the Coulomb gauge and with this choice, the boundary condition applicable to A can be further sharpened:  A · nds = 0. ∂Ω

In closing this section we now include for completeness the fundamental tensor of curl curl A − k 2 A = 0, namely: (I + k −2 grad x divx )Gk (x, y).

(6.5.7)

where Gk (x, y) = Gk (x, y)I.

6.6 Transmission and Boundary Conditions We specify the distinct constitutive parameters and field quantities in two different media Ω1 and Ω2 in R3 with an interface Σ; curl E1 = iωμ1 H1 , curl H1 = (−iωε1 + σ1 )E1 + J01 in Ω1 , curl E2 = iωμ2 H2 , curl H2 = (−iωε2 + σ2 )E2 + J02 in Ω2 .

(6.6.1)

Let n be the surface unit normal of Σ pointing from Ω1 to Ω2 . The transmission conditions can be derived by using the integral form of the Maxwell equations (see Dautray and Lions [95, Section 4]) which are as follows: (ε2 E2 − ε1 E1 ) · n = Σ , (μ2 H2 − μ1 H1 ) × n = −JΣ , (μ2 H2 − μ1 H1 ) · n = 0

(6.6.2)

(ε2 E2 − ε1 E1 ) × n = 0 on Σ. Here, Σ and JΣ denote the charge and current densities concentrated on Σ. At the interface between two lossless dielectric media, it is usually assumed that both JΣ and Σ are identically zero. On the other hand, when Σ is the interface between two lossy dielectrics, the surface current JΣ is identically zero because E is finite in both media. The surface charge density Σ may or may not be zero.

6.7 Boundary Value Problems

329

In the special case where Ω1 is a perfect conductor then E1 = 0 and H1 = 0 and (6.6.2) is reduced to ε2 E2 · n = Σ ,

μ2 H2 × n = −JΣ ;

μ2 H2 · n = 0,

ε2 E2 × n = 0,

(6.6.3)

where the surface charge Σ and surface current density JΣ in general are unknowns. The surface current and surface charge are not normally encountered except when one of the materials is a good conductor. At high frequencies there is a well known effect which confines current largely to surface regions. This is known as the skin effect MacCamy and Stephan [278, 279] in common conductors and is often sufficiently small to be neglected. The case Ω2 = Ω c = R3 \Ω where Ω is a perfect conductor with boundary Γ = ∂Ω = Σ leads to the boundary conditions n · H|Γ = 0 and n × E|Γ = 0.

(6.6.4)

6.7 Boundary Value Problems We classify the boundary value problems into two classes according to applications. We begin with scattering problems followed by a discussion of eddy current problems. 6.7.1 Scattering problems Let Ω ⊂ R3 be a bounded domain with a regular C 1+α boundary Γ = ∂Ω. • Interior problems Find the electromagnetic complex–valued vector fields E and H in Ω having a perfect conducting boundary Γ = ∂Ω such that (E, H) ∈ C 1 (Ω)6 ∩ C 0 (Ω)6 are satisfying the Maxwell equations (6.4.1)–(6.4.4) with J0 = 0 and  = 0 together with the boundary conditions (n × E)(x) = f (x) for x ∈ Γ

(6.7.1)

where f (x) = −n × Ein (x) for a given incident field Ein (x). • Exterior problems Find the electromagnetic fields E and H in Ω c := R3 \ Ω where ∂Ω = Γ is a perfectly conducting surface such that (E, H) ∈ C 1 (Ω c )6 ∩ C 0 (Ω c ∪ Γ )6 satisfy the Maxwell equations (6.4.1)–(6.4.4) in Ω c with J = 0 and  = 0, the boundary conditions (6.7.1) and the Silver–M¨ uller radiation conditions

330

6. Electromagnetic Fields

x × (curl E(x) + ik|x|E(x) = 0, |x|→∞  lim x × (curl H(x) + ik|x|H(x) = 0 lim



(6.7.2)

|x|→∞

uniformly as |x| → ∞. They can also be written as x k ×H+ E = o(|x|−1 ), |x| ωμ x ωμ ×E− H = o(|x|−1 ), |x| k uniformly as |x| → ∞. • Transmission problems Let Ω c := R3 \ Ω be an exterior domain. Then the transmission problem reads: Given the functions f and g on the interface Γ . Find the two pairs of vector fields (E, H) and (Ec , Hc ) satisfying the Maxwell equations Ω and Ω c , respectively, and the transmission conditions n × (E − Ec ) = f , n × (H − Hc ) = g on Γ.

(6.7.3)

In addition, the exterior field (Ec , Hc ) should satisfy the Silver–M¨ uller radiation conditions (6.7.2). 6.7.2 Eddy current problems Many applications in high electromagnetic energy problems are governed by the Maxwell equations in the low frequency region. In such a case the electrical displacement can be neglected in the Maxwell–Ampere’s law (see Ammari, Buffa and Nedelec [12]). This leads to the eddy current model. As stated by Faraday’s law, a time variation of the magnetic field generates an electric field. Therefore, a current density J = σE is induced in each of the conductors and is called the eddy current. The system of Maxwell equations obtained by disregarding the displacement current term then iωεc E is the eddy current model of the time–harmonic Maxwell equations, which can be justified by a dimensional analysis as in Rodriguez and Valli [366, p.7] if we let L = diam(Ω c ) be the reference length, we see that  ωε    |ωεc E| = |ω 2 εc μc L2 | |curl H| and |ωεc E| =   |σE|. σ Hence, under the assumptions |ω 2 εc μc L2 | # 1 and |σ −1 εc ω| # 1, the electric field ωεc E in the time–harmonic Ampere’s equations can be neglected in comparison with curl H and σE, respectively (see, e.g. Rodriguez and Valli [366, p.7]). (a) The eddy current problem in the bounded domain Ω ⊂ R3 Let Ω c with Ω c ⊂ Ω c be a conductor, and ΩI := Ω \ Ω c be a dielectric

6.7 Boundary Value Problems

331

medium. Then the pair (E, H)t satisfies the eddy current Maxwell equations: curl Hc − σc Ec = 0, (6.7.4) in Ω c curl Ec − iωμc Hc = 0 and

⎫ curl H = J0 , ⎪ ⎬ curl E − iωμH = 0 and ⎪ ⎭ div (εE) = 0

in ΩI

(6.7.5)

where the density J0 is given such that div J0 = 0 in ΩI .

(6.7.6)

In addition we need the transmission conditions n × μc Hc − n × μH = 0 and (6.7.7) n × Ec − n × E = 0 on the interface Σ := ∂Ω c ∩ ∂Ω, and also the boundary conditions n × E = 0 on ∂Ω.

Ωc σ, μ F J0 σ = 0, 0 , μ0

Fig. 6.1. The geometry of the bounded eddy current problem

(6.7.8)

332

6. Electromagnetic Fields

(b) The eddy current problem in R3 Here Ω = R3 and in addition to Equations (6.2.4)–(6.7.8) the electromagnetic field must satisfy the Silver–M¨ uller radiation conditions (6.7.2) instead of boundary conditions on ∂Ω.

6.8 Uniqueness In the following, we include some of the uniqueness results of the boundary value problems. For presenting the proofs of uniqueness we follow E. Meister in [293, Section 5.4]. 6.8.1 The cavity problem In this problem let Ω ⊂ R3 be a bounded Lipschitz domain and (E, H) ∈ C 1 (Ω)6 ∩C 0 (Ω)6 be a solution of the Maxwell equations (6.4.1)–(6.4.4) satisfying the boundary conditions (6.7.1). Since the system as well as the boundary conditions are linear, for uniqueness, it suffices to consider the homogeneous problem with J0 = 0 ,  = 0 in Ω, and f = 0 on Γ . Theorem 6.8.1. The homogeneous cavity problem (6.4.1)–(6.4.4), (6.7.1) with J0 = 0 and  = 0 in Ω and f = 0 on Γ has a) for σ > 0 in Ω only the trivial solution E = 0 and H = 0. b) If in addition σ ≡ 0 then the homogeneous cavity problem in Ω has only the trivial solution if and only if k 2 is not an eigenvalue of the problem ΔE + k 2 E = 0 and div(εE) = 0 in Ω, and n × γ0 E(x) = 0 for x ∈ Γ, Proof: (See Meister [293, Theorem 5.17]). From (6.7.1) with f = 0 we get  (n × E) · Hds = 0 Γ

implying

 n · (E × H)ds

0= Γ

and with Gauss’s theorem    0 = div(E × H)dx = (H · curl E)dx − (E · curl H)dx. Ω

Ω

Inserting (6.4.1) and (6.4.2) yields

Ω

6.8 Uniqueness



 H · (−iωμH)dx −

0= Ω

333

E · (−iωε + σ)Edx Ω





(ε|E|2 − μ|H|2 )dx −

= iω Ω

σ|E|2 dx. Ω

case a): Taking the real part of this relation gives  σ|E|2 dx = 0 Ω

which implies E = 0 in Ω since σ > 0. Then  −iω μ|H|2 dx = 0 Ω

with μ > 0 also H = 0 in Ω which completes the proof for σ > 0. If n × H|Γ = 0, vanishing of E and H in Ω follow with the same arguments. case b): Since σ = 0 there holds with constant ε > 0 , divE = 0 due to (6.4.3) because J0 = 0 and  = 0. Therefore ΔE + k 2 E = −curl curl E + k 2 E = 0 and the first Green’s identity yields     2 2 k |E| dx = Ecurl curl E dx = curl E·curl E dx− curl E·(n×E)ds. Ω

Ω

Ω

Since (n × E)|Γ = 0, 

∂Ω

 curl E · curl E dx =

Ω

k 2 |E|2 dx.

(6.8.1)

Ω

As will be seen, the inhomogeneous problem for E with boundary condition (6.7.1) and f ∈ C α (Γ ) defines a Fredholm mapping f → E ∈ C 1+α (Ω) ∩ C α (Ω) of index zero. Then the homogeneous problem above only admits the trivial solution E = 0 except for a countable set of real eigenvalues −1 −1 0 < kj2 = λ−1 j ≤ λj+1 , λj → ∞. The eigenvalue problem reads −ΔE = k 2 E and divE = 0 in Ω ∈ C 1,α , n × γ0 E = 0 on Γ = ∂Ω, and σ ≡ 0 in Ω, J0 ,  = 0 where k 2 = εμω 2 > 0.

(6.8.2) 

334

6. Electromagnetic Fields

Theorem 6.8.2. (see also Meister [293, Theorem 5.20] and for Ω a Lipschitz domain see Clayes and Hiptmair [71],[72]). Let Ω be simply connected with a C 1+α boundary Γ . Then (6.8.2) has nontrivial eigensolutions for a countable number of eigenvalues 0 < kj2 = −1 λ−1 j ≤ λj+1 , j ∈ N. The linear operator associated with (6.8.2) is a Fredholm operator of index zero. Proof: From (6.8.1) we conclude that k 2 is real and positiv. Moreover, any solution of (6.8.2) admits the representation  2 E(y) = k G0 (x − y)E(y)dy −



Ω





G0 (x − y)(γt curl E)dsy + Γ

(n · E)(y)grad y G0 (x − y)dsy Γ



(γt E)(y) × grad y G0 (x − y)dsy .

+

(6.8.3)

Γ

Since γ t E|Γ = n × E|Γ = n × γ0 E = 0 on Γ , the last Integral vanishes. Let ϕ := n · E and v := γ t curl E on Γ . Then on the boundary Γ we obtain from (6.8.3) for x ∈ Γ :   ϕ(x) = k 2 n(x) · G0 (x − y)E(y)dy − n(x) · G0 (x − y)v(y)dsy Ω

Γ

 − n(x) ·

ϕ(y) grad y G0 (x − y)dsy ,

(6.8.4)

Γ

v(x) = γ t curl E(x) = n(x) × curl E(x)|Γ  = k 2 n(x) × (curl x G0 (x − y)E(y)dy)|Γ Ω − n(x) × curl x

G0 (x − y)v(y)dsy Γ



− n(x) × (curl x  E(x) = k

ϕ(y) grad y G0 (x − y)dsy )|Γ , Γ



G0 (x − y)E(y)dy −

2 Ω

 −

G0 (x − y)v(y)dsy Γ

ϕ(y) grad y G0 (x − y)dsy for x ∈ Ω. Γ

(6.8.5)

(6.8.6)

 This is a system of three integral equations for the triple (ϕ, v, E) ∈ C α (Γ )× (C α (Γ ))2 × C 1+α (Ω) .

6.8 Uniqueness

335

In the first equation of the system, (6.8.4), use grad y G0 (x − y) = −grad x (x − y), then   −n(x) · (ϕ(y)grad y G0 (x − y)dsy = n(x) · grad x G0 (x − y)ϕ(y)dsy Γ

∂ = ∂ny



Γ

1 G0 (x − y)ϕ(y)dsy |Γ = I + K  ϕ on Γ 2

Γ

due to (1.2.5). Then (6.8.4) becomes 1 I − K  ϕ = k 2 n(x) · 2

 G0 (x − y)E(y)dy Ω

− n(x) ·

G0 (x − y)v(y)dsy . Γ

 The operator 12 I − K  is invertible on H −1/2 (Γ ) (see (5.6.46) and (5.6.48)) and on C α (Γ ) as well, and (6.8.4) is equivalent to  1 −1  2 ϕ = I − K k n(x) · G0 (x − y)E(y)dy 2 Ω  (6.8.7)  − n(x) · G0 (x − y)v(y)dsy . Γ

Similarly, for the last term in (6.8.5), we find  n(x) × curl x ϕ(y) grad y G0 (x − y)dsy  = −n(x) ×

Γ

curl x grad x G0 (x − y)ϕ(y)dsy = 0 Γ

since tangential derivatives commute with surface integration. Then (6.8.5) takes the form  v(x) = k 2 n(x) × curl x G0 (x − y)E(y)dy|Γ (6.8.8) Ω

 − n(x) × curl x

G0 (x − y)v(y)dsy for x ∈ Γ. Γ

Using the local orthonormal vectors (e1 , e2 , n) at x ∈ Γ , the differential operator in (6.8.8) becomes with

336

6. Electromagnetic Fields

curl (w1 e1 + w2 e2 ) = −∂n w2 e1 + ∂n w1 e2 + (∂1 w2 − ∂2 w1 )n, n × curl w = −∂n w1 e1 − ∂n w2 e2 ,  v(x) = k 2 n(x) × curl x

G0 (x − y)E(y)dy|Γ + Ω



∂ ∂nx



 G0 (x−y)v(y)dsy Γ

Γ

1 = k n(x) × curl x G0 (x − y)E(y)dy|Γ + v(x) + K  v(x) , 2 Ω  1 −1   = k2 I − K  n × curl x G0 (x − y)E(y)dy|Γ for x ∈ Γ 2 2

Ω

(6.8.9)  2 and v ∈ (C α )2 ⊂ H −1/2 (Γ ) . Therefore the eigenvalue problem (6.8.2) is equivalent to the eigenvalue problem:  1 −1   n(x) × curl x G0 (x − y)E(y)dy|Γ , (6.8.10) v(x) = k 2 I − K  2 Ω  1  −1 ϕ+ I −K n(x) · G0 (x − y)v(y)dsy 2 Γ     1 −1 2  I −K n(x) · G0 (x − y)E(y)dy =k 2 Ω

 

 G0 (x − y)v(y)dsy +

E(y) + Γ

ϕ(y)grad y G0 (x − y)dsy Γ

G0 (x − y)E(y)dy in Ω.

= k2 Ω

This is a triangular the Riesz–Schauder type  system of integral operators of 2  2 α α 1+α where all the operators in the space X = C (Γ ) × C (Γ ) × C (Ω) on the right hand side multiplied by k 2 are defining a compact mapping in X. Then there exists a countable number of eigenvalues of finite multiplicity λ−1 j = kj → ∞ for j → ∞ (see Zeidler [443]) and due to (6.8.1) they are real and strictly positive.  Lemma 6.8.3. Under the conditions of Theorem 6.8.2, for 0 < k 2 < λ−1 1 the problem (6.8.2) has only the trivial solution E = 0 in Ω. As a consequence, for any k 2 = kj2 , (6.8.2) has only the trivial solution E = 0 in Ω.

6.8 Uniqueness

337

Proof: Since (6.8.2) with k = 0 is equivalent to the triangular system (6.8.10) with G0 which is invertible, the analyticity with respect to k 2 assures invertibility up to k 2 < λ−1 1 and Fredholm’s theorem implies invertibility for 0 < k12 not an eigenvalue.  Corollary 6.8.4. Theorem 6.8.2 and Lemma 6.8.3 are also true for Ω a 2  3  bounded Lipschitz domain and X := H 1/2 (Γ ) × H 1/2 (Γ ) × H 1 (Ω) since the triangular operator matrix on the left of (6.8.10) in X is continuous and  3  continuously invertible, and because of G0 (x − y)E(y)dy ∈ H 2 (Ω) , the Ω

operator vector on the right in (6.8.10) is mapping X continuously into Y := 2  3  1−ε (Ω) × H 1−ε (Γ ) × H 1−ε (Ω) with 0 < ε < 12 and the imbedding H Y → X is dense and compact, the propositions of Theorem 6.8.2 and Lemma 6.8.3 for the Riesz–Schauder system (6.8.10) again are valid. 6.8.2 Exterior problems Here the solution (E, H) ∈ C 1 (Ω c )6 of the Maxwell equations (6.4.1)– (6.4.4) are considered in the exterior domain Ω c satisfying one of the boundary conditions (6.7.1) together with the Silver–M¨ uller radiation conditions (6.7.2) as |x| → ∞ . Again, for the uniqueness it is sufficient to consider the homogeneous problem. We consider two cases with σ = 0 or σ > 0. The case for σ = 0 is particularly relevant in scattering problems for perfect dielectrics. For the case σ = 0 we shall need the following Theorem by Rellich ([293, Theorem 4.12]) for the uniqueness proof. Theorem 6.8.5. Let Φ in Ω c be a regular solution of the homogeneous Helmholtz equation ΔΦ + k 2 Φ = 0 in Ω c with k > 0 and satisfying the decay condition  |Φ(Ry)|2 do1 = o(R−2 ) for R → ∞ (6.8.11) |y|=1

then Φ = 0 in Ω c . Theorem 6.8.6. The homogeneous exterior problem (6.4.1)–(6.4.4), (6.7.1) with J0 = 0 and  = 0 in Ω c and f = 0 on Γ satisfying the Silver M¨ uller radiation conditions (6.7.2) for σ > 0 on Ω c and also for σ = 0, only has the trivial solution E = 0 and H = 0. Proof: (See Meister [293, Theorem 5.18].) Let R > 0 be a parameter chosen c := Ω c ∩ BR . so large that Ω ⊂ BR := {x ∈ R3 with |x| < R} and let ΩR Then

338

6. Electromagnetic Fields





1 x · (E × H)doR − R

div(E × H)dx = c ΩR

|x|=R



|x|=R

 H · (n × E)doΓ

Γ



H·(

=

n · (E × H)doΓ Γ

1 H · ( x × E)doR − R

=



1 x × E)doR R

|x|=R

since n × E|Γ = 0. Now we consider also the domain exterior to BR where    div(E × H)dx = H · curl Edx − E · curl Hdx |x|≥R

|x|≥R

|x|≥R





(μ|H|2 − ε|E|2 )dx −

= iω |x|≥R

σ|E|2 dx |x|≥R

due to (6.4.1), (6.4.2) with J0 = 0. Employing the Gauss theorem gives with the Silver–M¨ uller radiation conditions (6.7.2),   1 div(E × H)dx = H · ( x × E)doR R |x|≥R

|x|=R

= k(ωε + iσ)

−1

 |H|2 doR + o(1)

|x|=R

=







k1 ωε + k2 σ + i(k2 ωε − k1 σ) |ωε + iσ|



−2

|H|2 doR + o(1),

|x|=R

k = k1 + ik2 , for R → ∞. Now insert the surface integral on |x| = R into the two previous equations to obtain for the real parts   −2 2 |H| doR + o(1) = − σ|E|2 dx ≤ 0, 0 ≤ (k1 ωε + k2 σ)|ωε + iσ| c ΩR

|x|=R

(6.8.12) and for the imaginary parts (k2 ωε − k1 σ)|ωε + iσ|−2



 |H|2 doR + o(1) =

|x|=R

Hence, the relation (6.8.11) implies  lim σ|E|2 dx = 0 R→∞ c ΩR

{μ|H|2 − ε|E|2 }dx. c ΩR

6.8 Uniqueness

339

which yields E = 0 in Ω c for σ > 0. Then (6.4.1) gives H = 0 in Ω c as proposed. If σ = 0 then (6.4.8) gives k 2 > 0 and Im k = k2 = 0. Hence, (6.8.12) now implies with y := R1 x   2 |H| doR = lim |H(Ry)|2 do1 R2 = 0. (6.8.13) lim R→∞ |x|=R

R→∞ |y|=1

Since J0 = 0, H satisfies the homogeneous Helmholtz equation (6.4.7) in Ω c and therefore is analytic up to x = ∞ and must hence satisfy (6.8.12) which implies (6.8.1). The Rellich Theorem 6.8.5 then implies H = 0 in Ω c and E = 0 follows from (6.4.2).  6.8.3 The transmission problem Theorem 6.8.7. (See Meister [293, Theorem 5.19]) and Koshlyakov, Smirnov and Gliner [238, p.91] Let Ω be a bounded simply connected Lipschitz domain and Ω c := R3 \ Ω. (E, H) and (Ec , Hc ) are solutions of the homogeneous Maxwell equations (6.4.1)–(6.4.4) with J0 = 0 in Ω and Ω c , respectively, satisfying the transmission conditions (6.7.3) with f = 0 and g = 0 on the transmission surface Γ = ∂Ω. In the exterior domain Ω c the field (Ec , Hc ) satisfies the Silver–M¨ uller radiation conditions (6.7.2) with n the exterior normal field on ∂Ω. a) If σ ≥ 0 in Ω and σ c > 0 in Ω c or b) if σ ≥ 0 in Ω and σ c = 0 in Ω c then this homogeneous transmission problem only has the trivial solution E = 0 , H = 0 , Ec = 0 , Hc = 0. c , respectively, gives Proof: The Gauss Theorem in Ω and in ΩR    div(E × H)dsx = n · (E × H)dsΓ = H · (n × E)doΓ ,

Ω

Γ c



div(E × H )dx = − c

c ΩR

Γ c



H · (n × E )dsΓ + Γ

c

1 c x · (Ec × H )dsR , R

|x|=R

and with (6.4.1) and (6.4.2), div (E × H) = iωεμ|H|2 − (iωε + σ)|E|2 we obtain

340

6. Electromagnetic Fields



 iωμ|H| dx − 2

Ω





iωμc |Hc |2 dx − c ΩR

 H · (n × E)dsΓ ,

2

(iωε + σ)|E| dx = Ω

Γ



|Ec |2 (iωεc + σ c )dx = − c ΩR

Hc · (n × Ec )dsΓ Γ



x · (Ec × Hc )dsR . R

+ |x|=R

Multiplying the first equation with μμε c εc , adding to the second equation (6.7.3), and using the homogeneous transmission conditions (6.7.3) yields      1 c x·(Ec × H )dsR = iω μc |Hc |2 dx− εc |Ec |2 dx − σ c |Ec |2 dx R |x|=R

 + iω

c ΩR

c ΩR



(μ|H|2 dx − ε|E|2 ) dx −

με μ c εc

Ω

c ΩR

 σ|E|2 dx

(6.8.14)

Ω

For the surface integral over the sphere |x| = R, we employ the SilverM¨ uller radiation conditions (6.7.2) to obtain   1 c x · (Ec × H )doR = k c (ωεc + iσ c )−1 |Hc |2 do + o(1) R |x|=R



=

|x|=R



 (k1c ωεc + k2c σ c ) + i(k2c ωεc − k1c σ c ) |ωεc + iσ c |−2 |Hc |2 do + o(1)

|x|=R

(6.8.15) for R → ∞. Here k c = k1c + ik2c . Taking the real parts of (6.8.14) and (6.8.15) results in   με 2 0≥− c c σ|E| dx − σ c |Ec |2 dx μ ε (k1c ωεc

=

c ΩR

Ω



+

k2c σ c )|ωεc

(6.8.16) c −2

+ iσ |

|H | do + o(1) c 2

|x|=R

and R → ∞ yields με μ c εc



 σ c |Ec |2 dx = 0.

σ|E|2 dx + Ω

Ωc

Case a): In case of σ c > 0 this implies Ec = 0 in Ω c

6.9 Representation Formulae

341

and Equation (6.4.1) gives Hc = 0 in Ω c . Then n × E|Γ = 0 = n × Ec |Γ and n × H|Γ = 0 = n × Hc |Γ and Theorem 6.8.1 in Subsection 6.8.1 implies also E = 0 and H = 0 in Ω. Case b): In case σ c = 0 we obtain from (6.8.16) now  lim |Hc |2 do = 0. (6.8.17) R→∞ |x|=R

Hc satisfies the homogeneous Helmholtz equation (6.4.7) in Ω c and is analytic around ∞. Then (6.8.17) implies Hc (x) = o(|x|−2 ) for |x| → ∞. So, the Rellich Theorem 6.8.5 implies Hc = 0 in Ω c . Now, Equation (6.4.2) yields Ec = 0 in Ω c implying n × Ec |Γ = 0 as well as n × E|Γ = 0 , n × H|Γ = 0, n × Hc |Γ = 0. The transmission conditions and Theorem 6.8.1 gives E = 0 and H = 0 in Ω. This completes the proof of Theorem 6.8.7. 

6.9 Representation Formulae To be self contained we begin with the Stratton–Chu formulae [409], Stratton [408, p.466] and Schechter [377]. From the Maxwell equations (6.4.1)–(6.4.4) in the bounded domain Ω one obtains curl curl E = k 2 E + iωμJ0 , curl curl H = k 2 H + curl J0

(6.9.1)

with k 2 given in (6.4.8). Green’s second formula for a pair (P, Q) of continuously diffentiable vector fields reads as   (Q · curl curl P − P · curl curl Q)dy = {P × curl Q − Q × curl P} · n ds. Ω

Γ

(6.9.2) Let Q = Gk a with a constant vector a, and where Gk (x − y) =

1 eik|x−y| 4π |x − y|

is the fundamental solution of the Helmholtz equation (see (2.1.4) and (6.2.4)). We see that for x = y, curl Q = grad Gk × a, curl curl Q = ak 2 Gk + grad (a · grad Gk ). Letting Ωδ := Ω \ {y ∈ Ω ∧ |y − x| ≤ δ} , δ > 0

(6.9.3)

342

6. Electromagnetic Fields

and substituting Q and P = E into Green’s formula (6.9.2) in Ωδ , we obtain   Gk iωμJ0 dy · a − E · grad (a · grad Gk )dy Ωδ



Ωδ



 iωμGk (n × H) + (n × E) × grad Gk dsy · a.

= ∂Ωδ

With  E · grad (a · grad Gk )dy Ωδ



 div{(E(a · grad Gk )}dy −

= Ωδ





(divE)grad Gk dy · a Ωδ

(n · E)grad Gk dsy · a −

= Ωδ

(divE)grad Gk dy · a ∂Ωδ

and with (6.4.3) we finally get   {iωμGk J0 + grad Gk }dy · a = ε Ωδ  {iωμGk (n × H) + (n · E)grad Gk + (n × E) × grad Gk }dsy · a ∂Ωδ

for arbitrary a ∈ R3 . Now, using tacitly the identity E = (n × E) × n + (E · n)n, i.e. E = πt E + (E · n)n (see Section 4.5), we notice that if grad y Gk (x − y) is replaced by its normal component (grad y Gk (x − y) · n)n, then the last two terms in the integrand on the right–hand side of the above integral can (x−y) be simplified as E(y) ∂Gk∂n . Hence, a simple computation shows that for y δ → 0 one has the Stratton–Chu representation formula in the interior domain for the electric field,  1 E(x) = [Gk (x − y)iωμJ0 (y) + (y)grad y Gk (x − y)]dy (6.9.4) ε  Ω  − {iωμGk (x − y) n(y) × H(y) + (n · E)grad y Gk (x − y) Γ  + n(y) × E(y) × grad y Gk (x − y)}dsy for any x ∈ Ω. In a quite analogous way, one derives the corresponding formula of the Stratton–Chu representation formula for the magnetic field,

6.9 Representation Formulae

343

 J0 (y) × grad y Gk (x − y)dy

H(x) = Ω

 − Γ

(6.9.5)

 (−iωε + σ)Gk (x − y) n(y) × E(y) + (n · H)grad y Gk (x − y)   + n(y) × H(y) × grad y Gk (x − y) dsy for any x ∈ Ω, 

where we have employed the formula   Gk (x − y)curl J0 (y)dy = J0 (y) × grad y Gk (x − y)dy Ω

Ω



+



n × J0 (y) Gk (x − y)dsy

Γ

by using the identity 

 n × F(y)dsy

curl F(y)dy = Ω

Γ

(see Stratton [408, p.604]) . In the exterior domain Ω c = R3 \ Ω one obtains  1 E(x) = [Gk (x − y)iωμJ0 (y) + (y)grad y Gk (x − y)]dy (6.9.6) ε c Ω    + iωμGk (x − y) n(y) × H(y) + (n · E)grad y Gk (x − y) Γ

and

  + n(y) × E(y) × grad y Gk (x − y) dsy for x ∈ Ω c ,

 J0 (x) × grad y Gk (x − y)dy

H(x) =  + Γ



(6.9.7)

Ωc

 (−iωε + σ)Gk (x − y) n(y) × E(y) + (n · H)grad y Gk (x − y)   + n(y) × H(y) × grad y Gk (x − y) dsy for any x ∈ Ω c .

Here we used the Silver–M¨ uller radiation conditions (6.7.2) and assume that J0 has compact support, supp J0  Ω c . We remark that in the representation formulae (6.9.4), (6.9.5), (6.9.6) and (6.9.7) n × H and n × E are the Cauchy data for E(x) and H(x) while the terms n · E and n · H can be replaced by what are called the surface

344

6. Electromagnetic Fields

divergence of n × H and n × E, respectively. From vectorial calculus, we see that div(n × E) = E · curl n − n · curl E = −n · curl E, since the field of normals is a gradient, which implies curl n = 0. Then from (6.4.1), we obtain −div(n × E) . n·H= iωμ Similarly, div(n × H) = −n · curl H and from (6.4.2), we obtain n · E = (σ − iωε)−1 {n · curl H − n · J0 } = −(σ − iωε)−1 {div(n × H) + n · J0 )}. In fact, the surface divergence of a tangent vector field v is defined by its  as divΓ v = div v |Γ . In this sense, associated lifting v div(n × E)|Γ = divΓ (n × E) and div(n × H)|Γ = divΓ (n × H).

6.10 Boundary Integral Equations for Electromagnetic fields We begin with the Stratton–Chu representation formulae Stratton and Chu [409, (19),(20) Chap.8] (6.9.4), (6.9.5) for the solution of the Maxwell equations (6.4.1)–(6.4.5) within a given domain Ω. If x ∈ Ω tends to the boundary ∂Ω = Γ , then from standard jump relations of potential theory (see (6.10.8)– (6.10.11)), we arrive at   1 n(x) × E(x) = TE (n × H) (x) + n(x) × E(x) − R(n × E) (x) 2 (6.10.1) + n(x) × Ep (x) for x ∈ Γ . where Ep (x) = (σ − iωε)  +



−1



grad y Gk (x − y) n(y) · J0 (y)dsy



Γ

 1 Gk (x − y)iωμJ0 (y) + (y)grad y Gk (x − y) dy ε

Ω

is known for given J0 (y), and the boundary integral operators are defined by

6.10 Boundary Integral Equations for Electromagnetic fields

 TE (n × H)(x) = −iωμn(x) × + (−iωε + σ)−1 n(x) ×



 Gk (x − y) n(y) × H(y) dsy

345

(6.10.2)

Γ

 grad y Gk (x − y) divΓ (n(y) × H(y) dsy

Γ

  = − iωμn × Vk (n × H) + (−iωε + σ)−1 n × grad x Vk (divΓ (n × H) and the singular Calderon–Zygmund integral operator R is given by   { n(y) × E(y) × grad y Gk (x − y)}dsy R(n × E)(x) := p.v. n(x) ×  = n(x) × curl x

Γ \{x}

Gk (x − y)(n × E)dsy for x ∈ Γ.

(6.10.3)

Γ \{x}

Similarly we have from (6.9.5) for Ω  x → Γ ,  1 n(x) × H(x) = TH (n × E) (x) + {( I − R)(n × H)}(x) + n(x) × Hp (x), 2 (6.10.4) for x ∈ Γ where  Hp (x) := J0 (y) × grad y Gk (x − y)dy Ω

is known for given J0 (y). Moreover,  TH (n × E)(x) := −(−iωε + σ)n(x) × 1 n(x) × + iωμ



 Gk (x − y) n(y) × E(y) dsy

Γ

 grad y Gk (x − y)divΓ n(y) × E(y) dsy

(6.10.5)

Γ

   1 = − (−iωε + σ)n × Vk (n × E) + n × grad x Vk divΓ (n × E) . iωμ Now let us consider the Maxwell equations (6.4.1)–(6.4.5) in the exterior domain Ω c = R3 \ Ω for solutions (E, H) satisfying the Silver– M¨ uller radiation conditions (6.7.2) where J0 has compact support as well as . For Ω c  x → Γ , with the standard jump relations we obtain from (6.9.6) the equation on Γ : n(x) × E(x) (6.10.6)  1 = − TE (n × H) (x) + n(x) × E(x) + R(n × E)(x) + n(x) × Ecp (x) 2 for x ∈ Γ where

346

6. Electromagnetic Fields

Ecp (x)

−1



:= −(σ − iωε)  + Ωc



grad y Gk (x − y) n(y) · J0 (y)dsy

Γ

 1 Gk (x − y)iωμJ0 (y) + (y)grad y Gk (x − y) dy ε



Similarly, from (6.9.7) we find 1 n(x) × H(x) 2 + R(n × H)(x) + n(x) × Hcp (x)

n(x) × H(x) = −TH (n × E) +

for x ∈ Γ where

(6.10.7)

 J0 (y) × grad y Gk (x − y)dy.

Hcp (x) := Ωc

We now include derivations of three typical jump relations of the single– layer potential in electromagnetics (see, e.g. Hsiao and Kleinman [194], Nedelec [331, p.240] ). We follow the standard convention and assume here that Γ is a Lyapounov surface. The subscripts plus and minus denote limits from Ω c = R3 \ Ω and Ω, respectively. In the following derivations for simplicity we denote by J(y) := n × H(y), which can be replaced by M(y) = n × E(y) as well.  lim n(x) × curl x Gk (x − y)J(y)dsy ± x→Γ

Γ

1 = ± J(x) + n(x) × curl x 2

 Gk (x − y)J(y)dsy . Γ

(6.10.8) The derivation is based on the vector identity I (1) in the Appendix B:   n(x) × curl x Gk (x − y)J(y)dsy = n(x) × grad x Gk (x − y) × J(y)dsy  = Γ



Γ

Γ

 (n(x)·J(y)) (grad x Gk (x−y))dsy − (n(x)·grad x Gk (x−y)) J(y)dsy 



Γ

(n(x) − n(y)) · J(y) (grad x Gk (x − y))dsy

= Γ



−



n(x) · grad x Gk (x − y) J(y)dsy ,

Γ

where the limit of the last term on the right side is equal to

6.10 Boundary Integral Equations for Electromagnetic fields



1 ± J(x) − 2

Γ \{x}

∂ Gk (x − y)J(y)dsy , ∂nx

347

as x → Γ ± .

The second jump relation reads  Gk (x − y)J(y)dsy lim n(x) · curl x x→Γ ±

Γ



=



(n(x) − n(y)) × grad x Gk (x − y)) · J(y)dsy

Γ



Gk (x − y) curlΓ J(y)dsy

+

(6.10.9)

Γ

The derivation uses the vector identities I(3) , III(5) and V(6) in the Appendix B:   n(x) · curl x Gk (x − y)J(y)dsy = n(x) · grad x Gk (x − y) × J(y)dsy Γ



= Γ



= Γ

 +



Γ

J(y) · n(x) × grad x Gk (x − y) dsy  J(y) · (n(x) − n(y)) × grad x Gk (x − y) dsy  J(y) · n(y) × grad x Gk (x − y) dsy ,

Γ

where the last term on the right hand side is equal to    − J(y) · n(y) × grad Γ,y Gk (x − y) dsy Γ



 J(y) · curl Γ,y Gk (x − y)dsy =

= Γ

Gk (x − y) curlΓ J(y)dsy Γ

due to III (5) and VI (6) in the Appendix B. This completes the derivation of (6.10.9). The third jump relation is

348

6. Electromagnetic Fields

 lim n(x) × grad x

Gk (x − y)divΓ J(y)dsy

x→Γ ±

Γ



= n(x) × grad x

Gk (x − y)divΓ J(y)dsy ,

(6.10.10)

Γ



(n(x) − n(y)) × (grad x Gk (x − y))divΓ J(y)dsy

= Γ

 −

Gk (x − y) (curl Γ,y divΓ J(y)) dsy

(6.10.11)

Γ

The derivation makes use of formulae III(1), IV(2) in the Appendix B:  n(x) × grad x Gk (x − y)divΓ J(y)dsy Γ

 = n(x) ×

grad x Gk (x − y)divΓ J(y)dsy Γ



= n(x) ×



grad Γ,x Gk (x − y) + nx

∂ Gk (x − y) divΓ J(y)dsy , ∂nx

Γ

which has the limit (6.10.10): 1 ∓ n(x) × (n(x)divΓ J(x)) + n(x) × grad x 2

 Gk (x − y)divΓ J(y)dsy Γ



= n(x) × grad x

Gk (x − y)divΓ J(y)dsy , Γ

since n(x) × n(x) = 0. On the other hand, the form of (6.10.11) can be derived as follows:

6.10 Boundary Integral Equations for Electromagnetic fields

349

 n(x) × grad x

Gk (x − y)divΓ J(y)dsy Γ

 (n(x) − n(y)) × grad x Gk (x − y)divΓ J(y)dsy

= Γ

 n(y) × grad x Gk (x − y)divΓ J(y)dsy

+ Γ



(n(x) − n(y)) × grad x Gk (x − y)divΓ J(y)dsy

= Γ

 −

 =

(n(y) × grad y Gk (x − y))divΓ J(y)dsy Γ



n(x) − n(y) × grad x Gk (x − y)divΓ J(y)dsy

Γ

 +

 curl Γ,y gk (x − y)divΓ J(y) dsy

Γ



−  =



Gk (x − y)curl Γ,y divΓ J(y)dsy Γ

(n(x) − n(y) × grad x Gk (x − y)divΓ J(y)dsy

Γ

 −

 Gk (x − y)curl Γ,y divΓ J(y) dsy

Γ

due to V I (2) in the Appendix B. As will be seen, these three formulae will be useful concerning boundary integral equations in electromagnetics. In particular, (6.10.8)–(6.10.10) have been employed repeatedly in the derivation of the equations (6.10.1)–(6.10.7). 6.10.1 The Calderon projector and the capacity operators From the homogeneous boundary integral equations (6.10.1) and (6.10.4) on Γ where J0 = 0 one obtains for all solutions of the homogeneous Maxwell system (6.4.1)–(6.4.5) the identity   1   n×E n×E I − R, TE 2 = 1 n×H n×H TH , 2I − R where the boundary integral operator 1 I − R, CΩ = 2 TH ,



TE 1 I 2 −R

(6.10.12)

350

6. Electromagnetic Fields

is known as the Calderon projector from the space C α (Γ )3 × C α (Γ )3 into C α (Γ )3 × C α (Γ )3 with the porperty C Ω ◦ C Ω = CΩ

(6.10.13)

which implies the operator relations: TE ◦ TH = 14 I − R2 , R ◦ TE = −TE ◦ R, TH ◦ R = −R ◦ TH , TH ◦ TE = 14 I − R2 .

(6.10.14)

If we define the operator AΩ by CΩ =

1 I + AΩ , AΩ = 2 

hence



n×E n×H

AΩ

=



 −R TE , TH −R

(6.10.15)



 n×E . n×H

1 2

for (E, H) , the solutions of the homogeneous Maxwell equations in Ω. With   TE −R (6.10.16) BΩ = −R TH we arrive with the identities (6.10.13) at the equations 4A2Ω = I

(6.10.17)

and  4AΩ ◦ BΩ = This implies

 BΩ

0 I

 I BΩ = AΩ , 0 1 2 = A2Ω = I BΩ 4 I 0



 0 I . I 0

(6.10.18)



=

0 I

and the celebrated relation       n×H n×E n×E = 2AΩ = 2BΩ n×E n×H n×H

(6.10.19)

(6.10.20)

for all solutions (E, H) of the homogeneous Maxwell equations in Ω. The operator   TE , −R 2BΩ = 2 (6.10.21) −R, TH

6.10 Boundary Integral Equations for Electromagnetic fields

351

is called the capacity operator since it maps the (Neumann, Dirichlet) data to the (Dirichlet, Neumann) data of any solution pair (E, H) (see also Cessenat [63, p.65], Nedelec [331, p.116]). Now let us consider the Maxwell equations (6.4.1)–(6.4.5) in the exterior domain Ω c −R3 \Ω for solutions (E, H) satisfying the Silver–M¨ uller radiation conditions (6.7.2). Then for J0 = 0 and  = 0, we find the corresponding Calderon projector    1    n × Es I+R −TE n × Es n × Es 2 c CΩ := = (6.10.22) 1 n × Hs n × Hs n × Hs −TH 2 (I + R for all solutions (E, H) of the homogeneous Maxwell equations (6.4.1)– (6.4.5) which satisfy the Silver–M¨ uller radiation conditions (6.7.2). Note, that CΩ c + CΩ = I. If we now define AΩ c by CΩ c =

(6.10.23)

1 I + AΩ c 2

then AΩ c = −AΩ and BΩ c = −BΩ

(6.10.24)

is the Calderon or capacity operator for the exterior domain Ω c . Similar to the boundary value problems in the previous chapters, the Calderon projector provides us two boundary integral equations for unknowns either n × E or n × H, on the boundary. This is called the direct method for defining boundary integral equations for solving boundary value problems of the Maxwell’s system (6.4.1)–(6.4.4) in Ω or Ω c . 6.10.2 Weak solutions for a fundamental problem One of the difficulties in treating the solutions of boundary value problems in electromagnetics by using boundary integral equations is that the imbedding of the solution space H(curl , Ω) ⊂ L2 (Ω) is not compact in contrast to H1 (Ω) solutions of acoustic problems. To circumvent this difficulty one can employ the Helmholtz (or Hodge) decomposition of the solution of the Maxwell equations in Ω as well as the solution space of the corresponding integral equation on Γ . In this section we confine the analysis to simple model problems. 6.10.2.1 Interior Dirichlet problem in Ω. Let (E, H) ∈ H(curl , Ω) be the solution of the Dirichlet problem for the homogeneous Maxwell equations (6.4.1)–(6.4.4), namely

352

6. Electromagnetic Fields

curl E(x) = iωμH(x) curl H(x) = (−iωε + σ)E(x) n(x) × E(x)|Γ = m(x)

in Ω, on Γ,

(6.10.25)

where m on Γ is given. This problem is also called the cavity problem. For its weak solution it suffices to consider only the electric field E ∈ H(curl , Ω) of the equation (6.10.26) curl curl E − k 2 E = 0 where k 2 is given by (6.4.8). Then, according to Green’s first identity (App B(8)) the weak formulation of (6.10.26) reads: Given m ∈ H−1/2 (divΓ , Γ ) find E ∈ H(curl , Ω) such that m − n × E|Γ = 0 and    t curl E) · (curl Et ) − k 2 E · Et dy A(E, E ) := Ω

(6.10.27) (curl E) · (n × Et )dsy = 0

= Γ

for all test functions Et ∈ H0 (curl , Ω).  ∈ H(curl , Ω) be an extension of the boundary values First let E  Then E0 ∈ H0 (curl , Ω) satisfies  Γ into Ω and set E0 := E − E. m = n × E| the variational equation  Et ) for all Et ∈ H0 (curl , Ω) A(E0 , Et ) = −A(E,

(6.10.28)

 Et ) is a given linear functional operating on Et . To consider the where A(E, existence of a solution (6.10.28), we have to use the Helmholtz decomposition (see e.g. Ammari and Nedelec [9], Cessenat [63, p.52], Dautray and Lions [96]): H(curl , Ω) = M(Ω) ⊕ P(Ω) (6.10.29) where



M(Ω) := {u ∈ H0 (curl , Ω) with

u · grad pdx = 0 ∀ p ∈ H01 (Ω)}, Ω

P(Ω) := {v = grad p with p ∈ H01 (Ω)}. M(Ω) and P(Ω) are orthogonal with respect to the scalar product   (6.10.30) ((u, v)) := curl u · curl v dy + u · v dy, Ω

Ω

since for u ∈ M(Ω) and v ∈ P(Ω) there holds

6.10 Boundary Integral Equations for Electromagnetic fields



 (curl u) · (curl grad p)dy +

((u, v)) :=

353

Ω

u · grad pdy = 0. Ω

Moreover, p|Γ = 0 implies the tangential derivative of p, i.e. n × grad p|Γ = 0 as well. The bilinear equation (6.10.27) also reads ˜ Et ) + A(E, ˜ Et ) = 0 A(E, Et ) = A(E − E, ˜ ∈ H(curl , Ω) satisfies n × E ˜ = m on Γ , whereas E − E ˜ = E0 is the where E solution of the cavity problem and satisfies the boundary condition n×E0 |Γ = 0 and is determined by the system of equations for E0 = u + grad p with u ∈ M(Ω) and for p ∈ H01 (Ω). According to the Helmholtz decomposition (6.10.29) we test (6.10.28) first with ut ∈ M(Ω) and obtain for u ∈ M(Ω) the variational equation    ˜ ut ). {(curl u · curl ut dy} + u · ut dy − (1 + k 2 ) u · ut dy = −A(E, Ω

Ω

Ω

(6.10.31) for all ut ∈ M(Ω). Testing with the second component of the decomposition, with grad pt we obtain  1 ˜ grad pt )dy for all grad pt ∈ P(Ω). grad p · grad pt dy = 2 A(E, k Ω

(6.10.32) Note that div u = 0 in Ω

(6.10.33)

for u ∈ M(Ω) since   0 = u · grad pdy = − pdiv udy for all p ∈ H01 (Ω) Ω

Ω

due to p|Γ = 0. Theorem 6.10.1. Let Ω be a bounded Lipschitz domain in R3 . Then M(Ω) ⊂ H1 (Ω) and, hence, M(Ω) is compactly R3 imbedded into L2 (Ω). As a consequence we have G˚ arding’s inequality Re A(u + grad p, u + grad p) ≥ |||u + grad p|||2Ω − (1 + |k 2 |)||u + grad p||2L2 (Ω) (6.10.34) where 1 (6.10.35) |||u + grad p|||Ω := ((u + grad p, u + grad p)) 2 is the Hilbert space norm associated with the inner product((·, ·)) in (6.10.30).

354

6. Electromagnetic Fields

Proof of Theorem 6.10.1: For the proof we follow Girauld and Raviart  3 [146, Theorem 3.2]. Since u ∈ L2 (Ω) satisfying (6.10.33) we can extend  3 u to all of R3 with uext ∈ L2 (R3 ) having compact support and satisfying divuext = 0 in R3 . The Fourier transformed components of uext ,  u *j (ξ) := (2π)−3/2 e−ix·ξ uext,j (x)dx R3

are holomorphic in R3 and div uext = 0 in IR3 yields 3 

ξj u *j = 0.

(6.10.36)

j=1

3  * ∈ ( L2 (R3 ) such that curl ϕ = uext , i.e. Now, we must find a function ϕ *3 − ξ3 ϕ *2 ) , u *2 = 2πi(ξ3 ϕ *1 − ξ1 ϕ *3 ) , u *3 = 2πi(ξ1 ϕ *2 − ξ2 ϕ *1 ). u *1 = 2πi(ξ2 ϕ Note, the third equation is a consequence of the first two and (6.10.36). Hence, in order to fix ϕ we add the Coulomb gauge condition 3 

ξj ϕ *j = 0.

(6.10.37)

j=1

This implies ϕ *1 = (i|ξ|2 )−1 (ξ3 u *2 − ξ2 u *3 ), ϕ *2 = (i|ξ|2 )−1 (ξ1 u *3 − ξ3 u *1 ), *1 − ξ1 u *2 ). ϕ *3 = (i|ξ|2 )−1 (ξ2 u * ∈ L2 (R3 ) and |* ϕi | ≤ (2π|ξ|)−1 (|* uj | + |* uk |) there holds ϕ ∈ Since ξ ϕ  1 j 3j * is bounded at ξ = 0. Note, that u * (ξ) is holomorphic and H (Ω) if ϕ * (0) = 0 due to (6.10.36) and u u *j (ξ) =

3  =1

ξ

∂* uj (0) + O(|ξ|2 ) ∂ξ

* (ξ) is bounded near ξ = 0 and the inverse near ξ = 0. This implies that ϕ Fourier transform (3.1.14) gives  * (6.10.38) ϕ(x) = (2π)−3/2 eix·ξ ϕ(ξ)dξ. R3

Hence,

6.10 Boundary Integral Equations for Electromagnetic fields

355

curl ϕ = u and div ϕ = 0  3  3 due to (6.10.37) and u ∈ H1 (R3 ) = H1 (R3 ) |Ω . As a consequence, curl (curl ϕ) = −Δϕ + 0,

hence curl u ∈ (L2 (R3 )3

 3  3  3 which implies ϕ ∈ H 2 (Ω) = H 2 (R3 ) |Ω , therefore, u ∈ H 1 (Ω) and 3  3  M(Ω) ⊂ H 1 (Ω) which is compactly imbedded into L2 (Ω) as we have proposed.  Note that A (see (6.10.34)) is continuous with respect to the Hilbert space norm ||| · |||. As a consequence of the G˚ arding inequality (6.10.34), the Fredholm alternative is valid (see Chapter 5.3.3), uniqueness would imply the existence of a unique solution to (6.10.28) which satisfies the Maxwell equations (6.10.25) where curl E(x) = iωμH(x) defines the magnetic field H(x). 6.10.2.2 A reduction to boundary integral equations. The Dirichlet problem (6.10.25) or cavity problem, can be reduced to boundary integral equations based on the Stratton–Chu formulae (6.9.4) and (6.9.5). To be consistent with the previous chapters we first introduce the single layer and double layer potentials for the Maxwell equations as follows:  Vλ)(x) := −iωμ Gk (x − y)λ(y)dsy (6.10.39) (V Γ

+

1 (−iωε + σ)



grad y Gk (x − y) (divΓ λ(y))dsy ,

Γ

 Wϕ)(x) := (W



ϕ(y) × grad y Gk (x − y)dsy Γ

 Gk (x − y)ϕ(y)dsy for x ∈ Ω.

= curl Γ

These boundary potentials have the mapping properties V : H−1/2 (divΓ , Γ ) → H(curl , Ω), W : H−1/2 (divΓ , Γ ) → H(curl , Ω) continuously .

(6.10.40)

For the Dirichlet problem we seek the solution by the layer ansatz E(x) := (Vλ)(x) , x ∈ Ω.

(6.10.41)

356

6. Electromagnetic Fields −1/2

The boundary condition n × E|Γ = m ∈ H|| boundary integral equation

(divΓ , Γ ) then leads to the

n(x) × E(x) = (TE λ)(x) = m(x) for x ∈ Γ

(6.10.42)

and TE is given by (6.10.2), i.e., (TE )λ(x) = −iωμn(x) × (Vk λ)(x) 

− (−iωε + σ)

−1



(6.10.43)



n(x) × grad Γ Vk (divΓ λ) (x) , x ∈ Γ,

Gk (x − y)λ(y)dsy

Vk λ :=

(6.10.44)

Γ

(see (2.1.6) and Section 10.1.1). 1 1 Given m ∈ H− 2 (divΓ , Γ ), we seek λ ∈ H− 2 (divΓ , Γ ) by using the variational equation for (6.10.43) by testing (6.10.43) with n(x) × λt (x) to obtain    n(x) × Vk λ(x) · n(x) × λt (x) dsx (6.10.45) − iωμ Γ = −iωμ

 (Vk λ)(x) · πt λ (x)dsx = −iωμ

Γ

t

(Vk λ)(x) · λ (x)dsx .

t

Γ

Similarly,    grad (Vk divΓ λ) × n(x) · n(x) × λt (x) dsx Γ

 =−

(6.10.46)

t

πt grad (Vk divΓ λ) · λ (x)dsx Γ



=− 

grad Γ (Vk divΓ λ) · λt (x)dsx Γ

(Vk divΓ λ) · divΓ λt dsx .

= Γ

Collecting these relations for (6.10.43) gives the weak variational equation    m(x) × n(x) · λt (x)dsx m(x) · (n(x) × λt (x))dsx = Γ



= −iωμ

Γ

(Vk λ)(x) · λ (x)dsx + (−iωε0 + σ)−1



t

Γ

or, in short,

(Vk divΓ λ)divΓ λt (x)dsx Γ

6.10 Boundary Integral Equations for Electromagnetic fields

Vk λ, λt Γ −

1 −i n × m, λt Γ Vk divΓ λ, divΓ λt Γ = k2 ωμ

357

(6.10.47)

−1/2

for all λt ∈ H|| (divΓ , Γ ). For obtaining a G˚ arding inequality for the boundary integral equation (6.10.47), we again use a Helmholtz (Hodge) decomposition of the solution on the boundary in the following form (which in fact has no direct relation with the Helmholtz decomposition of the corresponding solution in the domain Ω): −1/2

H||

(divΓ , Γ ) = grad Γ H 3/2 (Γ )/C ⊕ curl Γ H 1/2 (Γ )/C.

(6.10.48)

The spaces on the right are orthogonal with respect to the L2 (Γ )–duality. In fact, we note that curl Γ H 1/2 (Γ ) ⊂ H−1/2 (Γ ) whereas grad Γ H 3/2 (Γ ) ⊂ H1/2 (Γ ) ⊂ H−1/2 (Γ ) compactly. The space on the left is purposely chosen in order to obtain ϕ ∈ H 3/2 (Γ ) and ψ ∈ H 1/2 (Γ ) so that the L2t (Γ )–duality grad Γ ϕ, curl Γ ψ is well defined. Therefore, in terms of the decomposition, we write λ = grad Γ ϕ + curl Γ ψ and λt = grad Γ ϕt + curl Γ ψ t . Then the equation (6.10.47), i.e. 1 Vk divΓ (grad Γ ϕ + curl Γ ψ), divΓ λt Γ k2 i n × m, λt  =− (6.10.49) ωμ

Vk (grad Γ ϕ + curl Γ ψ), λt  −

will first be tested by grad Γ ϕt to obtain Vk grad Γ ϕ, grad Γ ϕt  + Vk curl Γ ψ, grad Γ ϕt Γ 1 i − 2 Vk ΔΓ ϕ, ΔΓ ϕt Γ = − n × m, grad Γ ϕt Γ k ωμ

(6.10.50)

and then tested by curl Γ ψ t to yield Vk (grad Γ ϕ), curl Γ ψ t  + Vk curl Γ ψ, curl Γ ψ t Γ i =− n × m, curl Γ ψ t . ωμ

(6.10.51)

in view of divΓ curl Γ ψ t = 0. Multiplying the Equation (6.10.50) by −k 2 and adding to the Equation (6.10.51), the weak variational equation reads: Find (ϕ, ψ) ∈ H 3/2 (Γ ) × H 1/2 (Γ ) such that AΓ (ϕ, ψ; ϕt , ψ t ) := Vk ΔΓ ϕ, ΔΓ ϕt Γ + Vk curl Γ ψ, curl Γ ψ t Γ + Vk grad Γ ϕ, curl Γ ψ t , Γ − k 2 Vk grad Γ ϕ, grad Γ ϕt Γ − k 2 Vk grad Γ ϕ, curl Γ ψ t Γ i n × m, (k 2 grad Γ ϕt − curl Γ ψ t )Γ = (6.10.52) ωμ

358

6. Electromagnetic Fields

for all (ϕt , ψ t ) ∈ H 3/2 (Γ ) × H 1/2 (Γ ). This sesquilinear variational equation is called the Rumsey principle (see Nedelec [331, p.245]). From (2.1.9) we have the pseudohomogeneous expansion of the fundamental solution ik + Sk (x − y) 4π with the kernel function Sk of the remainder, given by Gk (x − y) = G0 (x − y) +

Sk (x − y) =

(6.10.53)

∞ m  (ik|x − y|) 1 . 4π|x − y| m=2 m!

(6.10.54)

Lemma 6.10.2. Based on the mapping properties of Vk in Theorems 5.6.2 and 5.6.8 Vk : H−1/2 (Γ ) → H1 (Ω), γ0 Vk : H−1/2 (Γ ) → H1/2 (Γ ), there holds the following G˚ arding inequality Re {Vk λ, λ + Ck λ, λ} ≥ α||λ||2H−1/2 (Γ )

(6.10.55)

with α > 0 and some compact linear operator Ck : H−1/2 (Γ ) → H1/2 (Γ ). In the weak varational formulation (6.10.52) the following compactness properties hold. Lemma 6.10.3. For ϕ ∈ H 3/2 (Γ ) and ψ ∈ H 1/2 (Γ ), the following linear operators are compact: Vk grad Γ : H 3/2 (Γ ) → curl Γ H 1/2 (Γ )  H−1/2 (Γ ), Vk curl Γ : H

1/2

(Γ ) → grad Γ H

1/2

−1/2

(Γ )  H

(Γ ).

(6.10.56) (6.10.57)

In view of Lemata 6.10.2 and 6.10.3 we obtain: Theorem 6.10.4. The sesquilinear form AΓ in (6.10.52) for Re k 2 > 0 is arding inequality strongly elliptic on H 3/2 (Γ ) × H 1/2 (Γ ) satisfying the G˚ Re AΓ (ϕ, ψ; ϕ, ψ) ≥ c0 {||ϕ||2H 3/2 + ||ψ||2H 1/2 (Γ ) } − c1 (||ϕ||2L2 (Γ ) + ||ψ||2L2 (Γ ) )

(6.10.58)

where c0 > 0 and c1 ≥ 0 are constants independent of (ϕ, ψ). The G˚ arding inequality (6.10.58) implies that the linear operator equation associated with the weak variational formula (6.10.52) is a Fredholm mapping with index zero. As a consequence, since σ > 0 and, hence, Imk > 0, uniqueness implies that the sesquilinear variational equation (6.10.52) has a unique solution (ϕ, ψ) ∈ H 3/2 (Γ ) × H 1/2 (Γ ) with λ = grad Γ ϕ + curl Γ ψ which generates via (6.10.41) a unique solution of the cavity problem of the Maxwell equations.

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

359

6.11 Application of the Electromagnetic Potentials to Eddy Current Problems In this section we consider a classical formulation of eddy current problems in terms of three fields A, ϕ and ψ which was introduced in Leon and Rodger [264, 90–93] and Bir´o and Preis [31, 3145–3159]. 6.11.1 The ’(A, ϕ) − (A) − (ψ)’ formulation in the bounded domain In what follows, we consider all the functions involved to be sufficiently smooth. Later on, the precise function spaces will be specified. As in the figure below, consider three bounded domains Ωc , ΩA and Ω in R3 such that Ω c ⊂ ΩA , Ω A ⊂ Ω and Ω ⊂ R3 , with the boundaries ∂Ωc = Γc , ∂ΩA = ΓA and ∂Ω = Γ . Here Ωc is the conducting domain, and a source current density J0 is given in ΩA \ Ω c . We denote by ΩD := Ω \ Ω c the subdomain of Ω occupied by dielectric material with  = 0 and constant μA and εA in ΩA \ Ω c and constant μ0 , ε0 in the free space Ω \ Ω A . The approach is based on the (A, ϕ) − (A) − (ψ) formulation in the conductor Ωc and dielectric material ΩD = Ω \ Ω c , respectively. Both A’s are magnetic vector potentials, while ϕ and ψ are respectively the electric and magnetic scalar potentials. For given J0 in ΩA \Ω c and the homogeneous boundary condition n×H = 0 on Γ , the time harmonic Maxwell equations (6.4.1)–(6.4.5) for the eddy current problem now read: ⎧ iωμc H in Ωc , ⎪ ⎨ curl E = iωμA H in ΩA \ Ω c , (6.11.1) ⎪ ⎩ iωμ0 H in Ωψ = Ω \ Ω A ; ⎧ ⎪ ⎨σE in Ωc (6.11.2) curl H = J0 in ΩA \ Ω c ⎪ ⎩ 0 in Ωψ . J0 is the given electric current distribution with div J0 = 0 and

where

div D = 0 in ΩD ,

(6.11.3)

⎧ ⎪ ⎨ εc E in Ωc D = εA E in ΩA \ Ω c , ⎪ ⎩ ε0 E in Ωψ .

(6.11.4)

div B = 0 in Ω,

(6.11.5)

Moreover,

360

6. Electromagnetic Fields

n

n

n Ωc Γc ΩA \ Ωc

ΓA

Ωψ := Ω \ ΩA Γ Fig. 6.2. An eddy current configuration

⎧ ⎪ ⎨ μc H in Ωc , B = μA H in ΩA \ Ω c , ⎪ ⎩ μ0 H in Ωψ .

where

(6.11.6)

From Equation (6.11.5) in Ωc we introduce a vector potential A such that B = curl A in Ωc . Then from Equation (6.11.1) we have curl (E − iωA) = 0 in Ωc . This implies that there exists a scalar potential ϕ such that E = iωA + iω grad ϕ in Ωc . Then from Equation (6.11.2) we have curl

1 curl A − σiω(A + grad ϕ) = 0 in Ωc , μc

(6.11.7)

and, consequently, div {σiω(A + grad ϕ)} = 0 in Ωc .

(6.11.8)

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

361

In ΩA \ Ω c , again from Equation (6.11.5) there exists a vector potential A with curl A = B. Hence, from Equation (6.11.2) we arrive at curl

1 (curl A) = J0 in ΩA \ Ω c . μA

(6.11.9)

Finally, in Ωψ = Ω \ Ω A , again from Equation (6.11.2), there exists a scalar potential ψ such that H = ω grad ψ and from Equation (6.11.5) we obtain from div B = 0 finally div(μ0 grad ψ) = 0 in Ωψ = Ω \ Ω A .

(6.11.10)

In order to ensure uniqueness of A in ΩA , we choose the simple Coulomb’s gauge function, namely divA = 0 in ΩA \ Γc .

(6.11.11)

In summery, our ’(A, ϕ) − (A) − (ψ)’ formulation consists of the following system of Maxwell equations: ⎧ 1 ⎪ curl curl A − σiω(A + grad ϕ) = 0, ⎪ ⎪ ⎪ μ ⎪ c ⎪ ⎪ ⎨ div{σiω(A + grad ϕ)} = 0 in Ωc ; (6.11.12) 1 ⎪ ⎪ ⎪ curl A = J0 in ΩA \ Ω c ; curl ⎪ ⎪ μA ⎪ ⎪ ⎩ div(μ0 grad ψ) = 0 in Ω \ Ω A ; together with the Coulomb gauge condition divA = 0 in ΩA \ Γc . For the transmission conditions on Γc , we have ⎧ 1 1 ⎪ ⎪ n × curl A− = n × curl A+ , ⎪ ⎨ μc μA n · σiω(A− + grad ϕ− ) = 0, ⎪ ⎪ ⎪ ⎩ n · A− − n · A+ = 0. On the interface ΓA , we have ⎧ 1 ⎪ ⎪ n× curl A− = n × ω grad ψ + , ⎪ ⎨ μA n · curl A− = n · ωμ0 grad ψ + , ⎪ ⎪ ⎪ ⎩ n · A− = 0.

(6.11.13)

(6.11.14)

(6.11.15)

362

6. Electromagnetic Fields

Finally, we impose the boundary condition on Γ : n · H|Γ = h ∈ H −1/2 (Γ )

(6.11.16)

We now add an existence theorem for the weak solution of the eddy current problem in the bounded domain Ω. Let us first specify the function spaces in the domains and on the Lipschitz boundaries: Z = H0 (div, ΩA ) ∩ H(curl , ΩA ) := {Z : Z · nA = 0 on ΓA and div Z ∈ L2 (ΩA ) , curl Z ∈ L2 (ΩA )} M = H1 (Ωc )/C; HΓ1 (Ωψ ) := {ψ ∈ H 1 (Ωψ ) ∧ grad Γ ψ := n × grad ψ × n = 0 on Γ }. Moreover, we also need the trace spaces of H(curl , ΩA ). Recall from Section 4.5 that γt (u) := n × u|ΓA and πt (u) := −n × (n × u|ΓA ) denote respectively the tangential trace and projection operators. Then the operators −1/2

γt : H(curl , ΩA ) → H||

(divΓ , ΓA )

−1/2

and πt : H(curl , ΩA ) → H⊥

(curl Γ , ΓA )

are linear, continuous, surjective and possess continuous right–inverses, where   −1/2 −1/2 H|| (divΓ , ΓA ) := λ ∈ H|| (ΓA ) | divΓ (λ) ∈ H −1/2 (ΓA ) ,   −1/2 −1/2 H⊥ (curl Γ , ΓA ) := λ ∈ H⊥ (curl(λ) ∈ H −1/2 (ΓA ) , Γ

are the trace spaces for H(curl , Ω) equipped with graph norms (see Theorem −1/2 −1/2 4.5.3). The duality pairing between H|| (divΓ , ΓA ) and H⊥ (curl Γ , ΓA ) is well defined by   γ t (u) · πt (v)ds = (curl u · v − u · curl v)dy for u, v ∈ H(curl , ΩA ). ΓA

ΩA

(6.11.17) Following Rodriguez [365], we now state and prove the following Lemma 6.11.1. (See Rodriguez [365, Theorem 6, p.104–107])  −1 (ΩA \ Ω c ) and h ∈ H −1/2 (Γ ) be given and E and H be Let J0 ∈ H a solution of the Maxwell equations (6.11.1), (6.11.2) defining (A, ϕ, ψ) ∈ Z × M × HΓ1 (Ωψ ) by E = iωA + iω grad ϕ and (6.11.7) in Ωc , (6.11.9) in ΩA \ Ω c and ψ with ω grad ψ = H satisfying (6.11.11), (6.11.12) and

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

363

(6.11.13) in Ω \ Γc together with the transmission and boundary conditions (6.11.14), (6.11.15) and H · n = h on Γ . Then (A, ϕ, ψ) fulfills weakly the variational equation    A (A, ϕ, ψ) , (Z, u, η) = J0 · Zdy + ω ημ0 hds (6.11.18) ΩA \Ω c

Γ

◦1

for all (Z, u, η) ∈ Z0 × M × H Γ (Ωψ ) with 1 curl Z = 0 in Ω \ Ω c }, μ A ◦1 H Γ (Ωψ ) := {ψ ∈ HΓ1 (Ωψ ) with grad ψ · n = 0 on Γ } Z0 := {Z ∈ Z satisfying curl

and where the bilinear form A is given by  A (A, ϕ, ψ) , (Z, u, η)  1 := {curl A · curl Z + (divA) · (divZ)}d y μ ΩA  2 μ0 grad ψ · grad η dy +ω Ωψ



σ(A + grad ϕ) · (Z + grad u) dy

− iω Ωc

 −ω

(6.11.19)



(grad ψ × n) · πt (Z)ds + ω

ΓA

(grad η × n) · πt (A)ds,

ΓA

where μ is defined by (6.11.24) below. Proof: For the proof let us begin with the variational equations in Ωc by testing (6.11.8) with u, i.e.  div{σiω(A + (grad ϕ)}udy = 0. Ωc

Integration by parts gives   (σiωA)u · nc ds − iωσA · (grad u)dy 

Γc

Ωc



σiω grad ϕ u · nc ds −

+ Γc

iωσgrad ϕ · (grad u)dy = 0; Ωc

364

6. Electromagnetic Fields

and (6.11.14) in Ωc yields  iωσ(A + grad ϕ) · (grad u)dy = 0 for all u ∈ H 1 (Ωc ).

(6.11.20)

Ωc

In Ωc we also have the first of the equations in (6.11.12) which in weak form with test fields Z ∈ Z := H0 (div, ΩA ) ∩ H(curl , ΩA ) reads as   1 1 {(grad ) × curl A} · Z dy + (curl curl A) · Z dy μc μc Ωc Ωc  − iωσ(A + grad ϕ) · Z dy = 0. Ωc

With Green’s first identity we find       1 1 grad × curl A · Z dy + grad × Z · curl A dy μc μc Ωc Ωc   1 + curl A · curl Z dy − iωσ(A + grad ϕ) · Z dy μc Ωc Ωc  1  + n· (curl A) × Z ds = 0. μc ∂Ωc

The commutator rule for the scalar triple product implies cancelling of the first two integrals and for the last surface integral with Equations (6.11.13) and (6.11.12) we have the relation  1 {curl A · curl Z + (div A) · (div Z)}dy (6.11.21) μc Ωc   1 − iωσ(A + grad ϕ) · Z dy + (n × curl A) · Zds = 0. μc Ωc

Γc

Next, in ΩA \ Ω c we have from (6.11.2) and (6.11.7) the relation curl

 1 curl A = J0 μA

which we test also with Z ∈ Z:   1 curl curl A · Z dy = μA ΩA \Ω c

 J0 · Z dy. ΩA \Ω c

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

365

Again with Green’s first identity and the chain rule together with the commutator property of the scalar triple product we obtain    1 1 ( curl A · curl Z dy + curl A) × Z · n ds μA μA ΩA \Ω c

∂(ΩA \Ω c )



J0 · Z dy for all Z ∈ Z.

= ΩA \Ω c

(6.11.22) On ΓA , (6.11.14) gives   1  1 (curl A) × Z ds = (n × n· curl A) · Z ds μA μA ΓA ΓA  = (n × ωgrad ψ + ) · Z ds ΓA

and on Γc :   1  1 − n· (curl A) × Z ds = − (n × curl A) · Z ds μA μA Γc Γc  1 = − (n × curl A) · Z ds. μc Γc

Adding (6.11.20), (6.11.21) and (6.11.22) with (6.11.11) and (6.11.15) results in  1 {curl A · curl Z + div A · div Z}dy μ ΩA  − iω σ(A + grad ϕ) · (Z + grad u) dy 

Ωc



(n × ω grad ψ + ) · Z ds =

+ ΓA

J0 · Z dy

(6.11.23)

ΩA \Ω c

for all u ∈ H 1 (Ωc ) , Z ∈ Z(ΩA ) and with  μc in Ωc , μ= . μA in ΩA \ Ωc

(6.11.24)

Finally in Ωψ , (6.11.10) yields with integration by parts and (6.11.15):

366

6. Electromagnetic Fields

 ω

 μ0 grad ψ · grad η dy = ω

2 Ωψ

ημ0 grad ψ · n ds

2 ∂Ωψ



=ω

η μ0 grad ψ · n ds

2 Γ

(6.11.25)



−ω

η curl A · n ds for all η ∈ HΓ1 (Ωψ ).

ΓA

For the last integral in (6.11.25) we apply (6.11.17) and obtain:   η curl A · nds = div (η curl A) dy ΓA

ΩA





grad η · curl Ady =

= ΩA

ΩA



=−

div(A × grad η)dy

γt ( grad η) · πt (A)ds.

ΓA

Hence, (6.11.25) becomes    2 ω μ0 grad ψ · grad η dy − ω γt (grad η) · πt Ads = ω ημ0 H · nds. ΓA

Ωψ

Γ

(6.11.26) Finally by adding (6.11.23) and (6.11.26) we find the desired variational equation (6.11.18) for (A, ϕ, ψ) ∈ Z × M × HΓ1 (Ωψ ). and all test functions ◦1

(Z, u, η) ∈ Z0 × M × H Γ (Ωψ ).



In order to prove continuity and coerciveness of A we shall need the following elementary lemma: Lemma 6.11.2. Let c > 0 and κ ∈ (0, 12 ). Then (c + 2d)2 > κc2 − 8κd2 for all d ∈ R.

(6.11.27)

Proof: Set dc = 12 (t − 1) or t = (1 + 2 dc ). Then the inequality (6.11.27) is equivalent to Φ(t) := (2κ + 1)t2 − 4κt + κ > 0 for all t ∈ R. Φ (t0 ) := 0 = 2(2κ + 1)t0 − 4κ = 0 implies that with t0 = 2κ(2κ + 1)−1 Φ(t) ≥ Φ(t0 ) = κ(2κ + 1)−1 > 0 for all t ∈ R which is equivalent with (6.11.27).



6.11 Application of Electromagnetic Potentials to Eddy Current Problems

367

Theorem 6.11.3. Rodriguez ([365, Theorem 6]) The sesquilinear form (6.11.9) is continuous on Z × M × HΓ1 (Ωψ ) satisfying  |A (A, ϕ, ψ) ; (Z, u, η) | (6.11.28) ≤ c(AZ + ϕM + ψHΓ1 (Ωψ ) ) · (ZZ + uM + ηHΓ1 (Ωψ ) ). o1

Moreover, for (Z, u, η) ∈ Z0 × M × H Γ (Ωψ ), we have the coerciveness estimate  A (Z, u, η), (Z, u, η) ≥ κ0 (ZZ + uM + ηHΓ1 (Ωψ ) )2 . (6.11.29) Proof: We begin with the inequality (6.11.28) and consider  |A (A, ϕ.ψ), (Z, u, η) |2   1 {curl A · curl Z + (div A) · (div Z)}dy + ω 2 =| μ0 grad ψ · grad ηdy μ ΩA

Ωψ



− iω

σ(A + grad ϕ) · (Z + grad u)dy

Ω

c − ω {(grad ψ × n) · πt (Z) + (grad η × n) · πt (A)}ds|2 ΓA

 1 ≤  L∞ (ΩA ) (curl AL2 (ΩA ) curl ZL2 (ΩA ) + div AL2 (ΩA ) divZL2 (ΩA ) ) μ + ω 2 μ0 L∞ (Ωψ ) grad ψL2 (Ωψ ) grad ηL2 (Ωψ ) + ωσL∞ (Ωc ) A + grad ϕL2 (Ωc ) Z + grad uL2 (Ωc ) + ωgrad ψ

1

H − 2 (ΓA )

πt (Z)

1

H 2 (ΓA )

+ ωgrad η

1

H − 2 (ΓA )

πt (A)

1

2

H 2 (ΓA )

1 ≤ c 2L∞ (ΩA ) {A2Z + Z2Z }2 μ + ω 4 μ0 2l∞ (Ωψ ) {ψ2H 1 (Ωψ ) + η2H 1 (Ωψ ) }2 Γ

+ +

Γ

ω σ2L∞ (Ωc ) {A2L2 (Ωc ) + ϕ2M + Z2L2 (Ωc ) + ω 2 {ψ2H 1 (Ωψ ) + η2H 1 (Ωψ ) + A2Z + Z2Z }2 Γ Γ 2

u2M }2

≤ c (AZ + ϕM + ψHΓ1 (Ωψ ) )2 (ZZ + uM + ηHΓ1 (Ωψ ) )2 , which implies (6.11.28). The constant c depends on ω, μ, μ0 and σ. For showing the coerciveness estimate (6.11.29) we shall employ Lemma 6.11.2.

368

6. Electromagnetic Fields

Let



1 |curl Z|2 + |divZ|2 dy, μ ΩA    2 |Z| + |grad u|2 dy, μ0 |grad η|2 dy , c := b :=

a :=

Ωψ

Ωc





σ Re (grad u · Z)dy , f =

e := Ωc

Im(grad η × nA · πτ (Z)ds ΓA

and d := e + f be the constants in the Lemma 6.11.2. Then a simple computation shows that    A (Z, u, η), (Z, u, η)) 2 = (a + ω 2 b)2 + ω 2 (c + 2d)2 ≥ (a2 + ω 4 b2 ) + ω 2 (κc2 − 8κd2 ). Since a ≥

K 2 μmax ZZ

and b ≥ μ0 min grad η2L2 (Ωψ ) ,

2    A (Z, u, η) , (Z, u, η) 2 ≥ K Z4Z + ω 4 μ20 min grad η2 2 L (Ωψ ) μ2max 1

+ ω 2 κσ 2 grad u4L2 (Ωc ) − 16ω 2 κ(e2 + f 2 ). Moreover, we see that for all ε > 0,     2 2 2 ε 1 2 2 |σgrad u · Z|dy ≤ σ|grad u| dy + σ|Z|2 dy e = 2 2ε Ωc

Ωc

Ωc

and with the trace theorem f 2 ≤ grad η × nA 2

−1 2

H||

(divΓ ,ΓA )

· πτ (Z)2

 ≤ c grad η4L2 (Ωψ ) + Z4Z

−1

H⊥ 2 (curl Γ ,ΓA )

with some constants c (independent of η and Z), we get  K2   2 κ  2 − C − 16ω ω κ Z4Z |A (Z, u, η), (Z, u, η) |2 ≥ μ2max 2ε  ε 1 + ω 2 κ−16ω 2 κ σ 2 grad u4L2 (Ωc ) + (ω 2 μ0 min −16c ω 2 κ)grad η4L2 (Ωψ ) 2 1 K2 1 1 4 ≥ ZZ + ω 2 κσ 2 grad u4L2 (Ωc ) + ω 2 μ20 min grad η4L2 (Ωψ ) 2μ2max 2 2   1 for ε = 16 and κ = min K 2 (32c ωμmax )−1 , (32c )−1 . 2 If c20 := min{ 2μKmax , 12 ω 2 κ, 12 ω 2 μ0 min } then

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

369

 |A (Z, u, η), (Z, u, η) |2 ≥ c20 {Z4Z + u4M + η4H 1 (Ωψ ) }  4 ≥ c20 3−3 ZZ + uM + ηH 1 (Ωψ ) 3

which yields (6.11.29) with κ0 := c0 3− 2 . This completes the proof of Theorem 6.11.3.   −1 (ΩA \ Ω c ) and h ∈ H − 12 (Γ ) be given. Then Theorem 6.11.4. Let J0 ∈ H there exists exactly one solution (A, ϕ, ψ) ∈ Z × M × HΓ1 (Ωψ ) of the variational equation (6.11.18) where the test function (Z, u, η) traces the space Z0 × M × HΓ1 (Ωψ ). Note that here ω > 0 is a necessary assumption.  Proof: The sesquilinear form A (A, ϕ, ψ), (Z, u, η) is continuous on the Hilbert space Z × M × HΓ1 (Ωψ ) and coercive on the closed subspace 0

Z0 × M × H 1 Γ (Ωψ ) due to Theorem 6.11.3. Then the Lax–Milgram theorem yields the existence of a unique solution to the variational equation as proposed.  Remark 6.11.1: Since ϕ ∈ M = H 1 (Ωc )/C, ϕ is uniquely determined only up to an additive complex constant, whereas E = iωA+iωgrad ϕ is uniquely determined in Ωc . In terms of (A, ϕ, ψ), the magnetic field H of the eddy current problem is given by ⎧ 1 in Ωc , ⎨ μc curl A 1 H= curl A in ΩA \ Ω c , ⎩ μA ω grad ψ in Ωψ , whereas the electrical field is only determined in Ωc as E = iωA + iω grad ϕ in Ωc . In order to find E in ΩD , one needs to solve a transmission problem in terms of Ec on Γc . We will return to this topic when treating the exterior eddy current problem in the next section. 6.11.2 The  (A, ϕ) − (ψ) formulation in an unbounded domain In this section, we consider an eddy current problem in an unbounded domain. For simplicity, let Ωc be a bounded conducting domain in R3 with Lipschitz boundary Γc and ΩD := R3 \ Ωc is occupied with a lossless dielectric medium (see Figure 6.11.2). We assume that the electric conductivity σ and the magnetic permeability μ are in L∞ (Ωc ) such that  0 < μmin ≤ μc ≤ μmax for x ∈ Ωc . 0 < σmin ≤ σ ≤ σmax ,

370

6. Electromagnetic Fields

Moreover, the applied current density J0 is assumed to be in L2 (Ωc ). The magnetic permeability and the electric permittivity are assumed to be positive constants μ0 and ε0 , respectively in ΩD .

n

Ωc

ΩD := R3 \ Ωc

Γc

Fig. 6.3. An eddy current problem in unbounded domain.

For an exterior eddy current problem of this kind, it is most conveniently handled by a coupling of boundary and field equation methods (see, e.g., Gatica and Hsiao [139]). Our presentation follows the  (A, ϕ) − (ψ) formulation from Rodriguez and Valli [366]. That is, we use the field equations in Ωc for the magnetic vector potential A, and for an electric scalar potential ϕ. In ΩD , a symmetric boundary integral equation formulation for the scalar magnetic potential ψ is employed. For given J0 in Ωc , the time harmonic Maxwell equations of the magnetic field H and the electric field E for the eddy current problem read:  i ωμc H in Ωc , (6.11.30) curl E = i ωμ0 H in ΩD ,  σE + J0 in Ωc , curl H = (6.11.31) 0 in ΩD , div B = 0,

in

R3 ,

(6.11.32)

div D = 0,

in

ΩD ,

(6.11.33)

where Eq.(6.11.32) is the consequence of Eq.(6.11.30) and D = ε0 E in ΩD . For the boundary-field equation methods, we need also the transmission

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

conditions:

n · B− = n · B+

on

n × H− = n × H+ −

n×E =n×E

⎫ Γc , ⎬ ⎪ on Γc , ⎪ ⎭ on Γc

371

+

(6.11.34)

together with the condition at infinity: H(x) = O(|x|−1 ),

and E(x) = O(|x|−1 )

(6.11.35)

For the  (A, ϕ) − (ψ) formulation, we see from Equation (6.11.32) in Ωc , we may introduce a magnetic vector potential A such that B = curl A

in

Ωc .

Then from Equation (6.11.30) in Ωc we have curl (E − iωA) = 0. This implies that there exists an electric scalar potential ϕ such that E = iω(A + grad ϕ).

(6.11.36)

As a consequence of (6.11.31) in Ωc , we have curl

1 curl A − iσω(A + grad ϕ) = J0 μc

in

Ωc .

(6.11.37)

subject to the constraint div{−iσω(A + grad ϕ)} = div J0

in

Ωc .

(6.11.38)

In order to have unique vector potential A, as usual we choose the Coulomb gauge condition div A = 0 in Ωc . Next, from Equation (6.11.31) in ΩD , there exits a magnetic scalar potential ψ such that H = grad ψ and from Equation (6.11.31) we obtain −div (μ0 grad ψ) = 0 or − Δ ψ = 0 in ΩD ,

(6.11.39)

since μ0 is constant. In summary, the  (A, ϕ) − (ψ) formulation for the eddy current problem consists of the following system of equations for the unknowns (A, ϕ, ψ): ⎧ 1 ⎪ ⎪ ⎪ ⎨curl μ c curl A − iσω(A + grad ϕ) = J0 in Ωc , (6.11.40) div {−iσω(A + grad ϕ)} = div J0 in Ωc , ⎪ ⎪ ⎪ ⎩ −Δψ = 0 in ΩD

372

6. Electromagnetic Fields

with the Coulomb gauge condition div A = 0 in Ωc ,

(6.11.41)

where ϕ is unique up to an additive constant. The corresponding transmission conditions on Γc now read ⎧ 1 ⎪ n× curl A− = n × grad ψ + , ⎪ ⎪ ⎪ μc ⎪ ⎪ ⎪ ⎨ 1 ∂ + n· ψ , curl A− = (6.11.42) μ ∂n 0 ⎪ ⎪ ⎪ ⎪ n · {−iσω(A− + grad ϕ− )} = n · J− ⎪ 0, ⎪ ⎪ ⎩ − n·A =0 together with conditions at infinity |ψ(x)| + |grad ψ(x)| = O(|x|−1 )

as

|x| → ∞.

(6.11.43)

We are interested here in the weak solutions of the eddy problem consisting of Equations (6.11.40), (6.11.41), (6.11.42) and (6.11.43). First, notice that we may remove (6.11.41) and consider the modified system of equations ⎧ 1 1 ⎪ ⎪ curl curl A − grad div A ⎪ ⎪ μ μ max ⎪ c ⎪ ⎪ ⎪ ⎪ −iσω(A + grad ϕ) = J0 in Ωc , ⎪ ⎪ ⎪ ⎪ ⎪ div {−iσ ω(A + grad ϕ)} = div J0 in Ωc , ⎪ ⎪ ⎪ ⎪ ⎪ −Δψ = 0 in ΩD ⎪ ⎪ ⎪ ⎨ 1 n× curl A− = n × grad ψ + on Γc (6.11.44) μ ⎪ c ⎪ ⎪ ⎪ 1 ∂ + ⎪ ⎪ ⎪ ψ n· curl A− = on Γc ⎪ ⎪ μ0 ∂n ⎪ ⎪ ⎪ ⎪ ⎪ n · {−iσω(A− + grad ϕ− )} = n · J− on Γc ⎪ 0 ⎪ ⎪ ⎪ − ⎪ n · A = 0 on Γc ⎪ ⎪ ⎪ ⎩ |ψ(x)| + | grad ψ(x)| = O(|x|−1 ) as |x| → ∞. It is easy to see that this modified system and the original system of equations (6.11.40), (6.11.41), (6.11.42) and (6.11.43) are equivalent. In particular, any solution to (6.11.44) will satisfy (6.11.41). This can be seen as follows: By taking the divergence of the first equation in (6.11.44), denoted by (6.11.44)1 , we obtain that Δ (div A) = 0 in Ωc in view of (6.11.44)2 . Then taking the scalar product of (6.11.44)1 with n and making use of the condition (6.11.44)6 , we see that

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

373

1 ∂ 1 1 div A|Γc = n · curl ( curl A)|Γc = −divΓc {n × curl A− } μmax ∂n μc μc = −divΓc {n × grad ψ + } = 0 on Γc . This implies div A is a solution of the Laplace equation in Ωc and satisfies the homogeneous Neumann boundary condition on Γc . Hence div A is a constant and the constant is zero from the condition (6.11.44)7 . To this end, it is worth mentioning that in order to determine solutions A (and ϕ) in Ωc , in addition to the differential equations for A and ϕ in Ωc , the natural boundary condition needed for A on Γc from variational point of view is the condition (6.11.44)4 of (6.11.44), namely n×

1 curl A− = n × grad ψ + = −curl Γc ψ + , μc

on Γc ,

which depends on ψ + = γ(ψ), the trace of ψ on Γc . On the other hand, ψ is the solution of the Laplace equation in the unbounded domain ΩD with a natural boundary condition in terms of the trace of A, namely ∂ + 1 ψ =n· curl A− on Γc , ∂n μ0 which is known, if the solution A is available. Thus, the eddy current problem in the unbounded domain defined by (6.11.44) should be reduced to a nonlocal boundary value problem in the bounded domain Ωc for the unknowns A, ϕ satisfying PDEs in the domain Ωc ,and the unknown ψ + = γ(ψ) satisfying a suitable BIE on the boundary Γc , the nonlocal boundary condition. If ψ ∈ Hloc (ΩD ) (or the weighted space H 1 (ΩD ; λ), see Section 4.5) satisfies the Laplace equation (6.11.44)3 of (6.11.44), we have the Green representation formula of the solution in terms of single– and double–layer boundary potentials, V and W, respectively:   ∂ ∂ E(x, y)ψ + (y)dsy − E(x, y) ψ + (y)dsy ψ(x) = ∂ny ∂n Γc Γc

=: W(ψ + )(x) − V(

∂ψ + )(x) , x ∈ ΩD , ∂n

(6.11.45)

where E(x, y) = 1/(4π|x − y|) is the fundamental solution of −Δ. The Cauchy data (ψ + , ∂ψ + /∂n)T of the solution ψ satisfy the relations: ⎛ + ⎞ ⎛ 1 ⎞⎛ + ⎞ ψ ψ −V 2I + K ⎝ ⎠=⎝ ⎠⎝ ⎠ on Γc , (6.11.46) 1  ∂ψ + ∂ψ + I − K −D 2 ∂n ∂n where the boundary integral operators K, V, D, K  are related to the limits of the boundary potentials V and W from ΩD which are defined explicitly in

374

6. Electromagnetic Fields

Section 1.4.1. Their mapping properties are given in Theorems 1.2.3 and 5.6.2. In terms of the transmission conditions (6.11.44)5 , we have from (6.11.46) 1 1 ψ + (x) = ( I + K)ψ + (x) − V (n · curl A− )(x) , x ∈ Γc , 2 μ0 1 1 0 = D(ψ + )(x) + ( I + K  )(n · curl A− )(x) , x ∈ Γc . 2 μ0

(6.11.47) (6.11.48)

As will be seen, in the transmission condition (6.11.44)4 , ψ + is replaced by the right hand of Equation (6.11.47), while Equation (6.11.48) serves as a nonlocal boundary condition and it is a boundary integral equation of the + first kind for the unknown ψ + provided n · μ−1 0 curl A is given. We are now in a position to define the nonlocal boundary value problem which are equivalent to the eddy current problem in the unbounded domain formulated in (6.11.44). Let us first specify the function spaces in the domain Ωc and on the Lipschitz boundary Γc for the unknowns (A, ϕ, ψ + ) ∈ Z × H 1 (Ωc )/C × H 1/2 (Γc )/C, where the function space Z is defined as in Section 6.11.1 by Z = H(curl , Ωc ) ∩ H0 (div, Ωc ) := {Z ∈ L2 (Ωc ) : Z ∈ L2 (Ωc ), div Z ∈ L2 (Ωc )

and Z · n = 0

on Γc }

equipped with the norm 2 Z||Z := Z20,Ωc + divZ20,Ωc + curl Z20,Ωc .

The nonlocal boundary value problem for the eddy current problem in the unbounded domain now reads: Given J0 ∈ L2 (Ωc ), find (A, ϕ, ψ + ) ∈ E × H 1 (Ωc )/C × H 1/2 (Γc )/C such that ⎧ 1 1 ⎪ ⎪ curl curl A − grad div A ⎪ ⎪ μc μmax ⎪ ⎪ ⎪ ⎪ ⎪ −iσ ω(A + grad ϕ) = J0 in Ωc , ⎪ ⎪ ⎪ ⎪ ⎪ div {−iσ ω(A + grad ϕ)} = div J0 in Ωc , ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ +  − ⎪ ⎨μ0 D(ψ ) + ( I + K )(n · curl A ) = 0 on Γc , 2 (6.11.49) 1 ⎪ ⎪ curl A− = − curl Γc ψ + on Γc with n× ⎪ ⎪ μc ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ψ + = ( I + K)ψ + − V (n · curl A− ) on Γc , ⎪ ⎪ 2 μ ⎪ 0 ⎪ ⎪ ⎪ ⎪ on Γc , n · {−iσω(A− + grad ϕ− )} = n · J− ⎪ 0 ⎪ ⎪ ⎩ − n · A = 0 on Γc .

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

375

We now consider the existence theorem for the weak solution of the nonlocal boundary value problem. To obtain a variational formulation of the nonlocal boundary value problem (6.11.49), notice that from (6.11.49)1 , and (6.11.49)4 , we have that 1 μc curl A ∈ H(curl , Ωc ). Multiplying (6.11.49)1 by the test function Z ∈ Z and integrating by parts lead to    1 1 J0 · Z dx = ( curl A) · (curl Z) dx + div A div Z dx μc μmax Ωc Ωc Ωc   1 − iω σ(A + grad ϕ) · Z dx + (n × curlΓ A) · Z ds μc Ωc Γc  1 − div A(n · Z)ds. (6.11.50) μmax Γc

Since Z ∈ Z , the last term on the right hand side vanishes, and from (6.11.49)4 , we see that    1 + (n × curl A) · Z ds = − curl Γc ψ · Z ds = − ψ + curlΓc Z ds μc Γc Γc Γc    1 1 + − =− curl Z · ndx I + K)ψ − V (n · curl A (6.11.51) 2 μ0 Γc

.

= − n · curl Z,

1 2

I + K)ψ + −

/ 1 V (n · curl A− . μ0 Γc

Recall the notation  , Γc denotes the duality pairing between H −1/2 (Γc ) and H 1/2 (Γc ) (cf. Section 4.5). Similarly, multiplying (6.11.49)2 by the test function u ∈ H(Ωc )/C and integrating by parts, we obtain   div J0 u dx = div {−iσ ω(A + grad ϕ)}u dx  = iω

Γc

Γc



σ(A + grad ϕ) grad u dx − i ω

Ωc

σn · (A + grad ϕ)u ds. Γc

Hence, using (6.11.49)5 , we obtain  −i ω

 σ(A + grad ϕ) grad u dx = −i ω

Ωc

 − Ωc



σn · (A + grad ϕ)u ds Γc

J0 · grad u dx.

div J0 udx = Ωc

(6.11.52)

376

6. Electromagnetic Fields

Finally, multiplying (6.11.49)3 by the test function η ∈ H 1/2 (Γc )/C, and integrating over the boundary Γc yield .

μ0 Dψ + +

1 2

I + K

  / n · curl A , η

= 0.

(6.11.53)

Γc

Collecting (6.11.50) , (6.11.51), (6.11.52) and (6.11.53), we have the weak formulation of (6.11.49): For given J0 ∈ L2 (Ωc ), find (A, ϕ, ψ + ) ∈ Z × H 1 (Ωc )/C × H 1/2 (Γc )/C such that  ⎧ 1 1 ⎪ curl A) · (curl Z) dx + (div A) (div Z) dx ( ⎪ ⎪ ⎪ μc μmax ⎪ ⎪ Ωc ⎪Ωc ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ − iω σ, (A + grad ϕ) · Z dx ⎪ ⎪ ⎪ ⎪ ⎪ Ωc ⎪ ⎪  ⎨ .  / 1 1  + n · curl Z , −( I + K ψ + + V n · curl A = J0 · Z dx, ⎪ 2 μ0 Γc ⎪ ⎪ ⎪ Ωc ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ − iω σ(A + grad ϕ) · (grad u) dx = J0 · grad u dx, ⎪ ⎪ ⎪ Ωc ⎪ ⎪ Ωc ⎪ ⎪ ⎪ .   / 1 ⎪ ⎪ +  ⎩ μ0 Dψ + n · curl A , η I +K =0 2 Γc (6.11.54) for all (Z, u, η) ∈ H := Z × H 1 (Ωc )/C × H 1/2 (Γc )/C . In order to discuss existence and uniqueness of the weak solution, we rewrite (6.11.54) in a more compact form: A ((A, ϕ, ψ), Z, u, η) =  ((Z, u, η))

for all (Z, u, η) ∈ H ,

(6.11.55)



where

J0 · (Z + grad u) dx

 ((Z, u, η)) := Ωc

is the linear functional and A is the sesquilinear form defined on H by

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

377

A ((A, ϕ, ψ), (Z, u, η))    1 1 := ( curl A) · (curl Z) + (div A)(div Z) dx μc μmax Ωc  − iω σ(A + grad ϕ) · (Z + grad u)dx .

Ωc

  / 1 . 1 / − n · curl Z, I + K ψ+ I + K  (n · curl A) , η + 2 2 Γ0 Γ0 . . / / 1 + + n · curl Z, V (n · curl A) + μ0 Dψ , η . μ0 Γc Γc (6.11.56) We now state and prove the main result in this section. Theorem 6.11.5. (Rodriguez and Valli [366, Theorem7.5]) The sesquilinear form A (·, ·) in (6.11.56) is continuous on H Γc = (Z, u, η) ∈ H with    udx = 0 and ηds = 0 satisfying Ωc

Γc

A((A, ϕ, ψ), (Z, u, η))| ≤ M (A, ϕ, ψ)H Γc (Z, u, η)H Γc . |A

(6.11.57)

Moreover, for (Z, u, η) ∈ HΓc , we have the coerciveness estimate 2 A((Z, u, η), (Z, u, η))| ≥ α(Z, u, η)H |A Γc

for all (Z, u, η) ∈ HΓc (6.11.58) with α > 0. Consequently, there is a unique weak solution of (6.11.55) in the function space HΓc . Proof: In order to apply the Lax-Milgram lemma to establish the existence and uniqueness of the weak solution of (6.11.55), we need to check the continuity of the sesquilinear form A (··) and the boundedness of the functional  (·). These are clear and in the following, we only concentrate on the coercivity of A (··). We begin with the examination of the real part of A ((Z, u, η), (Z, u, η)), Re A ((Z, u, η), (Z, u, η)) containing the following terms:   1 1 1 2 ( curl Z) · (curl Z) dx, + (div Z) (div Z) dx ≥ k1 ZZ , μc μmax μmax Ωc

Ωc

/ 1. 1 V (n · curl Z), (n · curl Z) ≥ k2 n · curl Z2H −1/2 (Γc ) , μ0 μ0 Γc . / μ0 Dη, η ≥ μ0 k3 η2H 1/2 (Γc ) . (6.11.59) Γc

For the inequality of (6.11.59)1 , see for instance Girault and Raviart [147, Lemma I.3.6], while for inequalities (6.11.59)1 and (6.11.59)2 see Section 5.5. Thus, we have the estimates:

378

6. Electromagnetic Fields

| Re A ((Z, u, η), (Z, u, η))| 1 2 k1 ZZ + k2 n · curl Z2H −1/2 (Γc ) + k3 η2H 1/2 (Γc ) ≥ μmax   2 ≥ k4 ZZ + n · curl Z2H −1/2 (Γc ) + η2H 1/2 (Γc ) , (6.11.60) , k , k }. where k4 := min{k1 /μ−1 max 2 3 For the imaginary part of A ((Z, u, η), (Z, u, η)), it is a little more involved. In view of the negative sign −iω in (6.11.56), we begin with − Im A ((Z, u, η), (Z, u, η))  = ω σ (Z + grad u) · (Z + grad u) dx Ωc

 / . 1 I +K η + 2 Im n · curl Z, 2 Γ0   (|grad u|2 + |Z|2 ) + 2 ≥ ω σmin Ωc

.

+ 2 Im

n · curl Z,

1 2



(6.11.61)

Re (Z · grad u) dx

 / I +K η . Γ0

We now examine the real and imaginary terms in (6.11.61) separately as follows.     2 Re (Z · grad u) dx ≤ 1 grad u20,Ω + 2 Z20,Ω , (6.11.62) c c 2 Ωc

while for the imaginary term there exists a constant k5 > 0 such that .  / 1 2| Im n · curl Z, I +K η | ≤ k5 ηH 1/2 (Γc ) n · curl ZH −1/2 (Γc ) 2 Γ0 k5 k5 ≤ η2H 1/2 (Γc ) + n · curl Z2H −1/2 (Γc ) . 2 2 (6.11.63) Substituting (6.11.62) and (6.11.63) into (6.11.61) yields the estimate: − Im A ((Z, u, η), (Z, u, η)) 1 ≥ ωσmin grad u20,Ωc − ωσmin Z20,Ωc 2 k5 k5 − η2H 1/2 (Γc ) − n · curl Z2H −1/2 (Γc ) . 2 2

(6.11.64)

From the Poincar´e inequality, there is a constant k6 > 0 such that grad u20,Ωc ≥ k6 u2H 1 (Ωc ) .

(6.11.65)

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

These together with (6.11.64) imply that    − Im A ((Z, u, η), (Z, u, η)) 1 ≥ ωσmin k6 u2H 1 (Ωc ) − ωσmin Z20,Ωc 2 k5 k5 − η2H 1/2 (Γc ) − n · curl Z2H −1/2 (Γc ) . 2 2

379

(6.11.66)

Then for any positive parameter q, we see that A(Z, u, η), (Z, u, η))| (1 + q)|A ≥ | Re A ((Z, u, η), (Zu, η))| + q| − Im A ((Z, u, η), (Z, u, η))| k6 2 + ωσmin qu2H 1 (Ωc ≥ (k4 − ω σmin q) ZZ 2 k5 k5 2 + (k4 − q)ηH 1/2 (Γc ) + (k4 − q)n · curl Z||2H −1/2 (Γc ) . 2 2

(6.11.67)

Let 0 < q < 1 be sufficiently small so that for fixed k4 , k5 , and k6 , we require that k5 (k4 − ω σmin q) > 0 and (k4 − q) > 0. 2 Hence if we choose any q such that 0 < q < min{

2k4 k4 , } ωσmin k5

and define α :=

1 k6 k5 min{k4 − ωσmin q, ωσmin q, k4 − }, 1+q 2 2

we arrive at the desired result (6.11.58). This completes the proof.



Remark 6.11.2: We should remark that our time-harmonic Maxwell equations formulation are in terms of the factor e−iω t , and hence the timeharmonic Faraday equation (see for instance (6.11.68)1 ) has a different sign from conventional engineering literature.

6.11.3 Electric field in the dielectric domain Ω D . It ist noteworthy to mention that in the conductor domain Ω, once the magnetic vector potential A and electrical potential ϕ are found, the electric field E is known and given by formula (6.11.35), namely Ec = −iω(A + grad ϕ) in Ωc

380

6. Electromagnetic Fields

However, for the eddy current problem in the dielectric domain ΩD , E is still unknown, and we need to find E in ΩD by solving the following system: ⎧ curl E − i ω μ0 H = in ΩD , ⎪ ⎪ ⎪ ⎪ ⎪ div (ε0 E) = 0 in ΩD , ⎪ ⎪ ⎪ ⎪ ⎪ n × E = n × Ec on Γc , ⎪ ⎪ ⎨  (6.11.68) n · (ε0 E) dx = 0, ⎪ ⎪ ⎪ ⎪ Γc ⎪ ⎪ ⎪ ⎪ ⎪ E(x) = O(|x|−1 ) as |x| → ∞, ⎪ ⎪ ⎪ ⎩ curl E(x) = O(|x|−1 ) as |x| → ∞. This is an exterior Dirichlet problem for the electric field E. It seems that it is best to reduce the problem to a boundary integral equation on Γc . To do so, first set  H(y) dy, Ep (x) := i ωμ0 curl 4π|x − y| ΩD

and define E0 := E − Ep . Then we see that curl curl E0 = 0,

and div E0 = 0.

This simplifies the problem for E0 by solving a modified exterior problem in the exterior domain ΩD : ⎧ in ΩD , curl curl E0 = 0 and div E0 = 0 ⎪ ⎪ ⎨ on Γc πt E0 = πt (Ec − Ep ) := f (6.11.69) E0 = O(|x|−1 ) as |x| → ∞. ⎪ ⎪ ⎩ curl E(x) = O(|x|−1 ) as |x| → ∞. In order to solve (6.11.69) by boundary integral equations, it is useful, if we have some information concerning the subject of vector potentials for the Laplacian (6.11.70) ΔE0 := grad (divE0 ) − curl curl E0 . in three–dimensional Lipschitz-domains. A short digression to vector potentials. Let Γ be a closed Lipschitz boundary in R3 . We denote its interior and exterior domains by Ω − and Ω + , respectively. Suppose that u is a vector field satisfying Δu = 0,

and div u = 0 in R3 \ Γ

u = O(|x|−1 ) and curl u = O(|x|−1 ) as |x| −→ ∞.

(6.11.71)

Based on the formula (6.11.70) in an analogue to the Stratton-Chu formulas for the Maxwell’s equations, we have the representation formula for the solution of (6.11.71) in the form

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

 u(x) =

 − grad

 G(x − y)[n × u]Γ dsy +

curl

381

Γ

G(x − y)[n × curl u]Γ dsy Γ

G(x − y)[n · u]Γ dsy for x = Γ,

(6.11.72)

Γ

where the fundamental solution for the - Laplacian in R3 is denoted by G(x − y) =

1 1 . 4π |x − y|

λ]Γ = λ + − λ − stands In the representation formula (6.11.72), the notation [λ + − for the jump of λ across Γ from Ω and Ω , respectively. Similar to the scalar case, we now introduce the potentials for x = Γ :  ⎧ ⎪ ⎪ λ λ(y) dsy , vectorial single layer potential , Vλ (x) := G(x − y)λ ⎪ ⎪ ⎪ ⎪ ⎪ Γ ⎪ ⎪  ⎪ ⎨ μ(x) := curl μ)(y) dsy , vectorial double layer potential , Wμ G(x − y)γt (μ ⎪ ⎪ Γ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ Qϕ(x) := grad G(x − y)ϕ(y)dsy . nonstandard potential , ⎪ ⎪ ⎩ Γ

(6.11.73) together with the vectorial Newton potential:  Nf (x) := G(x − y) f (y)dy,

(6.11.74)

R3

which defines a continuous mapping 1 3 3 N : H−1 comp (R ) −→ Hloc (R ).

As we know from (4.2.30), the simple layer potentials and the Newton potenλ) = Nγ ∗0 λ = N(λ λ ⊗δΓ ). tials are related for special distributions, namely, V(λ For smooth test function ϕ ∈ D (R3 ), it follows that λ, ϕ )L23 = (Nλ λ ⊗ δΓ ) , ϕ )R3 = λ λ, πt (Nϕ ϕ)Γ = (Nπt∗ (λ λ), ϕ )R3 . (V(λ R

Hence, we obtain that V∗ = πt ◦ N and V = N ◦ πt∗ by a density argument, and moreover, the mapping

(6.11.75)

382

6. Electromagnetic Fields −1/2

V = N ◦ πt∗ : H||

(Γ ) −→ H1loc (R3 ) −1/2

is continuous, since the L2 -adjoint operator πt∗ : H|| (Γ ) −→ H1loc (R3 ) is continuous (see Figure 6.4 below). −1/2 We note that for λ ∈ H|| (Γ ), if we require that divΓ λ = 0, then λ) = 0 in R3 \ Γ . This follows from the following result. div V(λ −1/2 Lemma 6.11.6. If λ ∈ H|| (divΓ , Γ ), then

 G(x − y) divΓ λ (y) dsy for x ∈ R3 \ Γ.

λ= div Vλ Γ

Proof: The proof of this lemma is based on the Stokes formula,   grad Γ G(x − y) · λ dsy + G(x − y) divΓ λ dsy = 0, Γ

since



Γ

divΓ wdsy = 0 for any tangential field w. (See, e.g. Colton and Kress

Γ



[75, Theorem 2.29] and MacCamy and Stephan [278, Lemma 2.3]).

−1/2 We may now restrict λ to the space H|| (divΓ , Γ ) with divΓ λ = 0 , −1/2

λ(x) = 0 for x = which will be denoted by H|| (divΓ 0, Γ ) . Since ΔVλ λ)(x) = 0 as well for x = Γ . In view of Γ , it follows that curl curl V(λ λ)(x) = 0, we apply Green’ s formula to V(λ λ) in Ω ∓ with the curl curl V(λ  ∞ 2 3 test function ϕ ∈ C0 (R ) to obtain  . / . / λ) · curl curl ϕ dx = [γtV (λ λ)]Γ , πt curl ϕ − [γt curl V(λ λ)]Γ , πtϕ V (λ t

R3

. / λ)]Γ , πtϕ , = − [γt curl V (λ

(6.11.76)

t

λ) = 0, the left–hand side of and by making use of the condition div V(λ (6.11.76) is also equal to  . / λ) · Δϕ ϕ dx = − λ, (V∗ ◦ Δ) ϕ . − V(λ t

R3

This implies that / . / . / . λ))]Γ , πtϕ = λ , (V∗ ◦ Δ)ϕ ϕ = − λ , πtϕ , [γt (curl V(λ t

t

t

the last step following from the definition of V∗ in (6.11.75). Hence,

t

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

λ))]Γ = −λ λ in H−1/2 [γt (curl V(λ (divΓ , Γ ), ||

383

(6.11.77)

3  −1/2 since C0∞ (R3 ) is dense in H(curl , Ω ∓ ) and πt u ∈ H⊥ (curl Γ , Γ ) ∓ for u ∈ H(curl , Ω ) . As will be seen, the property of the vectorial double–layer potential will depend upon properties of the vectorial simple–layer potential V as well as the nonstandard potential Q. So, let us first examine the property of Q. This non–standard potential has the form Q (ϕ) = grad V(ϕ), where V(ϕ) is the scalar simple layer potential and from the property of V in Chapter 5 we can conclude that Q : H −1/2 (Γ ) −→ H(curl , R3 ) is a continuous operator.

πt H 1loc (R3 )

1/2

H || (Γ) j −1

N π ∗t

3 H −1 comp (R )

−1/2

H ||

(Γ)

Fig. 6.4. Mapping property of the simple–layer potential

It remains now to consider the vectorial double-layer potential. The vectorial double layer potential for a tangential vectorial field μ is defined by μ) := curl V (γtμ )). W(μ

(6.11.78)

μ), then μ = [πt (u)] ∈ H−1/2 (Γ ) for u ∈ It can be shown that if u = W(μ ⊥ H(curl , Ω − ) ∪ Hloc (curl , R3 ). From the mapping property of the vectorial simple-layer operator V, this implies that −1/2

W : H⊥

(Γ ) −→ H(div, R3 )

(6.11.79)

is a continuous operator. For obtaining further properties of the double layer potential, similar to the scalar case in Chapter 5, we need some information

384

6. Electromagnetic Fields

concerning the solution of the aforementioned Dirichlet problem of the transmission problem consisting of (6.11.71) together with the given Dirichlet boundary condition: −1/2

πt (u) = f ∈ H⊥

(curl Γ , Γ ) on Γ,

(6.11.80)

To this end, let S be the solution operator of the Dirichlet problem on both sides of Γ defined by −1/2

S : H⊥

(curl Γ , Γ ) → W (curl 2 , Ω − ∪ Ω + ) ∩ H(div 0; Ω − ∪ Ω + ) : S : f → u = S(f ),

(6.11.81)

where W (curl , Ω) and W (curl 2 , Ω) are weighted Sobolev spaces defined by   v ∈ L2 (Ω), curl v ∈ L2 (Ω) W (curl , Ω) := v ∈ D(Ω) | 8 1 + |x|2   W (curl 2 , Ω) := v ∈ W (curl , Ω)|curl curl v ∈ L2 (Ω) (6.11.82) (cf. Section 4.4 for the weighted Sobolev space Hm (Rn ; λ ) with n = 3, m = 1. λ = 0, κ = −1). Then from (6.11.72), we have the representation of the solutions in the form:     S(f ) = ±W(f ) + V [γt ( curl S(f ) )]Γ − grad V [γn (S(f ))]Γ in Ω ± , (6.11.83) from which, the vectorial double-layer potential W can be explicitly expressed in terms of the solution operator S. For instance, in Ω + ,   W = I − V ◦ γt ◦ curl − grad ◦ V ◦ γn ◦ S. This indicates that the properties of W can be deduced from those of S. A careful analysis with the help of continuity properties of potentials V and Q and of the trace operators, one can show that −1/2 W : H⊥ (curl Γ ; Γ ) −→ H( div 0 ; Ω ± ) ∪ H(curl ; Ω ± )

is continuous. Now we are in a position to define the boundary integral operators from the potentials defined in (6.11.73). We consider here only the simple, doublelayer and the nonstandard boundary integral operators, πt (V), πt± (W) and πt± (Q). We summarize the results in the following theorem. Theorem 6.11.7. The following boundary integral operators are continuous: ⎫ 1/2 V) : H−1/2 πt (V (divΓ 0, Γ ) −→ H|| (divΓ 0, Γ ) ⎪ ⎪ || ⎬ −1/2 −1/2 ± (6.11.84) W) : H⊥ (curl Γ , Γ ) −→ H⊥ (curl Γ , Γ ) πt (W ⎪ ⎪ ⎭ −1/2 ± −1/2 πt (Q) : H (Γ ) −→ H⊥ (Γ ).

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

385

Proof: The continuity properties for the vectorial simple–layer and double– layer boundary integral operators, πt (V) and πt± (W) should be clear. We shall consider here only the nonstandard boundary integral operator πt± (Q): Vϕ)(x) = grad Γ (V ϕ)(x), for x ∈ Γ, πt± Q(ϕ)(x) = πt± grad (V  where (V ϕ)(x) = G(x − y)ϕ(y)dsy is the scalar simple–layer boundary Γ

integral operator:

V : H −1/2 (Γ ) → H 1/2 (Γ ).

Then, as a mapping, −1/2

πt± (Q) : ϕ ∈ H −1/2 (Γ ) −→ grad Γ V ϕ ∈ H⊥

(curl Γ , Γ ),

−1/2

is continuous, and since grad Γ V ϕ ∈ H⊥ (Γ ), this completes the proof of the desired continuity property of the nonstandard boundary integral opera tor πt± Q(ϕ) (cf. Section 4.5). It should be mentioned that as in the scalar case, there are two other λ) and γt± (curl W)(μ μ), vectorial boundary integral operators, γt± (curl V)(λ the vectorial adjoint and hypersingular boundary integral operators. These boundary integral operators will be needed for the Neumann problems, which will not be included in this short digression; however, their development should be followed in the same manner as in the scalar case. For readers interested in these operators, we refer them to the comprehensive paper by Claeys and Hiptmair [72]. In closing this short digression, we now state and prove the following useful theorem for the Dirichlet problem to be considered in (6.11.68). λt , induced by the boundλ, πt Vλ Theorem 6.11.8. The sesquilinear form λ −1/2 ary integral operator πt V is H|| (div0, Γ ) − elliptic, that is, there exists a positive constant α such that . / −1/2 λ ≥ αλ λ2 −1/2 e λ, πt Vλ for all λ ∈ H|| (div 0, Γ ). (6.11.85) H (div0,Γ ) t



−1/2 λ in Ω − ∪ Ω + . Then Proof: For given λ ∈ H|| (div 0, Γ ), let u = Vλ u ∈ W (curl 2 , Ω − ∪ Ω + ) is a solution of the transmission problem:

curl curl u = 0, div u = 0 in Ω − ∪ Ω + , [πt u]Γ = 0 and [γt (curl u)]Γ = λ u = O(|x|−1 ), curl u = O(|x|−1 ) as |x| −→ ∞. An application of Green’s formula yields . / 0 / λ, πt Vλ λ = [γt u], πt u = curl u2L2 (Ω − ∪Ω + ) .

(6.11.86)

386

6. Electromagnetic Fields

To complete the proof, we need a similar generalized first Green’s formula (e.g. Lemma 5.1.1) to dominate the norm of λ . In particular, for −1/2 v ∈ H⊥ (curlΓ , Γ ), Green’s formula implies that |γt (curl u), πt Zvt | = |(curl u, curl Zv)L2 (Ω) | ≤ ccurl uL2 (Ω) vH−1/2 (curl Γ ,Γ ) . ⊥

Thus, γt (curl u)H−1/2 (divΓ , Γ ) = ||

|γt (curl u), πt Zvt |

sup v

H

=1 −1/2 (curl Γ ,Γ ) ⊥

≤ ccurl uL2 (Ω) for Ω = Ω ± , where −1/2

Z : H⊥

(curl Γ , Γ ) → H(curl , Ω)

is a continuous right inverse of the operator πt . Consequently, λ2 −1/2 λ H (div ||

Γ

, Γ)

= [γt (curl u)]2H−1/2 (div ||

Γ

, Γ)

≤ ccurl u2L2 (Ω)

and we obtain the result with α = c−1 .



We now return to the problem for the electric field in the dielectric domain ΩD defined by (6.11.69). Let f = πt (Ec −Ep ) be given. We are going to solve the problem by using the direct method in terms of the boundary integral equation of the first kind. For given Dirichlet boundary data πt (E0 ) = −1/2 f ∈ H⊥ (curl Γ , Γ ), the solution assumes a representation in the form   λ(y)dsy − grad x G(x − y)(n · E0 )(y)dsy E0 (x) = G(x − y)λ Γ

Γ



G(x − y)(n × f )dsy ,

+ curl x

(6.11.87)

Γ −1/2

where λ = γt (curl E0 ) ∈ H|| (Γ ) and ϕ = n · u ∈ H −1/2 (Γ ) are the unknowns. The corresponding boundary integral equation is a first kind boundary integral equation for λ and ϕ in the form: λ)(x) − grad Γ V (ϕ)(x) = q(x) , πt (Vλ

for x ∈ Γ,

(6.11.88)

where q is given by 1 q(x) := − f + πt curl 2



−1/2

G(x − y)(n × f )dsy ∈ H⊥ Γ

(curl Γ , Γ ).

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

387

For the weak solution of (6.11.88), multiplying with test functions ξ ∈ −1/2 H|| (divΓ , Γ ) yields the variational equation: λt − grad Γ V ϕ, ξ t = q, ξ t , ∀ ξ ∈ H−1/2 ξξ , πt Vλ (divΓ , Γ ) ||

(6.11.89)

It is worth mentioning that (6.11.89) can not be solved without an additional condition, since in contrast to the EFIE in electromagnetics, it contains more than one unknown (while the EFIE has n × H and n·E as the unknowns, but the latter can be replaced by divΓ (n × H)) . One way to circumvent this difficulty is to solve the equation (6.11.89) by a two–step procedure. Note that from the Stokes formula, we obtain grad Γ V ϕ, ξ t = −V ϕ, divΓ ξ t , where divΓ ξ ∈ H −1/2 (Γ ). Here are the two steps: first testing (6.11.89) by −1/2 −1/2 ξ ∈ H|| (divΓ 0, Γ ) and next testing the equation by ξ ∈ H|| (divΓ , Γ ) , we obtain two equations as follows: −1/2

λt = q, ξ t , ξξ , πt Vλ

∀ ξ ∈ H||

V ϕ, divΓ ξ t = q, ξ  − ξξ , πtVλ t ,

∀ ξ ∈ H||

−1/2

(divΓ 0, Γ )

(6.11.90)

(divΓ , Γ ) −1/2

The first equation in (6.11.90) is uniquely solvable for given q ∈ H⊥ (curl Γ , Γ ) −1/2 in view of Theorem 6.11.8. After solving for λ ∈ H|| (divΓ , Γ ), the second equation in (6.11.90) is uniquely solvable for ϕ as well for given λ ∈ −1/2 −1/2 H|| (divΓ , Γ ), and for given q ∈ H⊥ (curl Γ , Γ ). Since V is H −1/2 (Γ )– elliptic. In this way, we obtain the complete electric field E = Ec + Ep in the domain ΩD . Coupling procedures for eddy current problems in unbounded domains have been developed by many individuals directly in terms of magnetic fields H ( see for instance Hiptmair [181], Meddahi and Selgas [292], Girault and Raviart [147] and Malgrange [280]). 6.11.4 Vector potentials — revisited We remark that the electric filed E0 in the dielectric field satisfying a special form of the three–dimensional vector potential equation or the Hodge– Laplace equation in R3 . The two–step method for solving the electric field in the dielectric domain ΩD is in fact equivalent to solve a system of three simple–integral equations of the first kind for the three unknowns γt (curl E0 ) and γn (E0 ) from the saddle–point of view in Claeys and Hiptmair [72]. This motivates us following Claeys and Hiptmair [72] to re-examining the Hodge– Laplace equation as follows:

388

6. Electromagnetic Fields

Again let Γ be the closed Lipschitz boundary in R3 . We denote its inside and outside domains by Ω − and Ω + , respectively. Suppose u is a vector field satisfying the Hodge–Laplace equation ΔHL u := grad div u − curl curl u = 0

in

Ω − ( or Ω + ).

(6.11.91)

We begin with the solution u in Ω − . For u ∈ H(curl , Ω − ) ∩ H(div, Ω − ), from the Green formula, u admits the representation:   u(x) = − G(x − y)(n × curl u) dsy + grad x G(x − y)(n · u) dsy Γ



− curl x

 G(x − y) (n × πt u) dsy +

Γ

Γ

G(x − y)n(y) div u(y) dsy Γ

x ∈ Ω−, (6.11.92) where VHL and WHL are the corresponding single– and double–layer potentials for the Hodge–Laplacian defined by  ⎧ ⎪ VHL (γt curl u, γn u)(x) := − G(x − y)(n × curl u) dsy ⎪ ⎪ ⎪ ⎪ ⎪ Γ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ + grad G(x − y)(n · u) dsy ⎪ x ⎪ ⎪ ⎨ Γ (6.11.93) ⎪ ⎪ ⎪ WHL (πt u, div u)(x) := curl x G(x − y) (n × πt u) dsy ⎪ ⎪ ⎪ ⎪ ⎪ Γ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ − G(x − y)n div u dsy . ⎪ ⎩ = VHL (γt curl u, γn u)(x) − WHL (πt u, div u)(x),

Γ

Here in contrast to the scalar  Laplacian, we have four Cauchy data :  −1/2 −1/2 πt u, div u, γt curl u, γn u ∈ H⊥ (curl Γ , Γ )×H 1/2 (Γ )×H (divΓ , Γ )× H −1/2 (Γ ). We now define four basic boundary integral operators as in the case of the scalar Laplacian:  −    γ curl u πt V t (x) := VHL (γt curl u, γn u) div γn u   ⎞ ⎛ −πt G(x − y)(n × curl u) dsy + grad Γ G(x − y)(n · u) dsy Γ Γ ⎠  =⎝ − G(x − y) divΓ (n × curl u) dsy Γ  π  + t := VHL (γt curl u, γn u) , div

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

389



    πt u πt K WHL (πt u, div u ) (x) := div div u   ⎛ ⎞ πt curl x G(x−y)(n × πt u) dsy − πt G(x − y) n div u dsy Γ ⎜ ⎟ Γ \{x}  =⎝ ⎠ −divx G(x − y)n div u dsy Γ \{x}

(6.11.94)     γt curl u γt curl K VHL (γt curl u, γn u) (x) := γn u γn  ⎛ ⎞ G(x − y)(n × curl u) dsy −γt curl ⎜ ⎟  Γ \{x}  =⎝ −n · G(x − y)(n × curl u)dsy + (∂G(x − y)/∂nx )(u · n)dsy ⎠ 



Γ \{x}

Γ

−   γt curl πt u (x) := − WHL (πt u, div u) D div u γn ⎞ ⎛   curl Γ G(x−y)divΓ (n × πt u)dsy +n × G(x−y) curl Γ (divu)dsy ⎟ ⎜ Γ   Γ  =⎝ ⎠ divΓ n× G(x−y)(n)×(πt u)(y)dsy +n · G(x − y)ndivudsy 



 =−

Γ

γt curl γn



+

Γ

WHL (πt u, div u)

In the formulations above, we have adopted the notations:  1 {v}− + {v}+ . 2 A simple computation shows that the Cauchy data from Ω − on Γ lead to the equations ⎛ ⎛ ⎞ ⎞ πt u πt u   1 ⎜ divu ⎟ ⎜ divu ⎟ V 2 I − K, ⎜ ⎜ ⎟ ⎟ 1  ⎝γt curl u⎠ = ⎝γt curl u⎠ on Γ. (6.11.95) I + K D, 2 γn u γn u {v}∓ := γ ∓ v and {{v}} :=

For any solution of (6.11.91) in Ω − , the Cauchy data on Γ are reproduced by operators on the right–hand side of (6.11.95), namely by   1 I − K, V 2 , (6.11.96) C Ω − := 1  D, 2I + K which therefore defines the Calderon projector C Ω − for the Hodge–Laplacian with respect to Ω − . Similarly, for the solution u of (6.11.91) in the exterior domain Ω + , if u ∈ Hloc (curl , Ω + ) ∩ Hloc (div, Ω + ) satisfying the conditions at infinity u = O|x|−1 ) and curl u = O(|x|−1 ) as |x| −→ ∞,

(6.11.97)

390

6. Electromagnetic Fields

then u can be represented in the form     u(x) = curl x G(x − y)(n × πt u)dsy − G(x − y)n(y) div u(y) dsy −



Γ Γ   − G(x − y)(n × curl u)dsy + grad x G(x − y)(n · u)dsy Γ

Γ

= WHL (πt u, div u)(x) − VHL (γt curl u, γn u)(x),

for x ∈ Ω + . (6.11.98) In the same manner we may consider the Cauchy data from Ω + on Γ , which leads to the equations ⎛ ⎛ ⎞ ⎞ πt u πt u   1 ⎜ divu ⎟ ⎜ divu ⎟ −V 2 I + K, ⎜ ⎜ ⎟ ⎟ 1  ⎝γt curl u⎠ = ⎝γt curl u)⎠ on Γ. (6.11.99) −D, 2I − K γn u γn u Clearly, C Ω+ = I − C Ω− . As in the case for the scalar Laplacian, we may consider two basic boundary value problems for the Hodge–Laplacian in Ω + consisting of either boundary conditions of the Dirichlet type or boundary conditions of the Neumann type. In both cases, from (6.11.99), the boundary value problems can be reduced to a system of boundary integral equations of the first kind. Since our purpose is to find the electric field in the dielectric by solving the exterior problem (6.11.69), which belongs to the Dirichlet type of boundary value problem for the Hodge–Laplacian, in the following we shall only consider the Dirichlet type boundary value problem. For the Neumann type boundary value problem, we refer to the papers of Claeys and Hiptmair [72], and for natural boundary–value problems see Taylor [415, p.361]. For a comprehensive treatment of the general Hodge Laplacian, we refer to the monograph of Mitrea et al. [311]. The Dirichlet type boundary value problem states: Given −1/2

f ∈ H⊥

(curl Γ , Γ ) and g ∈ H 1/2 (Γ )

find u ∈ Hloc (curl , Ω + ) ∩ Hloc (div, Ω + ) satisfying (6.11.91) in Ω + , and the boundary conditions πt u = f and div u = g on Γ together with the conditions at infinity (6.11.97). We seek a solution in the form (6.11.98). This reduces to the boundary integral equation system of the first kind for the unknowns γt curl u and u · n from ( 6.11.99):      u  1 1 πt f γt curl u = −( I − K) (6.11.100) =− I −K V γn u div u g 2 2

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

391

since πt u = f and divu = g on Γ . This is a direct boundary integral formulation and hence the right–hand side of (6.11.100) is always in the range of V. The weak formulation of (6.11.100) now reads: −1/2 Given (f , g) ∈ H⊥ (curl Γ , Γ ) × H 1/2 (Γ ), find (γt curl u, u·n) −1/2 ∈ H (divΓ , Γ ) × H −1/2 (Γ ) such that 

   /  f  γ . 1 γt curl u γ curl v / curl v I −K , t , t =− g γn u γn v γn v 2 (6.11.101) −1/2 for all (γt curl v, v · n) ∈ H (divΓ , Γ ) × H −1/2 (Γ ). Or componentwise, we have explicitly ⎧. / . / ⎪ ⎨ γt curl v, V (γt curl u) + div Γ (γt curl v), V (u · n) = 1 (γt curl v) . / ⎪ ⎩ (v · n), V (divΓ (γt curl u)) = 2 (v · n) .

V

(6.11.102) ×H (Γ ) where V (ϕ) := for all (γt curl v, v · n) ∈  G(x − y)ϕ(y) dsy , whereas V = V I, and 1 and 2 are functionals defined −1/2 H (divΓ , Γ )

−1/2

Γ

by 

1 (γt curl v) 2 (v · n)

 :=

. 1 2

I −K

/  f  γ curl v , t g γn v

(6.11.103)

Note that this is a system of three equations for three unknowns since γt curl is two–dimensional. As previously mentioned the variational formulation (6.11.101) has a saddle–point structure which can be analyzed by the standard method for the mixed formulations by following Gatica [138] and Claeys and Hiptmair [72, Lemma 3.5 and Corollary 3.6]. Let us denote the σ , p) and the corresponding test functions by unknowns (γt curl u, u·n) by (σ (ττ , q). Then we may rewrite the system in a more familiar form : −1/2 Find (γt curl u, u·n) ∈ H (divΓ , Γ ) × H −1/2 (Γ ) such that 

σ , τ ) + b(ττ , p) = 1 (ττ ), a (σ σ , q) = 2 (q), b (σ

−1/2 for all τ ∈ H (divΓ , Γ ) =: H

for all q ∈ H −1/2 (Γ ) =: Q,

(6.11.104)

where a(•, •) on H × H and b(•, •) on H × Q are two continuous bilinear forms defined by 0 1 σ , τ ) := τ , V(σ σ) , a(σ (6.11.105) 1 0 (6.11.106) b(ττ , q) := q, V (div Γ τ ) , Associated with the bilinear form a(•, •), two linear continuous operators A : H → H and At : H → H are defined by

392

6. Electromagnetic Fields

0

σ, τ Aσ

1 H ×H

0 1 σ , τ ), = σ , Atτ H ×H = a (σ

∀ τ , σ ∈ H.

(6.11.107)

Similarly, associated with the bilinear form b(•, •), two linear operators B : H → Q and B t : Q → H are defined by . / . / t Bττ , q = τ , B q = b(ττ , q), ∀ τ ∈ H, ∀q ∈ Q. (6.11.108)   Q ×Q

H×H

The problem (6.11.104) can also be written as  σ + B t p = 1 in H , Aσ σ = 2 in Q . Bσ

(6.11.109)

We shall consider the formulations (6.11.104) and (6.11.109) to be the same, referring to one or the other according to the convenience. For the existence theorem for the solutions of this problem, we begin with the kernel of the bilinear form b(•, •) (or Ker B) in the space H :   H0 := τ ∈ H | b(ττ , q) = 0, for all q ∈ Q   (6.11.110) = τ ∈ H | divΓ τ = 0 . The bilinear form a(•, •) is coercive on H0 , that is, there exists a constant α0 > 0 such that a(ττ , τ ) ≥ α0 ττ 2H−1/2 (Γ ) = α0 ττ 2H for τ ∈ H0 .

(6.11.111)



Then let A0 be the restriction of A to H0 (or Ker B) which shows that A0 is an isomorphism from H0 to (H0 ) , which is equivalent to saying that there is an α1 > 0 such that inf

sup

τ 0 ∈H0 σ 0 ∈H0

σ0, τ 0) a(σ ττ 0 H ≥ α1 σ 0 H σ

(6.11.112)

which implies Ker At0 = 0 that is, A0 is surjective, and inf

sup

σ 0 ∈H0 τ 0 ∈H0

σ0, τ 0) a(σ ≥ α1 σ 0 H ττ 0 H σ

(6.11.113)

which implies Ker A0 = 0 that is, A0 is injective. σ , q), Before we consider the inf–sup condition for the bilinear form b(σ σ 0 , p0 ) ∈ we note that as in the potential theory for the scalar Laplacian, if (σ H×Q is a solution of the corresponding homogeneous equations of (6.11.104), consider σ 0 , p0 )(x) for x ∈ R3 \ Γ. σ ) = −VHL (σ u(σ

(6.11.114)

6.11 Application of Electromagnetic Potentials to Eddy Current Problems

393

Then u(x) satisfies a transmission problem for ΔHL u(x) = 0 in R3 \ Γ with jumping transmission conditions:   $  curl u % $ u % γt πt  = 0 and  = σ0 .  (6.11.115) div u  γn u p0  Γ

Γ

As a result, it can be shown that σ 0 = 0 and p0 ∈ Q0 ,

(6.11.116)

where Q0 = Ker B t is the kernel of the bilinear form b(•, •) in the space Q:  0 1  Q0 := p∗ ∈ Q | b(ττ , p∗) = div Γ τ , p ∗ = 0, ∀ τ ∈ H , (6.11.117) which is a finite–dimensional subspace of Q whose dimension equals the second Betti number of Γ , together with the Euler–Poincar´e characteristic (see Kress [244, 245, 246], Martensen [283] for κ = 0 Cessenat [63, p.335] and Rodriguez and Valli [366], Claeys and Hiptmair [72, Section 7.1] for κ = 0). This means, from Fredholm’s alternative, the right–hand side of (6.11.100) needs to satisfy the orthogonality condition   . 1  f  0 / 0 I −K , = for all q ∈ Q0 = Ker B t . g q 0 2 Γ This leads to the condition 1 0 1 0 1 1 ( I − K)g, q Γ = g, ( I − K  ) q Γ = 0 for all q ∈ Q0 = Ker B t , 2 2 (6.11.118) where  ∂ G(x − y) q(y) dsy , for x ∈ Γ K  (q)(x) = ∂nx Γ \{x}

is the transposed double–layer boundary integral operator for the three– dimensional scalar Laplacian (see, e.g. Equation (1.2.9) in Chapter 1).  be the orthogonal complement of Q0 . Since Q  equals Now let Q −1/2 divΓ H (divΓ , Γ ), it is a closed subspace of Q and, hence, the range of   = range (B). According to   . For 2 ∈ range (B), we get Q B is closed in Q Boffi, Brezzi and Fortin [34, §4.24], the operator B t is invertible and there is a constant β > 0 such that B t qH ≥ βqQ = βqQ/Q0 for all q ∈ Q. Thus, we have the inf–sup condition inf

sup

0= q ∈Q 0= τ ∈H

b(ττ , q) ≥β>0 ττ H qQ/Q0

(6.11.119)

According to Boffi, Brezzi and Fortin [34, Theorem 4.2.4], as consequences of (6.11.112), (6.11.113) and (6.11.119) we obtain the following existence result.

394

6. Electromagnetic Fields

Theorem 6.11.9. Under the condition (6.11.118), the problem (6.11.104) σ , p) where σ is uniquely determined, and p is determined up has a solution (σ to an element of the finite–dimansional Ker B t . We now return to the problem (6.11.69) for the electric field E0 in the dielectric domain ΩD (i.e. Ω + ). In terms of the notations in the problem σ , p) = (γt curl E0 , E0 · n) for the unknowns. The given (6.11.104), we have (σ data are πt E0 = f and div E0 = g = 0 on the boundary Γc . So this is a special case where 2 = 0 in the second equation of (6.11.104). From Green’s formula, it follows div E0 = 0 on Γc implies div E0 = 0 in ΩD . Hence, if we seek a solution E0 of (6.11.69) in the form σ , p)(x) E0 (x) = WHL (f , 0)(x) − VHL (σ  = curl x G(x − y) (n × f )(y) dsy −



 −

Γ

(6.11.120) 

σ (y) dsy + grad x G(x − y)σ Γ

 G(x − y) p(y)dsy for x ∈ ΩD

Γ

σ , p) satisfies the system of integral equations then (σ  σ , τ ) + b(ττ , p) = f (ττ ), for all τ ∈ H a (σ σ , q) = 0, for all q ∈ Q, b (σ

(6.11.121)

where 1 0 1 f (ττ ) := ( I − K)(f ), τ 2  G(x − y)(n × f )(y) dsy .

with K(f ) := πt curl x Γc \{x}

Or in terms of the operators A, B and B t , the system of boundary integral equations (6.11.109) can be written as ⎧ 1 ⎨Aσ σ + B t p = ( I − K) f in H , 2 (6.11.122) ⎩ σ B = 0 in Q . We end this section by including an existence and uniqueness result for the solutions of (6.11.121) or (6.11.122) as a corollary of Theorem 6.11.9 in the following σ , p) where Corollary 6.11.10. The equations (6.11.121) have a solution (σ σ is uniquely determined, and p is determined up to an element of the Ker(B t ). Moreover, setting

6.12 Applications of boundary integral equations to scattering problems

β :=

sup

inf

0= q ∈Q 0= τ ∈H

b(ττ , q) > 0, ττ H qQ/Q0

395

(6.11.123)

we have σ H ≤ σ

1 f ||H α1

and pQ/Q0 ≤

2Ma f H , α1 β

(6.11.124)

with Ma being the continuity constant such that σ H ττ H , σ , τ )| ≤ Ma σ |a(σ

for all σ , τ ∈ H.

The estimates (6.11.124) are straight forward and their derivations will be omitted. We remark that in this special case, the orthogonal condition (6.11.118) is automatically fulfilled.

6.12 Applications of boundary integral equations to scattering problems (See also the Monograph by D. Colton and R. Kress [75].) In all the basic scattering problems, we are given ω, ε+ μ+ , incident fields i E , Hi and Γ , and we must determine the total fields E = Ei + Es , H = Hi + Hs

(6.12.1)

so that the scattered fields Es , Hs satisfy Maxwell’s equations in Ω c : curl Es = iωμ+ Hs

(6.12.2)

curl Hs = −iωε+ Es

and the Silver–M¨ uller conditions (6.7.2) together with appropriate boundary conditions (see Agranovich [6, p.28]). We consider as an incident field any pair (Ei , Hi ) of either entire solutions of the homogeneous exterior Maxwell equations, or fields that solve the exterior Maxwell equations everywhere in Ω and also in Ω c \ Ω 0 , where Ω 0 is the support of the sources of the incident waves. We assume that all the sources originate in Ω c . In particular, for plane * we define the normalized fields: wave incidence in the direction of k * · x}, * exp{ik k Ei (x) = a * · x} , Y + = Hi (x) = Y + * b exp{ik k

< ε+ , μ+

* and * where the polarization vectors a b and the direction of propagation * * Note that the *×* k form a right–handed orthonormal system, i.e. a b = k.

396

6. Electromagnetic Fields

plane waves do not satisfy the Silver–M¨ uller radiation conditions (6.7.2). The incident fields (Ei , Hi ) will represent the total field produced by the sources in absence of any scatterers. They satisfy similar Stratton–Chu representation formulae, but in particular   i (x − y)n × H (y)ds + n · Ei (y)grad y G+ 0 = iωμ+ G+ y k k (x − y)dsy 

Γ



Γ



n × Ei (y) × grad y G+ k (x − y)dsy ,

+ Γ



0 = −iωε+ 

 i G+ k (x − y)n × E (y)dsy +

Γ



Γ



n × H (y) ×

+

i

n · Hi (y)grad y G+ k (x − y)dsy

grad y G+ k (x

− y)dsy for x ∈ Ω c ,

Γ

since the incident fields satisfy the Maxwell equations in the region occupied by Ω in the absence of the scatterer (see e.g. Dassios and Kleinman [94, p.54] and Hsiao and Kleinman [194]). From Equation (6.9.6) and (6.9.7) with J0 = 0 and  = 0, we arrive at   i E(x) = E (x) + iωμ+ Gk (x − y) n(y) × H(y) dsy (6.12.3)  + Γ



+

Γ



n(y) · E(y) grad y Gk (x − y)dsy



n(y) × E(y) × grad y Gk (x − y)dsy ,

Γ



H(x) = H (x) − i

 + Γ



+

 iωε+ Gk (x − y) n(y) × E(y) dsy

(6.12.4)

Γ



n(y) · H(y) grad y Gk (x − y)dsy



n(y) × H(y) × grad y Gk (x − y)dsy for x ∈ Ω c .

Γ

If the scatterer is a perfect electric conductor then (6.12.3) and (6.12.4) with (6.6.4) lead to the representation formulae

6.12 Applications of boundary integral equations to scattering problems

 i

E(x) = E (x) +  +

 iωμ+ Gk (x − y) n(y) × H(y) dsy

397

(6.12.5)

Γ



n(y) · E(y) grad y Gk (x − y)dsy , x ∈ Ω c ,

Γ



H(x) = Hi (x) +



n(y) × H(y) × grad y Gk (x − y)dsy , x ∈ Ω c .

Γ

(6.12.6) Now we consider the application of boundary integral equations to scattering problems. 6.12.1 Scattering by a perfect electric conductor, EFIE and MFIE (see also Mitrea, D., Mitrea, M. and Pipher [308], Buffa, Costabel and Sheen [52] and Buffa, Hiptmair, von Petersdorff and Schwab [53].) Let a perfect electric conductor occupy the bounded region Ω ⊂ R3 with the Lipschitz boundary Γ = ∂Ω which is immersed into an isotropic homogeneous medium with constant ε+ and μ+ . When illuminated by the incident field, Ei (x) and Hi (x), we consider the following boundary value problem together with the homogeneous boundary conditions (6.6.4), i.e. n · H|Γ = 0 , n × E|Γ = 0. This leads to the boundary integral equations EFIE   + iωμ n(x) × Gk (x − y) n(y) × H(y) dsy Γ

 +n(x) ×

n(y) · E(y)grad y Gk (x − y)dsy = −n × Ei ,

for x ∈ Γ,

Γ

(6.12.7) and MFIE 1 n × H (x) − n(x) × 2





n(y) × H(y) × grad y Gk (x − y)dsy

Γ

= n × Hi (x) ,

for x ∈ Γ.

(6.12.8)

Equation (6.12.7) is known as the electric field integral equation (EFIE) whereas (6.12.8) is the magnetic field integral equation (MFIE), Both are integral equations for n × H on Γ . We note in (6.12.7), that the term n · E can be replaced by n·E=

1 divΓ (n × H) on Γ, iωε+

(6.12.9)

398

6. Electromagnetic Fields

which is the equation of continuity on the boundary Γ (see Jones [220, (6.84)]). Then EFIE becomes an integral equation on Γ for n × H on Γ only. The integral equation MFIE also determines the only unknown n × H on Γ since n × E|Γ = 0 and n · H|Γ = 0 on Γ . Remark 6.12.1: The EFIE (6.12.7) and MFIE (6.12.8) result from a method based on the Green representation theorem of solutions of Maxwell equations, known as the direct method. We only need to seek the solution either from (6.12.7) or from (6.12.8), but not both. On the other hand, one may also obtain boundary integral equations by the indirect method or layer ansatz as used by Colton and Kress [75], Hsiao and Kleinman [194]. For instance, we may seek the scattered electric field Esˆ(x) in the form of either simple- or double–layer potentials in order to arrive at two boundary integral equations for the unknown moment or density. Again one needs only to solve one of the two boundary integral equations. Then the scattered magnetic field Hs (x) can be recovered from the corresponding Maxwell equations. In what follows, for simplicity, we will suppress + signs for μ+ and ε+ . Remark 6.12.2: Alternatively, the Stratton–Chu representation formula (6.12.5) can be rewritten in the form E(x) = Ei (x) −

1 (grad div + k 2 )A(x) , x ∈ Ω c , iωε

(6.12.10)



where

Gk (x − y)(n × H)(y)dsy .

A(x) :=

(6.12.11)

Γ

As will be seen, this formula is particularly useful for deriving boundary integral equations for scattering by a perfectly electric conducting thin wire or antenna. We include here a derivation of Equation (6.12.10). We concentrate on the term:   n · E(y) grad y Gk (x − y)dsy Γ

in the Stratton–Chu formula (6.12.5). We note that from (6.12.9) we get

6.12 Applications of boundary integral equations to scattering problems



399



n · E(y) grad y Gk (x − y)dsy

Γ

=

1 iωε



 grad y Gk (x − y)divΓ n × H(y) dsy

Γ

−1 grad x = iωε



 Gk (x − y)divΓ n × H(y) dsy

Γ

−1 grad x divx A(x), = iωε where we have employed formula   divA(x) = div Gk (x − y) n × H(y) dsy  =

Γ

 Gk (x − y) divΓ n × H(y) dsy

(6.12.12)

Γ

(see Colton and Kress [75, Theorem 2.2]). The formula (6.12.10) now follows immediately from the Stratton–Chu formula (6.12.3) and (6.12.12). Remark 6.12.3: (Pocklington’s integral equation). In the special case of a perfectly conducting wire or thin–wire antenna, equation (6.12.10) leads to a much simplified EFIE. For simplicity, we assume that the wire is a circular cylinder of radius a which is much smaller than its length l. Let us assume that an incident wave Ei impinges on the surface of a conducting wire. It induces a surface current density JΓ = n × H that in turn is used to find a scattered electric field Es (x) = −

1 (grad div + k 2 )A(x) iωε



with

Gk (x − y)JΓ (y)dsy ,

A(x) := Γ

Since the wire is a circular cylinder, it is convenient to use the cylindrical coordinate system where x = (, Φ, z) denotes the observation points and y = (a, ΦΓ , zΓ ) denotes the source points on the surface of the wire. A very reasonable assumption is that the surface current density JΓ (y) is entirely axially directed with azimuthal symmetry for a # . This assumption leads to the consideration of a linear z–directed current density, JΓz (y), and

400

6. Electromagnetic Fields

z

Incident Ei , Hi



y Scattered Es , Hs

x

2a Fig. 6.5. Plane incident wave on a conducting wire.

for the observation point of the wire surface only the z–component of Es (x) is needed. That is 1 (grad div + k 2 )Az (x) , x ∈ Γ, Esz (x) = − iωε by neglecting the edge effects, where  Gk (x − y)JΓz (y)dzΓ =

Az (x) =

/2  2π −/2

Γ

 JΓz (y)Gk (x − y)adΦΓ dzΓ .

0

Moreover, if the wire is very thin, then the surface current density JΓz is not a function of the azimuthal angle ΦΓ , and we can write JΓz = JΓz (zΓ ) and obtain /2   2π eikR Az (x) = dΦΓ adzΓ , JΓz (zΓ ) 4πR −/2

0

6.12 Applications of boundary integral equations to scattering problems

401

where R = |x − y| = {2 + a2 − 2a cos(Φ − ΦΓ ) + (z − zΓ )2 }1/2 . This shows that Esz (x)

1 (grad div + k 2 ) =− iωε

/2 JΓz (zΓ )

 2π eikR 4πR

−/2

 dΦΓ adzΓ .

0

In order to derive an integral equation for JΓz , we require the boundary condition on the tangent component of the total field over the surface of the wire: (6.12.13) Eiz + Esz = 0 for  = a, where Ei is the impregnate field assumed to be azimuthally symmetric. Also, we introduce the total current flowing is the zΓ –direction as J(zΓ ) = 2πaJΓz (zΓ ) and obtain /2 iωεEiz (a, Φ, z)

2

= (grad div + k )

J(zΓ )  2πa

−/2

2π 0

 eikR dΦΓ adzΓ , (6.12.14)  4π R 

;

where

(Φ − ΦΓ ) 2 ] . 2 Because of the symmetry of the scatterer, the integration over ΦΓ is independent of the reference point Φ, we may set Φ = 0 and obtain finally the integral equation  := R

(z − zΓ )2 + [2a sin

/2    d2 2 +k J(zΓ )gΓ (z − zΓ )dzΓ = iωεEiz (z) dz 2

(6.12.15)

−/2

where as in (6.12.13) and (6.12.14), Eiz (z)

:=

1 gΓ (z − zΓ ) := 2π

Eiz (a, 0, z) , ;

 Φ=0 = R0 := R|

2π

eikR0 dΦΓ , 4πR0

0

(z − zΓ )2 + 4a2 sin2

 ΦΓ . 2

We note that in (6.12.15) the second order partial differential operator grad div has been reduced to an ordinary differential operator with respect

402

6. Electromagnetic Fields

to z, since the integral is a function of z only. A further simplification results, if we impose the assumption ka # 1. Thus 8 0 ∼ R = (z − zΓ )2 + a2 and the integral equation (6.12.15) becomes /2   d2   eikRo 2 + k J(z ) dz = iωεEiz (z) Γ 0 p dz 2 4π R

(6.12.16)

−/2

with a nonsingular kernel. The integral equation (6.12.16), is known as the Pocklington’s integral equation developed for the thin wire or antenna (see e.g. Balanis [21, (8.22)] and Barelishli [24, p.301]). An existence theorem for EFIE To end this section, we now include an existence theorem for the solution for EFIE in the scattering by a perfect electric conductor. We begin with the representation of the electric field E(x), x ∈ Ω c by using III(2) in Appendix B:  i (6.12.17) E(x) = E (x) + iωμ Gk (x − y)(n(y) × H(y)) dsy 1 + iω%



Γ

grad y Gk (x − y) divΓ (n(y) × H(y)) dsy

Γ     1 grad x divx V (πt∗ J) (x), = Ei (x) + iωμ V (πt∗ J) (x) − iω%     1 i grad x divx V J (x), = E (x) + iωμ V J (x) − iω%     1 i grad x V divΓ J (x), = E (x) + iωμ V J (x) − iω%  2  2 where πt∗ J = n(y) × H(y), and where πt∗ : L2t (Γ ) −→ L2 (Γ ) is the adjoint operator of πt and V is the simple–layer potential defined by  1 eik|x−y| λ (y)dsy , x ∈ /Γ Vλ (x) := 4π |x − y| Γ

Remark 6.12.4: Some explanations should be in order. It has been shown −1/2 λ) = V(πt∗λ ) in Buffa, Costabel and Schwab [51] that for λ ∈ H|| (Γ ), V(λ λ) = γV(πt∗λ ) ∈ H1/2 (Γ )). Then one may replace n × H by the (Note: γV(λ −1

corresponding Jγ ∈ H|| 2 (divΓ , Γ ) in V, since V(n×H) = V(πt∗ Jγ ) = V(Jγ ). For simplicity, we suppress the subindex γ.

6.12 Applications of boundary integral equations to scattering problems

403

It is understood that here πt is really πt ◦ Z where Z is a right inverse of the trace operator γ, and the adjoint operator πt∗ is the adjoint operator of πt ◦ Z according to the notation in Buffa, Costabel and Schwab [51] as discussed in Remark 4.2.1 (see Section 4.5). In terms of the single–layer potential operator V, the EFIE from the rep−1/2 resentation formula (6.12.17) for the unknown function J ∈ H|| ( divΓ , Γ ) is given by     1 πt V (J) + 2 grad Γ V ( divΓ J) = g , x ∈ Γ, (6.12.18) k where V := γV is the single–layer boundary integral operator and 1/2 g = −πt Ei /iωμ ∈ H⊥ (curl Γ , Γ ) is given in terms of the incident field Ei (x). The weak formulation of (6.12.18) now reads: −1/2 −1/2 Given g ∈ H⊥ ( curl Γ , Γ ), find J ∈ H|| ( divΓ , Γ ) such that . / / . / 1. Jt , πt V(J) − 2 divΓ Jt , V ( divΓ J) = Jt , g k

(6.12.19)

−1/2

for all test fields Jt ∈ H|| ( divΓ , Γ ) (by making use of the Stokes formula). The variational formulation (6.12.19) is exactly the same as the one derived from the Rumsey Variational Principle (see Rumsey [368], Nedelec [331] and De La Bourdonnage [98]). In order to discuss the existence of the variational solution J, following Buffa, Costabel and Schwab [51], we need the Hodge decomposition of the −1/2 tangential vector space H|| ( divΓ , Γ ) introduced in Buffa and Ciarlet [50], −1/2

( divΓ , Γ ) = grad Γ H ⊕ curl Γ H 1/2 (Γ ) (6.12.20)   1 −1/2 and H(Γ ) := α ∈ H (Γ ) with ΔΓ α ∈ H (Γ ), α, 1Γ = 0 .

H

Set J := grad Γ ϕ + curl Γ ψ , Jt := grad Γ ϕt + curl Γ ψ t

(6.12.21)

and ΔΓ ϕ ∈ H −1/2 (Γ ) (see De la Bourdonnaye [98] and Nedelec [331]). Lemma 6.12.1. Since J = grad Γ ϕ + curl Γ ψ with ϕ ∈ H(Γ ) and ψ ∈ H 1/2 (Γ ), there holds (cf. Appendix B,V(3)) divΓ J = ΔΓ ϕ ∈ H −1/2 (Γ ). Proof: divΓ J = divΓ grad Γ ϕ + divΓ curl Γ ψ = ΔΓ ϕ − divΓ (γt grad Γ ψ)  = ΔΓ ϕ − div(n × grad Γ ψ) |Γ

404

6. Electromagnetic Fields

due to Appendix B,III(5). Then  div(n × grad Γ ψ) = grad Γ ψ · curl Γ n − n · curl Γ grad Γ ψ = grad Γ ψ · curl Γ n since curl Γ grad Γ ψ = 0 due to Appendix B, V(4). For Γ ∈ C 1 the exterior normal vector n can be written as n = grad Φ with a local implicit representation Φ = const of Γ , and curl Γ n = ∇×∇Φ = 0. This completes the proof.  Remark 6.12.5: For J we have J2H−1/2 (div 

Γ ,Γ )

= J2H−1/2 (Γ ) + divΓ J2H −1/2 (Γ ) 

= grad Γ ϕ2H−1/2 (Γ ) + curl Γ ψ2H−1/2 (Γ )+ΔΓ ϕ2H −1/2 (Γ ) 



and grad Γ ϕ2H−1/2 (Γ ) ≤ grad Γ ϕ2L2 (Γ ) ≤ cΔΓ ϕ2H−1/2 (Γ ) 

which implies that the norm curl Γ ϕH−1/2 (Γ ) + ΔϕH−1/2 (Γ ) 

is equivalent to the norm JH−1/2 (divΓ ,Γ ) . 

From (6.12.21), in terms of ϕ, ψ and the test functions ϕt , ψ t , the variational equation (6.12.19) becomes / /   . 1. πt V grad Γ ϕ + curl Γ ψ , grad Γ ϕt + curl Γ ψ t − 2 V (ΔΓ ϕ) , ΔΓ ϕt k / . (6.12.22) = g , grad Γ ϕt + curl Γ ψ t , from which we define the sesquilinear form .   / t A(ϕ, ψ; ϕt , ψ ) := πt Vk grad Γ ϕ + curl Γ ψ , grad Γ ϕt + curl Γ ψ t / 1. − 2 C k (ΔΓ ϕ) , ΔΓ ϕt k =: B( grad Γ ϕ + curl Γ ψ, grad Γ ϕt + curl Γ ψ t ) 1 − 2 C k (ΔΓ ϕ, ΔΓ ϕt ) k t = B0 ( grad Γ ϕ + curl Γ ψ, grad Γ ϕt + curl Γ ψ ) 1 (6.12.23) − 2 C 0 (ΔΓ ϕ, ΔΓ ϕt ) + R(ϕ, ψ; ϕt , ψ t ) k

6.12 Applications of boundary integral equations to scattering problems

405

where t

B0 (•, •) = πt V0 (grad Γ ϕ + curl Γ ψ) , (grad Γ ϕt + curl Γ ψ ) C 0 (•, •) = V0 (ΔΓ ϕ) , ΔΓ ϕt  and the remaining term R is a compact perturbation of the principal parts. Note that A is the sesquilinear form of a system of three boundary integral equations for two functions ϕ, ψ depending on the variables on Γ . First, we show continuity. Lemma 6.12.2. The sesquilinear form A (ϕ, ψ; ϕt , ψ t ) is continuous on H × −1/2 −1/2 H where H = H −1/2 (Γ ) × H (Γ ) H (divΓ , Γ ) (see (6.12.20)).  −1/2  −1/2 1/2 −1/2 Proof: Since Vk H (Γ ) = H (Γ ) = H (Γ ) and Vk H⊥ (Γ ) =   1/2 −1/2 H⊥ (Γ ) = H⊥ (Γ ) , we have A(ϕ, ψ; ϕt , ψ t )| |A 1 t ≤ |πt Vk (grad Γ ϕ + curl Γ ψ), (grad Γ ϕt +curl Γ ψ )|+ 2 |Vk ΔΓ ϕ, ΔΓ ϕt | k   + πt Vk curl Γ ψ) 12 ≤ (πt Vk (grad Γ ϕ) 12 × H (Γ )

H (Γ )

1 (grad Γ ϕt  − 12 +(curl Γ ψ  − 12 )+ 2 Vk ΔΓ ϕ 12 ΔΓ ϕt  − 12 H (Γ ) H (Γ ) H (Γ ) H (Γ ) k t

the mapping properties of the single layer potential operator of the Helmholtz equation (see Theorem 5.6.2 and Theorem 5.8.1) assure the continuity estimate for any k > 0:  t A(ϕ, ψ; ϕt , ψ )| ≤ cV,k (grad Γ ϕH−1/2 (Γ ) + grad Γ ϕt H−1/2 (Γ ) ) |A    t × (curl Γ ψH−1/2 (Γ ) + curl Γ ψ H−1/2 (Γ ) ) 



  1 + 2 cV,k ΔϕH −1/2 (Γ ) · Δϕt H −1/2 (Γ ) k t

for all (ϕ, ψ) and (ϕt , ψ ) ∈ H(Γ ) × H −1/2 (Γ ) and with a positive constant cV,k depending only on k and Γ . Note that continuity of the principal parts B0 , C 0 and the remainder R follow as well.  Lemma 6.12.3. The sesquilinear forms t

πt Vk grad Γ ϕ, curl Γ ψ  , πt Vk curl Γ ψ, grad Γ ϕt  and πt Vk grad Γ ϕ, grad Γ ϕt  are compact for ϕ, ϕt ∈ H(Γ ), since grad Γ ϕ and grad Γ ϕt ∈ L2 (Γ ) and −1/2 curl Γ ψ and curl Γ ψ t ∈ H (Γ ).

406

6. Electromagnetic Fields

Proof: Since Vk maps H−1/2 (Γ ) into H1/2 (Γ ), one obtains t

t

|πt Vk grad Γ ϕ, curl Γ ψ | ≤ c1 πt Vk grad Γ ϕH1/2 (Γ ) · curl Γ ψ H−1/2 (Γ ) , 



t

≤ c2 grad Γ ϕH−1/2 (Γ ) curl Γ ψ H−1/2 (Γ ) , 



t

≤ c2 grad Γ ϕL2 (Γ ) curl Γ ψ H−1/2 (Γ ) . 

Since the imbedding L2 (Γ ) = H0 (Γ ) → H−1/2 (Γ ) is compact due to the Rellich Lemma Theorem 4.1.6, the sesquilinear form is compact. For the second sesquilinear form we have |πt Vk curl Γ ψ, grad Γ ϕt | ≤ c2 πt Vcurl Γ ψH1/2 (Γ ) grad Γ ϕt H−1/2 (Γ ) 



≤ c3 curl Γ ψH−1/2 (Γ ) grad Γ ϕ L2 (Γ ) t



and again compactness of L2 (Γ ) → H−1/2 (Γ ) implies that the sesquilinear form is compact. The third sesquilinear form implies |πt Vk grad Γ ϕ, grad Γ ϕt | ≤ πt Vk grad Γ ϕH1/2 (Γ ) grad Γ ϕt H−1/2 (Γ ) 



≤ cv grad Γ ϕH−1/2 (Γ ) grad Γ ϕt H−1/2 (Γ ) 



≤ cv grad Γ ϕL2 (Γ ) grad Γ ϕt L2 (Γ ) . and the compactness of L2 (Γ ) → H−1/2 (Γ ) again implies the compactness of the sesquilinear form.  Since in (6.12.19)

0

1 0 t t 1 J , πt V(J) = J, πt V(J )

we may rewrite (6.12.22) in the form 0

0 1 t 1 grad Γ ϕ + curl Γ ψ, πt V(grad Γ ϕt + curl Γ ψ ) + − ΔΓ ϕ, k −2 V (ΔΓ ϕt ) 0 t1 = g, grad Γ ϕt + curl Γ ψ

which motivates us to introduce a modified form in Theorem 6.12.4 below. Let e1 , e2 , n be an orthonormal basis of local coordinates at x ∈ Γ . Then grad Γ ϕ = ∂1 ϕe1 + ∂2 ϕe2 and curl Γ ψ = ∂2 ϕe1 − ∂1 ϕe2 . Theorem 6.12.4. The following G˚ arding inequality holds for the principal part of A , that is, there exist a compact linear operator C and a positive coerciveness constant c0 depending on k 2 and Γ such that

6.12 Applications of boundary integral equations to scattering problems

= Re



407

⎛ −2 ⎞> k V0 ΔΓ ϕ − ΔΓ ϕ, grad Γ ϕ , curl Γ ψ , Θ ⎝πt V0 grad Γ ϕ⎠ πt V0 curl Γ ψ

+ (−ΔΓ ϕ, grad Γ ϕ , curl Γ ψ), C (−ΔΓ ϕ, grad Γ ϕ , curl Γ ψ) (6.12.24)   ≥ c0 ΔΓ ϕ2H −1/2 (Γ ) + grad Γ ϕ2H−1/2 (Γ ) + curl Γ ψ2H−1/2 (Γ ) , 

where



⎛

⎞ 0 0 1 0⎠ . 0 1

−1 Θ=⎝ 0 0

The proof will be given in Section 12.9 and follows from strong ellipticity of A. Since the sesquilinear form πt Vk grad Γ ϕ, grad Γ ϕt  is compact on H×H due to Lemma 6.12.3 this sesquilinear form could be discarded in (6.12.24) for getting strong ellipticity of the remaining 2 × 2–system. However, G˚ arding’s inequality for the extended system (6.12.24) supports the convergence and error analysis for corresponding Galerkin boundary element approximation. From G˚ arding’s inequality (6.12.24) we conclude that the system of equations (6.12.19) defines a Fredholm mapping of index zero. However, there exist examples where the dimension of the mapping’s kernel is non zero for a whole family of integral equations (see Hellwig [175, Chap. 5.4.5], [176], [177]). Hence, in order the Fredholm alternative to hold for the system (6.12.22) we need in the Fredholm resolvent set for k2 > 0 at least one point in C where we have uniqueness. −1/2

Lemma 6.12.5. For 0 < k < k1 = λ1 6.8.2, the solution of EFIE is unique.

where λ1 is given in Theorem

Proof: Since EFIE is equivalent to (6.12.18) non uniquenes would imply that to all k ∈ (0, k1 ) there exists J0 = grad Γ ϕ0 + curl Γ ψ0 = 0 satisfying (6.12.19) with J = Jt = J0 and g = 0: 1 ΔΓ ϕ0 , V ΔΓ ϕ0  = 0. k2 Due to the invertibility of V and its coerciveness this implies J0 , πt VJ0  −

cV ΔΓ ϕ0 2H −1/2 (Γ ) ≤ ΔΓ ϕ0 , V ΔΓ ϕ0  = k 2 J0 , πt VJ0  where cV does not depend on k. Without loss of generality, J0 H−1/2 (divΓ ,Γ ) = 1 for all k ∈ (0, k1 ). Then 

for k → 0 and J0 , πt VJ0  ≤ cV J0 2

−1/2

H

(divΓ ,Γ )

= cV and 0 < k suffi-

ciently small, ΔΓ ϕ0 H −1/2 (Γ ) becomes arbitrarily small. For k = 0, ΔΓ ϕ0 = 0 , J0 = 0 in contrast to J0 H−1/2 (divΓ ,Γ ) = 1. But the resolvent set is open 

and contains 0 < k ∈ C where we still have uniqueness. Hence, the solution J in (6.12.18) for given g is unique. 

408

6. Electromagnetic Fields

Theorem 6.12.6. If k 2 > 0 is not an eigenvalue of the Dirichlet problem −1 for the interior Maxwell equations, for any given g ∈ H⊥ 2 ( curl Γ 0, Γ ) there −1

exists a unique solution J ∈ H|| 2 ( divΓ , Γ ) of the equation (6.12.18). Proof: The G˚ arding inequality (6.12.24) implies that for equation (6.12.18) the Fredholm alternative is valid. Since then uniqueness implies existence if k 2 is not an eigenvalue of the integral equation for small k 2 > 0, the equation (6.12.18) is uniquely solvable and therefore the Fredholm index, κ = α − β = 0 and α = β = 0. This implies unique solvability (6.12.22) for all k 2 not an eigenvalue of (6.12.18), which is the case for k 2 not an eigenvalue of the interior Dirichlet problem of Maxwell’s equation (see e.g. Maz‘ya [289], Buffa, Costabel and Sheen [52]) and one has the a priori estimate J

−1

H|| 2 ( divΓ ,Γ )

≤ cg

−1

H⊥ 2 ( divΓ ,Γ )



with a constant c(k 2 , Γ ) > 0.

A saddlepoint formulation for the construction of the solution of the EFIE Buffa, Costabel and Schwab [51] present also a saddle point formulation for solving the EFIE with a boundary element method where the incident field on Γ is supposed to be given in the form πt Ei = grad Γ α + curl Γ β with α ∈ H 1/2 (Γ ) , β ∈ H(Γ ). In addition to the unknown J = grad Γ ϕ + curl Γ ψ , ϕ ∈ H(Γ ) , ψ ∈ H 1/2 (Γ ) and test fields Jt = grad Γ ϕt + curl Γ ψ t , ϕt ∈ H(Γ ) , ψ t ∈ H 1/2 (Γ ) an additional unknown m = −ΔΓ ϕ and mt = −ΔΓ ϕt are introduced as well −1/2 (Γ ). The saddle point problem reads: as a Lagrangian λ and λt in H∗ Find (ϕ, ψ, m, λ) ∈ X X satisfying the EFIE saddle point equations 0 t 1 iωμ grad Γ ϕ + curl Γ ψ, πt Vk (grad Γ ϕt + curl Γ ψ ) t

−grad Γ λ, grad Γ ϕt  = −grad Γ α, grad Γ ϕt  − ΔΓ β, ψ , i m, πt Vk mt  + λ, mt  = 0, εω t t −grad Γ ϕ, grad Γ λ  + m, λ  = 0 −

X with the Hilbert space for all (ϕt , ψ t , mt , λt ) ∈ X

(6.12.25)

6.12 Applications of boundary integral equations to scattering problems −1/2

X X = H 1 (Γ )/C × H 1/2 (Γ )/C × H∗

409

(Γ ) × H 1 (Γ )/C.

With similar arguments as in Lemma 6.12.3 it follows that the sequilinear forms t

ωμgrad Γ ϕ + curl Γ ψ, −πt Vk grad Γ λ , ωμgrad Γ ϕ, πt Vk curl Γ ψt, ωμcurl Γ ψ, πt Vk grad Γ ϕt , λ, mt  and m, ϕt  are compact on X X. Hence, these terms in the System (6.12.25) are moved to the compact part of the sequilinear form, and the system now reads:  t t (6.12.26) B (ϕ, ψ, m, λ) , (ϕt , ψ , mt , λ ) := t

iωμgrad Γ ϕ, πt Vk grad Γ ϕt  + iωμcurl Γ ψ, πt Vk curl Γ ψ  −grad Γ λ, grad Γ ϕt  + C (ϕ, ψ, ϕt , ψ t ) = −grad Γ α, grad Γ ϕ − ΔΓ β, ψ, i − m, πt Vk mt  + λ, mt  = 0, εω t t −grad Γ ϕ, grad Γ λ  + m, λ  = 0. In order to obtain strong ellipticity, [51] find ⎛ 0 ⎜0 Θ=⎜ ⎝0 −1

the authors Buffa, Costabel and Schwab 0 1 0 0

0 0 −1 0

⎞ −1 0⎟ ⎟ 0⎠ 0

and the principal part modulo compact sesquilinear forms reads  B 0 (ϕ, ψ, m, λ) , Θ (ϕ, ψ, m, λ) ⎞ ⎛ iωμπt V0 (grad Γ ϕ) . ⎜ iωμπt V0 (curl Γ ψ) ⎟ / ⎟ (6.12.27) = (grad Γ ϕ, curl Γ ψ, m, grad Γ λ), Θ ⎜ ⎝ −(i/ωε)V0 (m) ⎠ grad Γ λ i m, πt V0 m + iωμgrad Γ ϕ, πt V0 grad Γ ϕ εω +iωμcurl Γ ψ, πt V0 curl Γ ψ +grad Γ λ, grad Γ λ+grad Γ ϕ, grad Γ ϕ. =

Hence, B 0 satisfies the coerciveness inequalities  (6.12.28) Im B 0 (ϕ, ψ, m, λ), Θ (ϕ, ψ, m, λ) ≥ c1 m2H −1/2 (Γ ) + ωμc2 grad Γ ϕ2H −1/2 (Γ ) + ωμc2 curl Γ ψ2H−1/2 (Γ ) , εω  Re B 0 (ϕ, ψ, m, λ), Θ (ϕ, ψ, m, λ) ≥ grad Γ λ2L2 (Γ ) + grad Γ ϕ2L2 (Γ ) t

t

and the sesquilinear form B satisfies a corresponding G˚ arding inequality. As a consequence, the Galerkin approximation of the equations (6.12.25) is stable

410

6. Electromagnetic Fields

in X X and asymtotically qusioptimal convergent as analysed also by Buffa, Costabel and Schwab in [51]. Since ΔΓ ϕ = m ∈ H −1/2 (Γ ) is on Γ explicitly available if the system (6.12.26) is numerically apprimated with boundary element Galerkin methods, also ϕ may be approximated on Γ by using surface Crouzeix –Raviard elements as investigated by Hailong Guo in [164]. −1/2 Note that in terms of the solution J ∈ H|| (divΓ , Γ ) of the EFIE, the electric field E is given from the representation formula (6.12.5) in Ω c , namely:  E(x) = Ei (x) + iωμ Gk (x − y)J(y)dsy 1 + iωε

Γ



grad y Gk (x − y)divΓ J(y)dsy , Γ

and the magnetic field H can be obtained in terms of E(x) from the Maxwell system in the form: ? H(x) = curl E(x) iωμ  curl Ei (x) + curl Gk (x − y)J(y)dsy = iωμ Γ  i = H (x) + J(y) × grad y Gk (x − y)dsy , x ∈ Ω c , Γ

which is the same as in (6.12.6), where

? Hi (x) := curl Ei (x) iωμ.

On the other hand, we may alternatively begin with the representation formula of H(x) in Ω c given by (6.12.6):  i H(x) = H (x) + J(y) × grad y Gk (x − y)dsy Γ −1/2

with J ∈ H||

(divΓ , Γ ) being the solution of the MFIE given by (6.12.8):  1 J(y) − n(x) × J(y) × grad y Gk (x − y)dsy = γt Hi (x) , x ∈ Γ. (6.12.29) 2 Γ

In terms of the boundary integral operator  R(J)(x) := n(x) × J(y) × grad y Gk (x − y)dsy Γ



= n(x) × curl

Gk (x − y)dsy , x ∈ Γ Γ

6.12 Applications of boundary integral equations to scattering problems

411

from (6.12.7), the MFIE can be rewritten in the form: 1 I − R)J(x) = γt Hi (x) , x ∈ Γ. 2

(6.12.30)

Before we discuss the solvability of (6.12.30), we see that in terms of the solution J(x) of (6.12.30), the magnetic field H(x) is given from the representation formula (6.12.6) and the electric field E(x) in Ω c can be recovered from the Maxwell system: curl curl Hi (x) + E(x) = −iωε −iωε

 J(y) × grad y Gk (x − y)dsy . Γ

A simple computation shows that if Ei (x) = curl curl E(x) = E (x) + −iωε



then

Gk (x − y)J(y)dsy

i

= Ei (x) −

curl Hi (x) , −iωε

Γ

1  − Δx + grad x divx } iωε

 Gk (x − y)J(y)dsy Γ

 1 2 {k E(x) = E (x) − Gk (x − y)J(y)dsy iωε Γ   − grad y Gk (x − y))divΓ J(y)dsy i

Γ

 Gk (x − y)J(y)dsy

i

= E (x) + iωμ Γ

1 + iωε

 grad y Gk (x − y)divΓ J(y)dsy ,

Γ

which is exactly the same as (6.12.5), since 1 divΓ J(y) = n(y) · E(y). iωε Now let us consider the solution of (6.12.30). Solvability of MFIE For smooth Γ (say Γ ∈ C 1+α ), the kernel of R is weakly singular and hence R ∈ ψdO−1 . This can be seen as follows:

412

6. Electromagnetic Fields

 R(J)(x) = n(x) × curl  =

 n(x) × grad x Gk (x − y) × J(y) dsy

Γ

=

  

Γ

=

Gk (x − y)J(y)dsy for x ∈ Γ, Γ

 $ 

 ∂Gk n(x) · J(y) grad x Gk (x − y) − (x − y)J(y) dsy ∂nx

% n(x) − n(y) · J(y) grad x Gk (x − y)dsy

Γ

−

 $ % ∂Gk (x − y)J(y) dsy ∂nx

Γ

−1/2

Since n(y) · J(y) = 0 for J ∈ H||

(divΓ , Γ ). Thus, for smooth Γ , comp

R : H−1/2 (Γ ) → H−1/2 (Γ ) → H−1/2 (Γ ) is a compact operator and the MFIE is a Fredholm integral equation of the second kind. Hence, uniqueness implies existence. However, it is well-known that solutions of MFIE as well as EFIE for perfectly electric conducting scatterers are non–unique for irregular frequencies which correspond to interior Maxwell resonances of the scatterer. It is not our intent to address this question here. Various methods of eliminating these spurious resonances have been proposed (see, e.g. Colton and Kress [75], Mautz and Harrington [285], Yaghjian [441], and Jost [221]). For smooth boundary Γ , variational solutions of (6.12.30) in standard Sobolev spaces may be treated similar to boundary integral equations in Section 5.1.6. for classical solutions of (6.12.30), we refer to the work of Colton and Kress [75] (see also Hsiao, Kleinman and Wang [195]). It is noteworthy to mention that in order to apply the Fredholm alternative to an integral equation of the second kind such as the MFIE with non–unique solutions, we need to investigate the corresponding homogeneous adjoint (or rather transposed) equations. For (λ, ϕ) ∈ L2t (Γ ), we see that the transposed operator R of R with respect to the duality product  λ, ϕ := λ · ϕ ds, Γ

is given by R = γt Rγt and hence the L2 - adjoint operator R∗ λ = (R λ) (see Colton and Kress [75, 2.82]). This can be seen as follows:

6.12 Applications of boundary integral equations to scattering problems

 λ, R(ϕ) =

 λ(x) · n(x) × curl x (V(ϕ) dsx

Γ



 

=

curl x Γ

=

  grad x Gk (x − y) × ϕ(y)dsy · λ(x) × n(x) dsx

Γ

Γ



ϕ(y) · Γ

ϕ(y) · Γ

 ϕ(y) ·

= Γ

 

= Γ

$ 

% λ(x) × n(x) × grad x Gk (x − y)dsx dsy

Γ



=

  Gk (x − y)ϕ(y)dsy · λ(x) × n(x) dsx

Γ

  

=

413

  

  grad x Gk (x − y) × n(x) × λ(x) dsx dsy

Γ



− curl y

  Gk (x − y) n(x) × λ(x) dsx dsy

Γ

 n(y) × n(y) × ϕ(y) • 

    • curl y Gk (x − y) n(x) × λ(x) dsx dsy Γ

 = Γ

 n(y) × ϕ(y) •



 • curl y 

=− Γ



=−





  Gk (x − y) n(x) × λ(x) dsx × n(y)dsy

Γ

  n(y) × ϕ(y) · R(n × λ (y)dsy

  ϕ(y) · R(n × λ) × n(y) dsy

Γ

0 1 = ϕ(y) , n × R(n × λ) . Before we discuss the transposed equation of MFIE (6.12.30), we note that MFIE is a Fredholm equation of the second kind obtained from the direct method, which seems to have inherited some complications. So for simplicity, let us first consider the solution of the magnetic field H ∈ Ω c by using the indirect method based on a layer potential ansatz. We may seek a solution in the form:  ϕ(y)dsy in Ω c H(x) = curl x Gk (x − y)ϕ Γ

414

6. Electromagnetic Fields

in terms of an unknown density boundary field ϕ . This reduces to the boundary integral equation of the second kind in the form 1 I + R ϕ = γt Hinc (x) for x ∈ Γ. 2

(6.12.31)

However (6.12.31) appears startingly different from the corresponding equation (6.12.30) in that the equation involves the operator 12 I + R rather than 1 2 I − R (see Hsiao and Kleinman [194]). If (6.12.31) can be solved, we then define the associated electric field E by 1 E(x) = curl H(x) ∈ Ω c . −iωε Then H and E satisfy the Maxwell systems in Ω c automatically. In particular, we see that curl E(x) = = = =

=

−1 curl curl H(x) iωε −1 (−Δ + grad div)H(x) iωε 1 (ΔH(x)) iωε  1 curl Δx (Gk (x − y)ϕ(y))dsy iωε Γ  −k 2 curl Gk (x − y)ϕ(y)dsy iωε Γ

= iωμH(x). For the transposed equation of (6.12.30) let  H(x) = Hi (x) − curl Gk (x − y)ψ(y)dsy in Ω, Γ

be the magnetic field in Ω, where ψ = n × λ is a tangential vector on Γ to be determined. Then if πt H = 0 , we see that λ is a solution of the transposed equation of (6.12.31) in the form 1 ( I + R )λ = πt Hi on Γ. 2 Then the corresponding homogeneous transposed equation has only the trivial solution, provided k 2 is not an eigenvalue for the homogeneous interior Dirichlet problem: curl (εε1/2 E) = ik(μ1/2 H), in Ω curl (μ1/2 H) = −ik(ε1/2 E)

6.12 Applications of boundary integral equations to scattering problems

415

together with the corresponding boundary condition πt H = 0 on Γ. Thus, (6.12.31) has at most one solution if k 2 is not an eigenvalue. Conversely, if k 2 is an eigenvalue for the homogeneous interior Dirichlet problem, then the corresponding homogeneous transposed equation will have nontrivial solutions λ0 . In contrast to the direct method, the given datum γt Hi in (6.12.31) is required to satisfy the orthogonality condition:  γt Hi · λ0 ds = 0 Γ

which is a necessary condition for the existence of solutions of (6.12.31). Next, we return to the MFIE (6.12.30). It is quite surprising that the homogeneous transposed equation 1 ( I − R )λ = 0 on Γ 2

(6.12.32)

is not the correct homogeneous corresponding transposed equation for the equation (6.12.30) in the sense that (6.12.32) can not be identified as an integral equation for the solution of any homogeneous interior Maxwell boundary value problem. In fact, one may verify that the original integral equation (6.12.30) can be alternatively rewritten in terms of the tangential magnetic field strength H in J = n × H, whence the operator R as the transposed operator to R appears. This can be seen as follows. From (6.12.30), let J = (n × H). Then 1 (n × H) − R(n × H) = γt Hi , 2 or, 1 (n × H) − R(n × πt H) = γt Hi 2 since n × H = n × πt H. Then by applying −γt to the above equation, we obtain 1 (6.12.33) ( I + R )(πt H) = πt Hi , 2 That is, the equation (6.12.29) for the exterior Maxwell problem is equivalent to (6.12.33). This results from the corresponding homogeneous transposed equation for (6.12.30) (i.e. (6.12.33)) in the form 1 ( I + R)J(x) = 0 , x ∈ Γ, 2 which can be identified as a boundary integral equation reduced from an interior Maxwell problem with appropriate homogeneous boundary conditions.

416

6. Electromagnetic Fields

For details and non–uniqueness of the electromagnetic scattering integral equation solutions in general, we refer the readers to the paper by Langenberg [260]. To end this section, we remark that for Lipschitz continuous boundary Γ , it is known that R is no longer compact. Further investigation is needed. Either of EFIE or MFIE can be used to solve for the equivalent surface current J. However, the particular choice of EFIE or MFIE will in general be dictated by the geometry of the scatterer. The MFIE becomes useless when Γ shrinks to an infinitely thin scatterer, while EFIE is ideally suited for the thin cylinder. 6.12.2 Scattering by a dielectric body In this section, we concider the scattering from a dielectric body Ω with Lipschitz boundary surface Γ which is immersed into an isotropic homogeneous medium Ω c := R3 \ Ω. We denote the physical parameters and quantities as follows: √ ε, μ, k = ω με , H(x), E(x) in Ω 8 ε+ , μ+ , k + = ω μ+ ε+ , H+ (x), E+ (x) in Ω c , where all the physical parameters are assumed to be positive constants. The incidence electromagnetic fields are denoted by Ei or Hi . It is understood that the electric and magnetic fields as E(x), H(x) and E+ (x), H+ (x) satisfy corresponding Maxwell equations in Ω and Ω c , respectively. On the interface Γ , we require the transmission conditions: n × (E+ − E) = 0 , n × (H+ − H) = 0 on Γ together with the Silver–M¨ uller radiation conditions (6.7.2) for the scattered field as |x| → ∞. We consider two approaches: (a) The volume integral equation approach on Ω based on the equivalence principle in electromagnetics (see, e.g. Balanis [21, p.327], Barkeshli [24] and Harrington [172], to name a few), and (b) The boundary integral equation approach on Γ based on the Green representation theorem. For simplicity, we begin with (a) The volume integral equation approach Equivalence principles are extremely useful concepts in electromagnetics. Many source distributions outside a region may produce the same fields inside the region. Moreover, equivalence principles are based on the uniqueness theorem for solutions of the Maxwell equations. In general there are two classes of equivalence principles: Volumetric and Surface principles.

6.12 Applications of boundary integral equations to scattering problems

417

We will illustrate the volumetric equivalence principle here and refer to the references above for the surface equivalence principle. It is presumed that the dielectric body occupies the region Ω, and that μ and ε are the given permeability and permittivity as mentioned before; in the exterior the corresponding quantities are taken as μ+ and ε+ , respectively. An incident wave Ei , Hi strikes the dielectric body from Ω c := R3 \ Ω. As a result, the obstacle generates in Ω c a scattered field Es , Hs which is required to satisfy the Silver–M¨ uller radiation conditions at infinity. In addition, a total field E, H is transmitted into Ω. The transmission conditions are required for the total field as follows: n × (Ei + Es )+ = n × (E)− , n × (Hi + Hs )+ = n × (H)− .

(6.12.34)

Inside the electric body, E and H satisfy the Maxwell equations which can be rewritten in the form curl E − iωμ+ H = −iω(μ+ − μ)H ,

in Ω,

(6.12.35)

curl H + iωε E = −iω(ε − ε )E ,

in Ω.

(6.12.36)

+

+

With this modification made in the fields of these equations, all fields can now be interpreted as existing in the interior homogeneous region Ω bounded by Γ of constitutive parameters ε+ and μ+ together with dependent electric are magnetic current densities Je = −iω(ε − ε+ )E,

(6.12.37)

Me = −iω(μ − μ)H,

(6.12.38)

+

respectively. In this context, the influence of the dielectric body is represented as attributable to volume distributions of J and H in the same medium as in the exterior Ω c . The total field E, H can be expressed in terms of electromagnetic potentials introduced in Section 6.5. To do so, we first decompose the fields and obtain the corresponding scattered fields. By using superposition, one obtains the total scattered fields. For simplicity, let us first consider the case when μ = μ+ but ε = ε+ . Then from the magnetic vector potential, we see that  −1 s + 2 E (x) = (grad div + (k ) ) G+ (6.12.39) k (x − y)Je (y)dsy iωε+ Ω  s + H (x) = curl Gk (x − y)Je (y)dsy , (6.12.40) Ω

where k + := ω

8

+

μ + ε+ , G + k =

1 eik |x−y| . 4π |x − y|

418

6. Electromagnetic Fields

In the same manner, in the case where μ = μ+ , but ε = ε+ , we obtain  s G+ (6.12.41) E (x) = curl k (x − y)Me (y)dy, Ω

1  Hs (x) = grad div + (k + )2 + iωμ

 G+ k (x − y)Me (y)dy.

(6.12.42)

Ω

Finally, by superposition, we obtain



 E(x) = E (x) + grad div + (k + )2

G+ k (x − y)

i

Ω



(μ − μ+ )H(y)G+ k (x − y)dy,

+ iωcurl Ω

i



H(x) = H (x) + grad div + (k + )2  − iωcurl

 ε − 1 E(y)dy ε+





(6.12.43)

 μ − 1 G+ k (x − y)E(y)dy + μ

Ω

(ε − ε+ )G+ k (x − y)E(y)dy

(6.12.44)

Ω

(see Jones [220, p.382]). We notice equations (6.12.43) and (6.12.44) form a system of volume integral equations for determining E and H inside the dielectric body Ω, and provide a representation of the fields outside the dielectric body when x ∈ Ω c . It can be shown that the representation has continuous tangent components and therefore the transmission conditions (6.12.34) will be satisfied. Remark 6.12.6: One advantage of these volume integral equations is that μ, ε will not necessarily be constant, and we may concider an inhomogeneous dielectric body as well. Remark 6.12.7: In (6.12.37) and (6.12.38) the terms Je and Me are often referred to as equivalent volumetric currents, and can be rewritten in the form  ε Je = −iωε+ + − 1 E = −iωε+ (εr − 1)E,  με Mc = iωμ+ + − 1 H = iωμ+ (μr − 1)H, μ and (εr , μr ) are the relative permittivity and permeability of the medium. Thus, the presence of the dielectric material is actually equivalent to having the equivalent volumetric currents Je and Mc radiating in the unbounded space. These volumetric currents may be directly used to construct volumetric integral equations for scattering from a dielectric obstacle. The equivalent electric current inside the scatterer is often denoted by

6.12 Applications of boundary integral equations to scattering problems

Jeq

8 k+ = −i + χe E , k + = ω ε+ μ+ , z + = z

;

419

μ+ , ε+

where χe := εr − 1. In the special case when μ = μ+ , (6.12.43) reduces to the volumetric integral equation in the form of   k + χe i Jeq (x) − χe (k + )2 + grad div) G+ E (x) (x − y)J (y)dy = −i eq k z+ Ω

(6.12.45) for all x ∈ Ω. Equation (6.12.45) is the volumetric EFIE for scattering from a dielectric scatterer. (See Barelishli [24, p.307].) Remark 6.12.8: The existence theorem for solutions of (6.12.43), (6.12.44) and (6.12.45) are not currently known (see Jones [220, Section 6.28]). Next we consider (b) The boundary integral equation approach In the following, let E+ (x) and H+ (x) be the total fields in Ω c . (See Figure 6.6) and +

1 eik |x−y| 1 eik|x−y| , Gk (x − y) := , 4π |x − y| 4π |x − y| 8 √ = ω μ+ ε+ , k = ω με.

Gk+ (x − y) := k+

with

Then we have the integral equations for the scattering by a dielectric body as follows: 



 μ n × H+ (y) Gk+ (x − y) + + Gk (x − y) dsy μ Γ    1 ε+ Gk (x − y) dsy + n × divΓ n × H+ (y) grad y Gk+ (x − y) + + iωε ε Γ    n × E+ (y) × grad y Gk+ (x − y) + Gk (x − y) dsy , +n×

iωμ n × +

Γ

= −n × Ei (x) for x ∈ Γ, and

(6.12.46)

420

6. Electromagnetic Fields

 − iωε n × +

  ε Gk+ (x − y) + + Gk (x − y) n × E+ (y) dsy ε

Γ

=

   1 μ+ Gk (x − y) dsy n × divΓ n × E+ (y) grad y Gk+ (x − y) + + iωμ μ Γ    n × H+ (y) × grad y Gk+ (x − y) + Gk (x − y) dsy +n× Γ

= −n × Hi (y) for x ∈ Γ.

(6.12.47)

(E i , H i ) n (E s , H s ) Ω ε, μ Γ ε+ , μ +

Ωc = R3 \ Ω

Fig. 6.6. Scattering by a dielectric body

We have used the notation: E+ (y) and H+ (y) for the total fields in Ω c The boundary integral equations are in terms of E+ (x), H+ (x) in Ω c . To derive (6.12.46), (6.12.47), as usual, we start from the Stratton–Chu representation formulae for the fields in Ω and Ω c (c.f. (6.9.4), (6.9.5), (6.12.5) and (6.12.6) with J0 = 0 and  = 0 , σ = 0). In the representation of the fields in Ω, we first replace E, H in terms of E+ , H+ by making use of the transmission conditions. n × E = n × E+ , n × H = n × H+ on Γ, which imply that n · μH = n · μ+ H+ and n · εE = n · ε+ E+ on Γ.

6.12 Applications of boundary integral equations to scattering problems

421

Then combining the integral equations, we finally obtain the boundary integral equations for only E+ and H+ . For definiteness, we begin with the Stratton–Chu formulae in Ω and obtain the boundary integral equations:   1 (6.12.48) − n × E(x) = (iωμ)n × Gk (x − y) n × H(y) dsy 2  Γ  +n× n · E(y) grad y Gk (x − y)dsy Γ



+n×



n × E(y) × grad y Gk (x − y)dsy

Γ



 μ Gk (x − y) n × H+ (y) dsy + μ

= (iωμ )n × +



Γ



n · E+ (y)

+n× Γ



+n×

ε+ grad y Gk (x − y)dsy ε



n × E+ (y) × grad y Gk (x − y)dsy , x ∈ Γ,

Γ

and

  1 − n × H(x) = −(iωε)n × Gk (x − y) n × E(y) dsy 2 Γ   +n× n · H(y) grad y Gk (x − y)dsy Γ



+n×



n × H(y) × grad y Gk (x − y)dsy

Γ



= −(iωε )n × +



 ε Gk (x − y) n × E+ (y) dsy + ε

Γ



n · H+ (y)

+n× Γ



+n×

(6.12.49)

μ+ grad y Gk (x − y)dsy μ

n × H+ (y) × grad y Gk (x − y)dsy , x ∈ Γ.



Γ

On the other hand, the Stratton–Chu formulae in Ω c lead to the boundary integral equations:

422

6. Electromagnetic Fields

  1 i + n×E+ (x) = n × E (x)+(iωμ )n(x)× Gk+ (x−y) n×H+ (y) dsy 2 Γ   n · E+ (y) grad y Gk+ (x − y)dsy +n× Γ



+n×



n × E+ (y) × grad y Gk+ (x − y)dsy , x ∈ Γ,

Γ

(6.12.50)   1 n × H+ (y) = n × Hi (x) − (iωε+ )n × Gk+ (x − y) n × E+ (y) dsy 2 Γ   n · H+ (y) grad y Gk+ (x − y)dsy +n× Γ



+n×



n × H+ (y) × grad y Gk+ (x − y)dsy , x ∈ Γ.

Γ

(6.12.51) By adding (6.12.48) and (6.12.50), we obtain    μ + Gk+ (x − y) + + Gk (x − y) n × H+ (y) dsy (iωμ )n × μ Γ    ε+ Gk (x − y) dsy +n× n · E+ (y) grad y Gk+ (x − y) + ε Γ    n × E+ (y) × grad y Gk+ (x − y) + Gk (x − y) dsy +n× Γ

= −n × E+ (x) , x ∈ Γ.

(6.12.52)

Note

 1 divΓ n × H+ (y) on Γ. + iωε Similarly, by adding (6.12.49) and (6.12.51), we obtain    ε Gk+ (x − y) + + Gk (x − y) n × E+ (y) dsy −(iωε+ )n × ε Γ    μ+ Gk (x − y) dsy n · H+ (y) grad y Gk+ (x − y) + +n× μ Γ    +n× n × H+ (y) × grad y Gk+ (x − y) + Gk (x − y) dsy n · E+ (y) =

Γ

= −n × Hi (x) , x ∈ Γ

(6.12.53)

6.12 Applications of boundary integral equations to scattering problems

with n · H+ (y) =

423

 −1 divΓ n × E+ (y) on Γ. + iωμ

This completes the derivations of (6.12.52) and (6.12.53). 6.12.3 Scattering by objects with impedance boundary conditions When propagating a wave through an actual conductor, the electromagnetic wave amplitude falls off exponentially to e−dδ , where d is the distance into the conductor. Thus, in most cases, the field can be considered as not penetrating into good conductors. However, when it is necessary to account for dissipation losses in the conductor, we may proceed as follows: The electromagnetic field is found by assuming perfect conductivity with the boundary condition: of the conductor: n × E = 0, from which we obtain the approximate surface current density: J = n × H. Next, the perfect conductivity boundary condition is modified by assuming an inhomogenous boundary condition: n × E = Zs n × J, where Zs is the surface impedance of the conductor given by Zs =

1−i σδ

with δ being the skin depth and σ the conductivity. This eventually leads to the approximated boundary condition: −n × (n × E) = Zs n × H or equivalently, n × E = Zs n × (n × H) on Γ.

(6.12.54)

The condition (6.12.54) is called the impedance boundary condition. Remark 6.12.9: The surface of a conductor will not be perfectly conducting but the field will not penetrate deeply into the scatterer, particularly when it is coated. Then it might be enough to account for the properties by imposing an impedance boundary condition on the boundary instead of considering a transmission boundary value problem (see Jost [221]). For the scattering problem with a prescribed impedance boundary condition, the basic boundary integral equations are:

424

6. Electromagnetic Fields

 n×n×



 iωμ n × H(y) Gk (x − y)

Γ

  + Zs n × n × H(y) × grad y Gk (x − y)   1 divΓ n × H(y) grad y Gk (x − y) dsy +  iωε  + Zs n × Zs n × H(y) × grad y Gk (x − y) Γ

 1 divΓ Zs n × H(y) grad y Gk (x − y) iωμ    − (iωε) Zs n × n × H(y) grad y Gk (x − y) dsy   = −n × n × Ei (x) − Zs n × Hi (x) , x ∈ Γ. (6.12.55) −

and

  1 (n × H) − n × (n × H) × grad y Gk (x − y)dsy 2 Γ  − iωε Zs (n × (n × H) Gk (x − y)dsy Γ

1 − iωμ =n×H



Γ i

  divΓ Zs n × (n × H)grad y Gk (x − y)dsy for x ∈ Γ.

(6.12.56)

In the following,we provide the derivations of the boundary integral equations (6.12.55) and (6.12.56). We begin with (6.12.56) by using the Stratton– Chu formula for the magnetic field H in Ω c :   n × E(y) Gk (x − y)dsy H(x) = Hi (x) − iωε  + Γ



+

Γ



n · H(y) grad y Gk (x − y)dsy



n × H(y) × grad y Gk (x − y)dsy , x ∈ Ω c .

Γ

Now let x → Γ , and apply the impedance boundary condition (6.12.54). This results in the following relation:

6.12 Applications of boundary integral equations to scattering problems

425

  1 n × H(y) × grad y Gk (x − y)dsy n × H(x) − n × 2 Γ    − iωε Zs n × n × H(y) grad y Gk (x − y)dsy Γ

1 − iωμ



   divΓ Zs n × n × H(y) grad y Gk (x − y)dsy

Γ i

= n × H (x) for x ∈ Ω c . Next, the derivation of (6.12.55) is more involved. Let us first show that   1 Zs (n × H) + n × n × Zs n × (n × H) × grad y Gk (x − y)dsy 2 Γ  + iωμ(n × H)Gk (x − y))dsy Γ



+

 1 divΓ (n × H)grad y Gk (x − y))dsy iωε

Γ

= −n × (n × Ei ) for x ∈ Γ.

(6.12.57)

The Stratton–Chu formula for the E field in Ω c reads:   E(x) = Ei (x) + iωμ Gk (x − y) n × H(y)dsy 

Γ



n · E(y) × grad y Gk (x − y)dsy

+ Γ



+



n × E(y) × grad y Gk (x − y)dsy , x ∈ Ω c .

Γ

Again, let x → Γ and apply the impedance boundary condition (6.12.54). We obtain    1 Zs n × n × H(x) = n × Ei (x) + iωμn × Gk (x − y) n × H(y) dsy 2 Γ    1 n× div n × H(y) grad y Gk (x − y)dsy + iωε Γ    + n × Zs n × n × H(y) × grad y Gk (x − y)dsy , x ∈ Γ. Γ

By applying the cross–product n× to the above boundary integral equation once more, we obtain the desired equation:

426

6. Electromagnetic Fields

   1 Zs n × n × H(y) × grad y Gk (x − y))dsy Zs (n × H)(x) + n × n × 2 Γ   + iωμ Gk (x − y) n × H(y) )dsy Γ

1 + iωε



  divΓ n × H(y) grad y Gk (x − y)dsy

Γ

= −n × (n × Ei )(x) , x ∈ Γ, since     n × Zs n × n × H(x) = n × n × E(x) = −Zs n × H(x) . To complete the derivation of (6.12.55) we first multiply (6.12.56) by Zs :   1  Zs n × H(x) = Zs n × n × H(y) × grad y Gk (x − y)dsy 2 Γ    1 − divΓ Zs n × n × H(y) grad y Gk (x − y)dsy iωμ Γ     − iωε Zs n × n × H(y) Gk (x − y)dsy Γ

+ Zs (n × Hi )(x) , x ∈ Γ.

(6.12.58)

Substituting 12 Zs (n × H)(x) from (6.12.58) into (6.12.57) yields (6.12.55). This completes the derivations of (6.12.55) and (6.12.56). Remark 6.12.10: Equation (6.12.55) is the analogue of the EFIE and reduces to it when Zs = 0 (i.e. σ → ∞), while (6.12.56) reduces to MFIE. (See Jones [220, (6.134),(6.135)].) In closing the section, we note that the integral equations here are more complicated than those for perfect conductivity as well as for the transmission problems. Some uniqueness theorems are available (see Jones [220] and Colton and Kress [75]).In Picard and Mc Ghee [347], a unified Hilbert space approach for treating extended Maxwell systems is included. We refer to Colton and Kress [77] for inverse acoustic and electromagnetic scattering theory, and to Angell, Hsiao and Wen [15] for two–dimensional inverse scattering problems in electromagnetics. For optimization methods in electromagnetics, we refer to the monograph by Angell and Kirsch [14], while for computational methods see Christiansen [67], Jin [218], Monk [314], and monographs edited by Ciarlet et al. [113], and de Castro and Valli [78], respectively. Today, for treating exterior and transmission problems in electromagnetics, the coupling of boundary and finite element methods becomes a very effective and

6.12 Applications of boundary integral equations to scattering problems

427

popular solution procedure ( see e.g. Bendali [27, 28], Hiptmair [182], Hsiao [193], Hsiao, Monk and Nigam [200] and Langer, Pauly and Repin [200]. In Sterz and Schwab [407], an eddy current problem with impedance boundary condition has been investigated. The problem was reduced to a s calar hypersingular boundary integral equation on the surface Γ of the conductor. Strong ellipticity of the associated nonsymmetric hypersingular operator was established.

7. Introduction to Pseudodifferential Operators

The pseudodifferential operators provide a unified treatment of differential and integral operators. They are based on the intensive use of the Fourier transformation F (3.1.12) and its inverse F −1 = F ∗ (3.1.14). The linear pseudodifferential operators can be characterized by generalized Fourier multipliers, called symbols. The class of pseudodifferential operators form an algebra, and the operations of composition, transposition and adjoining of operators can be analyzed by algebraic calculations of the corresponding symbols. Moreover, this class of pseudodifferential operators is invariant under diffeomorphic coordinate transformations. As linear mappings between distributions, the pseudodifferential operators can also be represented as Hadamard’s finite part integral operators, whose Schwartz kernels can be computed explicitly from their symbols or as integro–differential operators and vice versa. For elliptic pseudodifferential operators, we construct parametrices. For elliptic differential operators, in addition, we construct Levi functions and also fundamental solutions if they exist. Since the latter provide the most convenient basis for boundary integral equation formulations, we also present a short survey on fundamental solutions.

7.1 Basic Theory of Pseudodifferential Operators In this introductory section for pseudodifferential operators we collect various basic results of pseudodifferential operators without giving all of the proofs. The primary sources for most of the assertions are due to H¨ormander’s pioneering work in [184, 185, 186, 188]. We begin with the definition of the classical symbols for operators on functions and distributions defined in some domain Ω ⊂ IRn (see [186]). Definition 7.1.1. For m ∈ IR, we define the symbol class S m (Ω × IRn ) of order m to consist of the set of functions a ∈ C ∞ (Ω × IRn ) with the property that, for any compact set K  Ω and multi–indices α, β there exist positive constants c(K, α, β) such that1 1

α For our readers, we here purposely do not  use  the notation D since in pseudo∂ α differential analysis Dα is reserved for −i ∂x .

© Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_7

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7. Introduction to Pseudodifferential Operators

      ∂ β ∂ α    a(x, ξ) ≤ c(K, α, β)ξm−|α|   ∂x  ∂ξ

(7.1.1)

for all x ∈ K and ξ ∈ IRn , where 1

ξ := (1 + |ξ|2 ) 2 .

(7.1.2)

Remark 7.1.1: In H¨ormander [188], the classical symbols need to satisfy (7.1.1) for all x ∈ IRn , ξ ∈ IRn . This approach eliminates some of the difficulties with operators defined only on Ω. Since we are dealing with problems where Ω is not necessarily IRn we prefer to present the local version as in [186]. A simple example is the polynomial  aα (x)ξ α . a(x, ξ) =

(7.1.3)

|α|≤m

A function a0m ∈ C ∞ (Ω × (IRn \ {0})) is said to be positively homogeneous of degree m (with respect to ξ) if it satisfies a0m (x, tξ) = tm a0m (x, ξ) for every t > 0 and all ξ ∈ IRn \ {0} .

(7.1.4)

If a0m is positively homogeneous of degree m then am (x, ξ) = χ(ξ)a0m (x, ξ) ∈ S m (Ω × IRn )

(7.1.5)

will be a symbol provided χ is a C ∞ –cut–off function with χ(η) = 0 in the vicinity of η = 0 and χ(η) ≡ 1 for |η| ≥ 1. In connection with the symbols a ∈ S m (Ω ×IRn ), we define the associated standard pseudodifferential operator A of order m. Definition 7.1.2. A(x, −iD)u := (2π)−n/2



eix·ξ a(x, ξ)* u(ξ)dξ for u ∈ C0∞ (Ω) and x ∈ Ω .

IRn

(7.1.6) Here u *(ξ) = Fx→ξ u(x) denotes the Fourier transform of u (see (3.1.12)). The operator in (7.1.6) can also be written in the form  A(x, −iD)u = Fξ−1 (7.1.7) →x a(x, ξ)Fy→ξ u(y) or as an iterated integral,   ei(x−y)·ξ a(x, ξ)u(y)dydξ . (7.1.8) A(x, −iD)u = (2π)−n IRn Ω

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431

The set of all standard pseudodifferential operators A(x, −iD) of order m will be denoted by OP S m (Ω × IRn ) and forms a linear vector space together with the usual linear operations. Note that the differential operator of order m, α   ∂ aα (x) −i , (7.1.9) A(x, −iD) = ∂x |α|≤m

with C ∞ –coefficients on Ω belongs to OP S m (Ω × IRn ) with the symbol (7.1.3). For a standard pseudodifferential operator one has the following mapping properties. Theorem 7.1.1. (H¨ormander [188, Theorem 18.1.6], Egorov and Shubin [109, p.7 and Theorem 1.4, p.18], Petersen [346, p.169 Theorem 2.4]) The operator A ∈ OP S m (Ω × IRn ) defined by (7.1.6) is a continuous operator A : C0∞ (Ω) → C ∞ (Ω) .

(7.1.10)

 s (K) The operator A can be extended to a continuous linear mapping from H s−m into Hloc (Ω) for any compact subset K  Ω. Furthermore, in the framework of distributions, A can also be extended to a continuous linear operator A : E  (Ω) → D (Ω) . The proof is available in textbooks as e. g. in Egorov and Shubin [109], Petersen [346]. The main tool for the proof is the well known Paley–Wiener– Schwartz theorem, Theorem 3.1.3. For any linear continuous operator A : C0∞ → C ∞ (Ω) there exists a distribution KA ∈ D (Ω × Ω) such that  Au(x) = KA (x, y)u(y)dy for u ∈ C0∞ (Ω) Ω

where the integration is understood in the distributional sense. Due to the Schwartz kernel theorem, the Schwartz kernel KA is uniquely determined by the operator A (see H¨ormander [188], Schwartz [382] or Taira [412, Theorem 4.5.1]). For an operator A ∈ OP S m (Ω × IRn ), the Schwartz kernel KA (x, y) has the following smoothness property. Theorem 7.1.2. (Egorov and Shubin [109, p.7]) The Schwartz kernel KA of A(x, D) ∈ OP S m (Ω × IRn ) is in C ∞ (Ω × Ω \ {(x, x) | x ∈ IRn }). Moreover, by use of the identity eiz·ξ = |z|−2N (−Δξ )N eiz·ξ with N ∈ IN , KA has the representation in terms of the symbol,

(7.1.11)

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7. Introduction to Pseudodifferential Operators

KA (x, y) = k(x, x − y) = |y − x|

−2N

(2π)

−n





(−Δξ )N a(x, ξ) ei(x−y)·ξ dξ

IRn

for x = y (7.1.12) ! " where N ≥ m+n + 1 for m + n ≥ 0 and N = 0 for m + n < 0. In the latter 2 case, KA is continuous in Ω × Ω and for u ∈ C0∞ (Ω) and x ∈ Ω we have the representation  (7.1.13) Au(x) = k(x, x − y)u(y)dy . Ω

C0∞ (Ω)

If n + m ≥ 0 let ψ ∈ be a cut–off function with ψ(x) = 1 for all x ∈ supp u. Then  cα (x)Dα u(x) (7.1.14) Au(x) = |α|≤2N



+

  k(x, x − y) u(y) − |α|≤2N

Ω

where cα (x) = (2π)−n

 

IRn Ω

 1 (y − x)α Dα u(x) ψ(y)dy α!

1 (y − x)α ψ(y)ei(x−y)·ξ dya(x, ξ)dξ . α!

Alternatively, Au(x) = (A1 u)(x) + (A2 u)(x) ; 

here

k1 (x, x − y)u(y)dy

(A1 u)(x) =

(7.1.15)

Ω



with k1 (x, x − y) = (2π)−n



1 − χ(ξ) a(x, ξ)ei(x−y)·ξ dξ ,

IRn

and the integro–differential operator  (A2 u)(x) = k2 (x, x − y)(−Δy )N u(y)dy

(7.1.16)

Ω

where k2 (x, x − y) = (2π)−n



|ξ|−2N χ(ξ)a(x, ξ)ei(x−y)·ξ dξ

IRn

and χ(ξ) is the cut–off function with χ(ξ) = 0 for |ξ| ≤ ε and χ(ξ) = 1 for |ξ| ≥ R.

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433

Remark 7.1.2: The representation (7.1.14) will be also expressed in terms of Hadamard’s finite part integrals later on. Remark 7.1.3: In the theorem we use the identity (7.1.11) to ensure that the integral (7.1.12) decays at infinity with sufficiently high order, whereas for ξ = 0, we employ the regularization given by the Hadamard’s finite part integrals. In fact, the trick (7.1.11) used in (7.1.12) leads to the definition of the oscillatory integrals which allows to define pseudodifferential operators of order m ≥ −n (see e.g. Hadamard [168], H¨ormander [188, I p.238], Wloka et al [440]). Proof: i) n + m < 0 : In this case, the integral has a weakly singular kernel, hence,   Au(x) = (2π)−n u(y) a(x, ξ)ei(x−y)·ξ dξdy IRn

IRn

and k(x, x − y) = (2π)−n

 a(x, ξ)ei(x−y)·ξ dξ .

IRn

For the regularity, we begin with the identity (7.1.11) setting z = x − y. Since (−Δξ )N a(x, ξ) ∈ S m−2N (Ω × IRn ) integration by parts yields    k(x, x − y) = |x − y|−2N (2π)−n (−Δξ )N a(x, ξ) ei(x−y)·ξ dξ . IRn

This representation can be differentiated 2N times with respect to x and y for x = y. Since N ∈ IN is arbitrary, k is C ∞ for x = y and continuous for x = y. ii) n + m ≥ 0 : Here, with   v(y) := u(y) − |α|≤2N

(7.1.14) has the form Au(x)

=



 1 (x − y)α Dα u(x) , α!

cα (x)Dα u(x)

|α|≤2N

+ (2π)−n

  v(y)ψ(y)ei(x−y)·ξ dya(x, ξ)dξ .

IRn Ω

Note that the latter integral exists since the inner integral defines a C ∞ function which decays faster than any power of ξ due to the Palay–Wiener– Schwartz theorem 3.1.3. With (7.1.11) we obtain

434

I

7. Introduction to Pseudodifferential Operators

:=

(2π)

−n



 v(y)ψ(y)ei(x−y)·ξ dy a(x, ξ)dξ

IRn supp(ψ)

=

(2π)

−n





  v(y)ψ(y)|y − x|−2 (−Δξ )ei(x−y)·ξ dy a(x, ξ)dξ

IRn supp(ψ)

and with integration by parts   $  −n I = lim (2π) v(y)ψ(y)|y − x|−2 ei(x−y)·ξ dy R→∞

|ξ|≤R supp(ψ)

−n





−(2π)

|ξ|=R  supp(ψ) 

+(2π)

−n



(−Δξ )a(x, ξ) dξ



v(y)ψ(y)|y − x|−2 i(x − y)ei(x−y)·ξ dy ·

ξ |ξ| a(x, ξ)dSξ

 v(y)ψ(y)|y − x|−2 ei(x−y)·ξ dy ∇ξ a(x, ξ) ·



ξ |ξ| dSξ

%

|ξ|=R supp(ψ)

= lim [I1 (R) + I2 (R) + I3 (R)] R→∞

Now we examine each of the Integrals Ij for j = 1, 2, 3 separately. Behaviour of I1 (R) : The inner integral of I1 reads   v(y)ψ(y)|y − x|−2 ei(x−y)·ξ dy Φ1 (x, ξ) := supp(ψ)⊂IRn

and is a C ∞ –function of ξ which decays faster than any power of |ξ|; in particular of an order higher than n+m−2, as can be shown with integration by parts and employing (7.1.11) with z and ξ exchanged. Hence, the limit    v(y)ψ(y)|y − x|−2 ei(x−y)·ξ dy (−Δξ )a(x, ξ) dξ lim I1 (R) = (2π)−n R→∞

IRn Ω

exists. Behaviour of I2 (R) : Similarly, the inner integral of I2 ,   v(y)ψ(y)|y − x|−2 (x − y)ei(x−y)·ξ dy , Φ2 (x, ξ) := i Ω ∞

is a C –function of ξ which decays faster than any power of |ξ| = R; in particular of an order higher than m + n − 2. Therefore, I2 (R) = O(Rm+n−1−ν ) → 0 for R → ∞ if ν > m + n − 1 .

7.1 Basic Theory of Pseudodifferential Operators

435

Behaviour of I3 (R) : The inner integral of I3 is Φ1 (x, ξ). Now, choose ν > m + n − 2, then lim I3 (R) = 0. R→∞

To complete the proof, we repeat the process by applying this technique N times. Finally, we obtain    −n v(y)ψ(y)|y − x|−2N ei(x−y)·ξ dy × A(vψ) = (2π) IRn supp(ψ)

 × (−Δξ )N a(x, ξ) dξ . Replacing v(y) by its definition, we find (7.1.14) after interchanging the order of integration. In the same manner as for m + n < 0 one finds that  (−Δξ )N a(x, ξ)ei(x−y)·ξ dσ IRn

is in C ∞ for x = y. The alternative formulae  (7.1.15) and (7.1.16) are direct  consequences of the Fourier transform of (−Δξ )N a(x, ξ) . For the symbol calculus in S m (Ω × IRn ) generated by the algebra of pseudodifferential operators it is desirable to introduce the notion of asymptotic expansions of symbols and use families of symbol classes. We note that the symbol classes S m (Ω × IRn ) have the following properties: a) For m ≤ m we have the inclusions S −∞@(Ω × IRn ) ⊂ S m (Ω × IRn ) ⊂  S m (Ω × IRn ) . S m (Ω × IRn ) where S −∞ (Ω × IRn ) := m∈IR  ∂ β  ∂ α b) If a ∈ S m (Ω × IRn ) then ∂x a ∈ S m−|α| (Ω × IRn ). ∂ξ 



c) For a ∈ S m (Ω ×IRn ) and b ∈ S m (Ω ×IRn ) we have ab ∈ S m+m (Ω ×IRn ). Definition 7.1.3. Given a ∈ S m0 (Ω ×IRn ) and a sequence of symbols amj ∈ S mj (Ω × IRn ) with mj ∈ IR and mj > mj+1 → −∞. We call the formal sum ∞

j=0

amj an asymptotic expansion of a if for every k > 0 there holds

a−

k−1  j=0

amj ∈ S mk (Ω × IRn ) and we write a ∼

∞ 

amj .

j=0

The leading term am0 ∈ S m0 (Ω × IRn ) is called the principal symbol. In fact, if a sequence amj is given, the following theorem holds. Theorem 7.1.3. Let amj ∈ S mj (Ω × IRn ) with mj > mj+1 → −∞ for j → ∞. Then there exists a symbol a ∈ S m0 (Ω × IRn ), unique modulo S −∞ (Ω × IRn ), such that for all k > 0 we have

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7. Introduction to Pseudodifferential Operators

a−

k−1 

amj ∈ S mk (Ω × IRn ) .

(7.1.17)

j=0

For the proof of this theorem one may set a(x, ξ) :=

∞ ξ  amj (x, ξ) χ tj j=0

(7.1.18)

by using a C ∞ cut–off function χ(η) with χ(η) = 1 for |η| ≥ 1 and χ(η) = 0 for |η| ≤ 12 and by using a sequence of real scaling factors with tj → ∞ for j → ∞ sufficiently fast. For the details of the proof see H¨ormander [186, 1.1.9], Taylor [416, Chap. II, Theorem 3.1]. With the help of the asymptotic expansions it is useful to identify in n S m (Ω×IRn ) the subclass S m c (Ω×IR ) called classical (also polyhomogeneous) symbols. A symbol a ∈ S m (Ω × IRn ) is called a classical symbol if there exist functions am−j (x, ξ) with am−j ∈ S m−j (Ω × IRn ) , j ∈ IN0 such that a ∼ ∞ j=0 am−j where each am−j is of the form (7.1.5) with (7.1.4) and is of homogeneous degree mj = m − j; i.e., satisfying am−j (x, tξ) = tm−j am−j (x, ξ) for t ≥ 1 and |ξ| ≥ 1 .

(7.1.19)

n The set of all classical symbols of order m will be denoted by S m c (Ω × IR ). n m We remark that for a ∈ S c (Ω × IR ), the homogeneous functions am−j (x, ξ) for |ξ| ≥ 1 are uniquely determined. Moreover, note that the asymptotic expansion means that for all |α|, |β| ≥ 0 and every compact K  Ω there exist constants c(K, α, β) such that N  ∂ β  ∂ α      a(x, ξ) − am−j (x, ξ)  ≤ c(K, α, β)ξm−N −|α|−1  ∂x ∂ξ j=0

(7.1.20) holds for every N ∈ IN0 . For a given symbol or for a given asymptotic expansion, the associated operator A is given via the definition (7.1.8) or with Theorem 7.1.3, in addition. On the other hand, if the operator A is given, an equally important question is how to find the corresponding symbol since by examining the symbol one may deduce further properties of A. In order to reduce A to the form A(x, D) in (7.1.6), we need the concept of properly supported operators with respect to their Schwartz kernels KA . Definition 7.1.4. A distribution KA ∈ D (Ω × Ω) is called properly supported if supp KA ∩ (K × Ω) and supp KA ∩ (Ω × K) are both compact in Ω × Ω for every compact subset K  Ω (Treves [420, Vol.I p.25]). We recall that the support of KA is defined by the relation z ∈ supp KA ⊂ IR2n if and only if for every neighbourhood U (z) there exists ϕ ∈ C0∞ (U(z))

7.1 Basic Theory of Pseudodifferential Operators

437

such that KA , ϕ = 0. In accordance with Definition 7.1.4, any continuous linear operator A : C0∞ → C ∞ (Ω) is called properly supported if and only if its Schwartz kernel KA ∈ D (Ω × Ω) is a properly supported distribution. In order to characterize properly supported operators we recall the following lemma. Lemma 7.1.4. (see Folland [126, Proposition 8.12]) A linear continuous operator A : C0∞ (Ω) → C ∞ (Ω) is properly supported if and only if the following two conditions hold: i) For any compact subset Ky  Ω there exists a compact Kx  Ω such that supp v ⊂ Ky implies supp(Av) ⊂ Kx . ii) For any compact subset Kx  Ω there exists a compact Ky  Ω such that supp v ∩ Ky = ∅ implies supp(Av) ∩ Kx = ∅. Proof: We first show the necessity of i) and ii) for any given properly supported operator A. To show i), let Ky be any fixed compact subset of Ω. Then define Kx := {x ∈ Ω |

there exists y ∈ Ky with (x, y) ∈ supp KA ∩ (Ω × Ky )}

where KA is the Schwartz kernel of A. Clearly, Kx is a compact subset of Ω since supp KA ∩ (Ω × Ky ) is compact because KA is properly supported. In order to show i), i.e. supp(Av) ⊂ Kx , we consider any v ∈ C0∞ (Ω) with supp v ⊂ Ky and ψ ∈ C0∞ (Ω \ Kx ). Then   KA (x, y)v(y)dyψ(x)dx Av, ψ = Ω Ω

 

=

KA (x, y)ψ(x)v(y)dxdy = 0 Ky Ω

since for x ∈ supp ψ ⊂ Ω \ Kx and all y ∈ supp v we have (x, y) ∈ supp KA ∩ (Ω ×Ky ). Hence, Av, ψ = 0 for all ψ ∈ C0∞ (Ω \Kx ) and we find supp(Av) ⊂ Kx . For ii), let Kx be any compact fixed subset of Ω. Then define Ky := {y ∈ Ω |

there exists x ∈ Kx with (x, y) ∈ supp KA ∩ (Kx × Ω)} ,

which is a compact subset of Ω. For v ∈ C0∞ (Ω \Ky ) and ψ ∈ C0∞ (supp(Av)∩ Kx ) we have   Av, ψ = KA (x, y)v(y)ψ(x)dydx = 0 Ω Ω

since for every y ∈ supp v we have y ∈ Ky and (x, y) ∈ supp KA ∩ (Kx × Ω) for all x ∈ supp ψ. Hence, Av, ψ = 0 for all ψ ∈ C0∞ (supp(Av) ∩ Kx ) which implies supp(Av) ∩ Kx = ∅.

438

7. Introduction to Pseudodifferential Operators

Next, we show the sufficiency of i) and ii). Let Kx be any compact subset of Ω and Ky the corresponding subset in ii) such that supp v ∩Ky = ∅ implies supp(Av) ∩ Kx = ∅. We want to show that supp KA ∩ (Kx × Ω) is compact. To this end, we consider ψ ∈ C0∞ (Ω) with supp ψ ⊂ Kx . Then   KA (x, y)v(y)ψ(x)dydx Av, ψ = Ω Ω





=

KA (x, y)v(y)dyψ(x)dx = 0 Ω Ω\Ky

∞ for  every v ∈ C0 (Ω) with supp v ∩ Ky = ∅ due to ii). This implies supp KA ∩ Kx × (Ω \ Ky ) = ∅; so,

supp KA ∩ (Kx × Ω) = supp KA ∩ (Kx × Ky ) ⊂ Kx × Ky . Hence, supp KA ∩ (Kx × Ω) is compact because Kx × Ky is compact. To show that for any chosen compact Ky  Ω, the set supp KA ∩ (Ω × Ky ) is compact we invoke i) and take Kx to be the corresponding compact set in Ω. Then for any v ∈ C0∞ (Ω) with supp v ⊂ Ky , we have supp(Av) ⊂ Kx . Hence, for every ψ ∈ C0∞ (Ω \ Kx ),   KA (x, y)v(y)dyψ(x)dx = Av, ψ = 0 . Ω Ω

As a consequence, we find supp KA ∩ (Ω × Ky ) = supp KA ∩ (Kx × Ky ) ⊂ Kx × Ky ; so, supp KA ∩ (Ω × Ky ) is compact. Thus, KA is properly supported.  Lemma 7.1.4 implies the following corollary. Corollary 7.1.5. The operator A : C0∞ (Ω) → C ∞ is properly supported if and only if the following two conditions hold: i) For any compact subset Ky  Ω there exists a compact Kx  Ω such that (7.1.21) A : C0∞ (Ky ) → C0∞ (Kx ) is continuous. ii) For any compact subset Kx  Ω there exists a compact Ky  Ω such that A : C0∞ (Kx ) → C0∞ (Ky ) is continuous.

(7.1.22)

Proof: Since A : C0∞ (Ω) → C ∞ (Ω) is continuous, i) in Lemma 7.1.4 is equivalent to (7.1.21).  (x, y) = KA (y, x) is the Now, if A is properly supported then KA Schwartz kernel of A and, hence, also properly supported. Therefore i)  (y, x) which, for this case, is in Lemma 7.1.4 is valid for KA (x, y) = KA

7.1 Basic Theory of Pseudodifferential Operators

439

equivalent to (7.1.22). Consequently, (7.1.21) and (7.1.22) are satisfied if A is properly supported. Conversely, if (7.1.21) and (7.1.22) hold, then i) in Lemma 7.1.4 is already satisfied. In order to show ii) let us choose any compact subset Kx of Ω and let Ky be the corresponding compact subset in (7.1.22). Let v ∈ C0∞ (Ω \ Ky ). Then Av ∈ C0∞ (Ω) due to (7.1.21). Now, let ψ ∈ C0∞ (Ω) be any function with supp ψ ⊂ Kx . Then supp A ψ ⊂ Ky because of (7.1.22). Hence, Av, ψ = v, A ψ = 0 for any such ψ. Therefore, supp(Av)∩Kx = ∅ which is the second proposition ii) of Lemma 7.1.4.  As an obvious consequence of Corollary 7.1.5, the following proposition is valid. Proposition 7.1.6. If A and B : C0∞ (Ω) → C ∞ (Ω) are properly supported, then the composition A ◦ B is properly supported, too. Theorem 7.1.7. If A ∈ OP S m (Ω × IRn ) is properly supported then a(x, ξ) = e−ix·ξ (Aeiξ• )(x)

(7.1.23)

is the symbol (see Taylor [416, Chap. II, Theorem 3.8]). To express the transposed operator A of A ∈ OP S m (Ω × IRn ) with the given symbol a(x, ξ) , a ∈ S m (Ω × IRn ) we use the distributional relation   Au, v = u, A v = Au(x)v(x)dx = u(x)A v(x)dx for u, v ∈ C0∞ (Ω) . Ω

Ω

Then from the definition (7.1.8) of A, we find   ei(x−y)·ξ a(y, −ξ)v(y)dydξ . A v(x) = (2π)−n IRn

(7.1.24)

Ω

From this representation it is not transparent that A is a standard pseudodifferential operator since it does not have the standard form (7.1.8). In fact, the following theorem is valid. Theorem 7.1.8. (Taylor [416, Chap. II, Theorem 4.2]) If A ∈ OP S m (Ω × IRn ) is properly supported, then A ∈ OP S m (Ω × IRn ). If, however, A ∈ OP S m (Ω × IRn ) is not properly supported, then A belongs to a slightly larger class of operators. Note that if we let 

a(x, y, ξ) := a(y, −ξ)

then we may rewrite A v in the form

440

7. Introduction to Pseudodifferential Operators 

−n

 

A v(x) = (2π)



a(x, y, ξ)ei(x−y)·ξ v(y)dydξ .

(7.1.25)

IRn Ω

Therefore, H¨ormander in [186] introduced the more general class of Fourier integral operators of the form   Au(x) = (2π)−n ei(x−y)·ξ a(x, y, ξ)u(y)dydξ (7.1.26) IRn Ω

with the amplitude function a ∈ S m (Ω × Ω × IRn ) and with the special phase function ϕ(x, y, ξ) = (x − y) · ξ. The integral in (7.1.26) is understood in the sense of oscillatory integrals by employing the same procedure as in (7.1.11), (7.1.12). (See e.g. H¨ormander [186], Treves [420, Vol. II p.315]). This class of operators will be denoted by Lm (Ω). Theorem 7.1.9. (H¨ormander [186, Theorem 2.1.1], Taira [412, Theorem 6.5.2]) Every operator A ∈ Lm (Ω) can be written as A = A0 (x, −iD) + R where A0 (x, −iD) ∈ OP S m (Ω × IRn ) is properly supported and R ∈ L−∞ (Ω) where ) L−∞ (Ω) := Lm (Ω) = OP S −∞ (Ω × Ω × IRn ) . m∈IR

Proof: A simple proof by using a proper mapping is available in [412] which is not constructive. Here we present a constructive proof based on the presentation in Petersen [346] and H¨ormander [188, Prop.18.1.22]. It is based on a partition of the unity over Ω with corresponding functions {φ } , φ ∈ C0∞ (Ω) ,  φ (x) = 1. For every x0 ∈ Ω and balls Bε (x0 ) ⊂ Ω with fixed ε > 0, the number of φ with supp φ ∩ Bε (x0 ) = ∅ is finite. Let us denote by I1 = {(j, ) | supp φj ∩ supp φ = ∅} and by I2 = {(j, ) | supp φj ∩ supp φ = ∅} the corresponding index sets. Then  Au(x) = φj (x)(Aφ u)(x) j,

=

 I1

φj (x)Aφ u +



φj (x)Aφ u .

I2

For every (j, ) ∈ I2 we see that the corresponding Schwartz kernel is given by φj (x)k(x, x − y)φ (y) which is C0∞ (Ω × Ω). Moreover, for any pair (x0 , y0 ) ∈ Ω × Ω, the sum  φj (x)k(x, x − y)φ (y) for (x, y) ∈ Bε (x0 ) × Bε (y0 ) I2

7.1 Basic Theory of Pseudodifferential Operators

441

for ε > 0 chosen appropriately small, has only finitely many nontrivial terms. Hence,  R := φj Aφ ∈ OP S −∞ (Ω × IRn ) . I2

The operator Q := I1 φj Aφ , however, is properly supported since it has the Schwartz kernel I1 φj (x)k(x, x − y)φ (y). Each term has compact support for fixed x with respect to y and with fixed y with respect to x. Now, if x ∈ K  Ω then, due to the compactness of K, we have supp φj ∩ K = ∅ only for finitely many indices j whose collection we call J (K). For every j ∈ J (K), there are only finitely many  with supp φj ∩ supp φ = ∅; we denote the collection of these  by J (K, j). Hence, if x traces K, then φj (x)k(x, x − y)φ (y) = 0 only for finitely many indices (j, ) with j ∈ J (K) and  ∈ J (K, j); and we see that   φj (x)K(x, x − y)φ (y) ∩ (K × Ω) supp j∈J (K) ∈J (K,j)

is compact since there are only finitely many terms in the sum. If y ∈ K  Ω, we find with the same arguments that   (K ∩ Ω) ∩ supp φj (x)k(x, x − y)φ (y) ∈J (K) j∈J (K,)

is compact. Hence, Q is properly supported.



We emphasize that Lm (Ω) = OP S m (Ω × IRn ) ⊂ Lm (Ω). The operators R ∈ L−∞ are called smoothing operators. (In the book by Taira [412] the smoothing operators are called regularizers.) Definition 7.1.5. A continuous linear operator A : C0∞ (Ω) → D (Ω) is called a smoothing operator if it extends to a continuous linear operator from E  (Ω) into C ∞ (Ω). The following theorem characterizes the smoothing operators in terms of our operator classes. Theorem 7.1.10. (Taira [412, Theorem 6.5.1, Theorem 4.5.2]) The following four conditions are equivalent: (i) A is a smoothing operator, ) Lm (Ω), (ii) A ∈ L−∞ (Ω) = m∈IR

(iii) A is of the form (7.1.26) with some a ∈ S −∞ (Ω × Ω × IRn ), (iv) A has a C ∞ (Ω × Ω) Schwartz kernel. Theorem 7.1.11. (H¨ormander [186, p.103], Taylor [416, Chap. II, Theorem 3.8]) Let A ∈ Lm (Ω) be properly supported. Then

442

7. Introduction to Pseudodifferential Operators

a(x, ξ) = e−ix·ξ A(eiξ·• )(x)

(7.1.27)

with a ∈ S m (Ω × IRn ) and A = A(x, −iD) ∈ OP S m (Ω × IRn ). Furthermore, if A has an amplitude a(x, y, ξ) with a ∈ S m (Ω × Ω × IRn ) then a(x, ξ) has the asymptotic expansion   1  ∂ α  ∂ α a(x, ξ) ∼ a(x, y, ξ) . (7.1.28) −i α! ∂ξ ∂y |y=x α≥0 Here we omit the proof. Theorems 7.1.9 and 7.1.11 imply that if A ∈ Lm (Ω) then A can always be decomposed in the form A = A0 + R with A0 ∈ OP S m (Ω × IRn ) having a properly supported Schwartz kernel and a smoothing operator R. Clearly, A0 is not unique. Hence, to any A and A0 we can associate with A a symbol a(x, ξ) := e−ix·ξ A0 (eiy·ξ )(x)

(7.1.29)

and the corresponding asymptotic expansion (7.1.28). Now we define σA

:=

the equivalence class of all the symbols associated (7.1.30) ? −∞ n m with A defined by (7.1.29) in S (Ω × IR ) S (Ω × IRn ) .

This equivalence class is called the complete symbol class of A ∈ Lm (Ω). Clearly, the mapping ? Lm (Ω)  A → σA ∈ S m (Ω × IRn ) S −∞ (Ω × IRn ) induces an isomorphism ? ? Lm (Ω) L−∞ → S m (Ω × IRn ) S −∞ (Ω × IRn ) . The equivalence class defined by σmA :=

the equivalence class of all the symbols associated ? with A defined by (7.1.29) in S m (Ω × IRn ) S m−1 (Ω × IRn ) (7.1.31) is called the principal symbol class of A which induces an isomorphism ? ? Lm (Ω) Lm−1 → S m (Ω × IRn ) S m−1 (Ω × IRn ) . As for equivalence classes in general, one often uses just one representative (such as the asymptotic expansion (7.1.28)) of the class σA or σmA , respectively, to identify the whole class in S m (Ω × IRn ). In view of Theorem 7.1.10, we now collect the mapping properties of the pseudodifferential operators in Lm (Ω) in the following theorem without proof.

7.1 Basic Theory of Pseudodifferential Operators

443

Theorem 7.1.12. If A ∈ Lm (Ω) then the following mappings are continuous (see Treves [420, Chap. I, Corollary 2.1 and Theorem 2.1], Taira [412, Theorem 6.5.9]): A : A : A :

C0∞ (Ω) E  (Ω) s Hcomp (Ω)

→ C ∞ (Ω) , → D (Ω) , s−m → Hloc (Ω) .

(7.1.32)

If in addition, A ∈ Lm (Ω) is properly supported, then the mappings can be extended to continuous mappings as follows [420, Chap. I, Proposition 3.2], [412, Theorem 6.5.9]: A A A A A A

: C0∞ (Ω) : C ∞ (Ω) : E  (Ω) : D (Ω) s (Ω) : Hcomp s : Hloc (Ω)

→ → → → → →

C0∞ (Ω) , C ∞ (Ω) , E  (Ω) , D (Ω) , s−m Hcomp (Ω) , s−m Hloc (Ω) .

(7.1.33)

Based on Theorem 7.1.9 and the concept of the principal symbol, one may consider the algebraic properties of pseudodifferential operators in Lm (Ω). The class Lm (Ω) is closed under the operations of taking the transposed and the adjoint of these operators. Theorem 7.1.13. If A ∈ Lm (Ω) then its transposed A ∈ Lm (Ω) and its adjoint A∗ ∈ Lm (Ω). The corresponding complete symbol classes have the asymptotic expansions α    1  ∂ α  ∂ σA ∼ σA (x, −ξ) (7.1.34) −i α! ∂ξ ∂x α≥0

and σA∗

∼

α  ∗  1  ∂ α  ∂ −i σA (x, ξ) α! ∂ξ ∂x

(7.1.35)

α≥0

where σA denotes one of the representatives of the complete symbol class. For the detailed proof we refer to [420, Chap. I, Theroem 3.1]. With respect to the composition of operators, the class of all Lm (Ω) is not closed. However, the properly supported pseudodifferential operators in Lm (Ω) form an algebra. More precisely, we have the following theorem [188, 18.1], [416, Chap.II, Theorem 4.4.]. Theorem 7.1.14. Let A ∈ Lm1 (Ω) , B ∈ Lm2 (Ω) and one of them be properly supported. Then the composition A ◦ B ∈ Lm1 +m2 (Ω)

(7.1.36)

444

7. Introduction to Pseudodifferential Operators

and we have the asymptotic expansion for the complete symbol class:     1  ∂ α ∂ α σA (x, ξ) −i σB (x, ξ) . (7.1.37) σA◦B ∼ α! ∂ξ ∂x α≥0

Here, σA (x, ξ) and σB (x, ξ) denote one of the respective representatives in the corresponding equivalence classes of the complete symbol classes. For the proof, we remark that with Theorem 7.1.9 we have either A = A0 + R and B = B0 or A = A0 and B = B0 + Q. Then A◦B

=

A0 ◦ B0 + R ◦ B0

A◦B

=

A 0 ◦ B 0 + A0 ◦ Q .

or

Since A0 and B0 are properly supported, R ◦ B0 or A0 ◦ Q are continuous mappings from E  (Ω) to C ∞ (Ω) and, hence, are smoothing operators. (Note that for A and B both not properly supported, the composition generates the term R ◦ Q, the composition of two regularizers, which is not defined in general.) For the remaining products A0 ◦ B0 , we can use that without loss of generality, A0 = A0 (x, D) ∈ OP S m1 (Ω) and B0 = B0 (x, D) ∈ OP S m2 (Ω). Hence, one obtains for C := A0 ◦ B0 a representation in the form (7.1.26) with the amplitude function   a(x, ξ)ei(x−y)·ξ b(y, η)ei(y−z)·η dydξ c(x, z, η) = IRn Ω

for which one needs to show c ∈ S m1 +m2 (Ω × Ω × IRn ) and, with the Leibniz rule, to evaluate the asymptotic expansion (7.1.28) for c. For details see e. g. [420, Chap. I, Theorem 3.2]. As an immediate consequence of Theorem 7.1.14, the following corollary holds. Corollary 7.1.15. Let A ∈ Lm1 (Ω) , B ∈ Lm2 (Ω) and one of them be properly supported. Then the commutator satisfies [A, B] := A ◦ B − B ◦ A ∈ Lm1 +m2 −1 (Ω) .

(7.1.38)

Proof: For A ◦ B and B ◦ A we have Theorem 7.1.14 and the commutator’s symbol has the asymptotic expansion    1  ∂ α ∂ α σA (x, ξ) −i σB (x, ξ) α! ∂ξ ∂x |α|≥1     ∂ α ∂ α σB (x, ξ) −i σA (x, ξ) . − ∂ξ ∂x

σ[A,B] ∼

7.1 Basic Theory of Pseudodifferential Operators

Hence, the order of [A, B] equals m1 − 1 + m2 .

445



To conclude this section, we now return to a subclass, the class of classical pseudodifferential operators, which is very important in connection with elliptic boundary value problems and boundary integral equations. Definition 7.1.6. A pseudodifferential operator A ∈ Lm (Ω) is said to be classical if its complete symbol σA has a representative in the class n n m Sm c (Ω × IR ), cf. (7.1.19). We denote by Lc (Ω × IR ) the set of all classical pseudodifferential operators of order m. n We remark that for Lm c (Ω × IR ), the mapping ? −∞ n m Lm (Ω × IRn ) c (Ω)  A → σA ∈ S c (Ω × IR ) S

induces the isomorphism ? −∞ ? −∞ n Lm (Ω) → S m (Ω × IRn ) . c (Ω) L c (Ω × IR ) S Moreover, we have L−∞ (Ω) =

)

(7.1.39)

Lm c (Ω) .

(7.1.40)

m∈IR

In this case, for A ∈ Lm c (Ω), the principal symbol σmA has a representative   ξ a0m (x, ξ) := |ξ|m am x, (7.1.41) |ξ| which belongs to C ∞ (Ω ×(IRn \{0})) and is positively homogeneous of degree m with respect to ξ. The function a0m (x, ξ) in (7.1.41) is called the homogeneous principal symbol of A ∈ Lm c (Ω). Note, in contrast to the principal symbol of A defined in (7.1.31), the homogeneous principal symbol is only a single function which represents the whole equivalence class in (7.1.31). Correspondingly, if we denote by   ξ 0 m−j (7.1.42) am−j x, am−j (x, ξ) := |ξ| |ξ| the homogeneous parts of the asymptotic expansion of the classical symbol σmA , which have the properties a0m−j (x, ξ) a0m−j (x, tξ)

=

am−j (x, ξ)

for |ξ|

≥

1

and

=

tm−j a0m−j (x, ξ)

for all t

>

0

and ξ = 0 ,

then σmA may be represented asymptotically by the formal sum

∞  j=0

a0m−j (x, ξ).

446

7. Introduction to Pseudodifferential Operators

Coordinate Changes Let Φ : Ω → Ω  be a C ∞ –diffeomorphism between the two open subsets Ω and Ω  in IRn , i.e. x = Φ(x). The diffeomorphism is bijective and x = Φ−1 (x ). If v ∈ D(Ω  ) then u(x) := (v ◦ Φ)(x) =: (Φ∗ v)(x)

(7.1.43)

defines u ∈ D(Ω) and the linear mapping Φ∗ : D(Ω  ) → D(Ω) is called the pullback of v by Φ. Correspondingly, the pushforward of u by Φ is defined by the linear mapping Φ∗ : D(Ω) → D(Ω  ) given by v(x ) := u ◦ Φ−1 (x ) =: (Φ∗ u)(x ) .

(7.1.44)

Now we consider the behaviour of a pseudodifferential operator A ∈ Lm (Ω) under changes of coordinates given by a C ∞ –diffeomorphism Φ. Theorem 7.1.16. (H¨ormander [188, Th.18.1.17]) Let A ∈ Lm (Ω) then AΦ := Φ∗ AΦ∗ ∈ Lm (Ω  ). Moreover, if A = A0 +R with A0 properly supported and R a smoothing operator then in the decomposition AΦ = Φ∗ A0 Φ∗ + Φ∗ RΦ∗ , Φ∗ A0 Φ∗ is properly supported and Φ∗ RΦ∗ is a smoothing operator on Ω  . The complete symbol class of AΦ has the following asymptotic expansion aΦ (x , ξ  )

∼

  1  ∂ α  ∂ α    a (x , y , ξ ) (7.1.45) − i Φ α! ∂ξ  ∂y  |y =x

0≤|α|

where aΦ (x , y  , ξ  )

=

 −1  x − y   ∂Φ −1  det Ξ  (x , y  ) (y) det ψ ∂y ε   −1  a x, Ξ  (x , y  ) ξ (7.1.46)

with x = Φ−1 (x ) , y = Φ−1 (y  ) and where the smooth matrix–valued function Ξ(x , y  ) is defined by Ξ(x , y  )(x − y  ) = Φ−1 (x ) − Φ−1 (y  ) which implies Ξ(x , x ) =



∂Φ ∂x

(7.1.47)

−1 .

(7.1.48)

−1  exists for |x − y  | ≤ 2ε. The constant ε > 0 is chosen such that Ξ(x , y  ) (7.1.45) yields for the principal symbols the relation

7.1 Basic Theory of Pseudodifferential Operators

  ∂Φ   σAΦ ,m (x , ξ  ) = σA,m Φ(−1) (x ) , ξ . ∂x

447

(7.1.49)

Here, ψ(z) is a C0∞ cut–off function with ψ(z) = 1 for all |z| ≤ ψ(z) = 0 for all |z| ≥ 1.

1 2

and

Remark 7.1.4: The relation between AΦ and A can be seen in Figure 7.1.1 D(Ω) Φ∗

A-

6

D(Ω  )

E(Ω) Φ∗

Φ∗AΦ∗ AΦ

?

E(Ω  )

Figure 7.1.1: The commutative diagram of a pseudodifferential operator under change of coordinates

Proof: i) If A0 is properly supported, so is Φ∗ A0 Φ∗ due to the following. With Lemma 7.1.4, we have that to any compact subset Ky  Ω  the subset Φ(−1) (Ky ) = Ky ⊂ Ω is compact. Then to Ky  Ω there exists the compact subset Kx  Ω such that for every u ∈ D(Ω) with supp u ⊂ Ky there holds supp A0 u ⊂ Kx . Consequently, for any v ∈ D(Ω  ) with supp v ⊂ Ky we have supp Φ∗ v ⊂ Φ(−1) (Ky ) = Ky and supp A0 Φ∗ v ⊂ Kx . This implies supp Φ∗ A0 Φ∗ v = supp A0Φ v = Φ(supp A0 Φ∗ v) ⊂ Φ(Kx ) =: Kx . On the other hand, if Kx  Ω  is a given compact subset then Kx := Φ (Kx ) is a compact subset in Ω to which there exists a compact subset Ky  Ω with the property that for every u ∈ D(Ω) with supp u ∩ Ky = ∅ there holds supp A0 u ∩ Kx = ∅. Now let Ky := Φ(Ky ) which is a compact subset of Ω  . Thus, if v ∈ D(Ω  ) with supp v ∩ Ky = ∅ then (−1)

supp Φ∗ v ∩ Ky = (Φ(−1) supp v) ∩ Ky = Φ(−1) (supp v ∩ Ky ) = ∅ and, since A0 is properly supported, supp A0 Φ∗ v ∩ Kx = ∅. Hence,  Φ(−1) Φ(supp A0 Φ∗ v) ∩ Φ(Kx ) = ∅ which yields supp Φ∗ A0 Φ∗ v ∩ Kx = ∅ . According to Lemma 7.1.4, this implies that Φ∗ A0 Φ∗ is properly supported. Clearly, for v ∈ D(Ω  ) we have    ∂Φ −1  dy  (Φ∗ RΦ∗ v)(x ) = kR Φ(−1) (x ) , Φ(−1) (y  ) v(y  ) det ∂y Ω

448

7. Introduction to Pseudodifferential Operators

with the Schwartz kernel kR ∈ C ∞ (Ω × Ω) of R. The transformed operator Φ∗ RΦ∗ has the Schwartz kernel  ∂Φ −1   kR Φ(−1) (x ) , Φ(−1) (y  ) det ∈ C ∞ (Ω  × Ω  ) . ∂y Hence, Φ∗ RΦ∗ is a smoothing operator. ii) It remains to show that the properly supported part Φ∗ A0 Φ∗ ∈ OP S m (Ω  × IRn ). For any v ∈ D(Ω  ) we have the representation  (Φ∗ A0 Φ∗ v)(x ) = (A0 Φ∗ v) Φ(−1) (x )   −n = (2π) ei(x−y)·ξ a(x, ξ)(v ◦ Φ)(y)dydξ IRn Ω

=

(2π)−n



 ei(x−y)·(Ξ



(x ,y  )ξ)

a(x, ξ)v(y  ) ×

IRn Ω  ∧|x −y  |≤2ε



×ψ where (Rv)(x)

=

(2π)

−n

 

IRn Ω 

x − y  ε

 −1 ∂φ det dy  dξ + Rv ∂y

  x − y   × ei(x−y)·ξ a(x, ξ)v(y  ) 1 − ψ ε  ∂φ −1  × det dy dξ ∂y

and y = Φ(−1) (y  ) , x = Φ(−1) (x ). With the affine transformation of coordinates, Ξ  (x , y  )ξ = ξ  , the first integral reduces to (AΦ,0 v)(x ) = (2π)−n

 







ei(x −y )·ξ v(y  )aΦ (x , y  , ξ  )dy  dξ 

IRn Ω 

with aΦ (x , y  , ξ  ) given by (7.1.46). Since a(x, ξ) belongs to S m (Ω×, IRn ), with the chain rule one obtains the estimates   ∂ β  ∂ γ  ∂ α      a (x , y , ξ )  ≤ c(K, α, β, γ)(1 + |ξ  |)m−|α|  Φ ∂x ∂y  ∂ξ  for all (x , y  ) ∈ K × K , ξ  ∈ IRn and any compact subset K  Ω  . Hence, aΦ ∈ S m (Ω  × Ω  × IRn ). Since the Schwartz kernel of AΦ,0 satisfies

7.1 Basic Theory of Pseudodifferential Operators

KAΦ,0 (x , x − y  ) = KΦ∗ A0 Φ∗ (x , x − y  )ψ

449

 x − y   ε

and Φ∗ A0 Φ∗ is properly supported, so is AΦ,0 . Now Theorem 7.1.11 can be applied to AΦ,0 . −1      ∂φ Since the function v(y  ) 1 − ψ x −y det vanishes in the vicinε ∂y ity of x = y  identically, we can proceed in the same manner as in the proof of Theorem 7.1.2 and obtain     x − y   Rv(x ) = (2π)−n × ei(x−y)·ξ v(y  ) 1 − ψ ε IRn Ω 

  ∂φ −1  × det (−Δξ )N a(x, ξ) dy  dξ . ∂y

Here, for N > m + n + 1 we may interchange the order of integration and find that R is a smoothing integral operator with C ∞ –kernel. Therefore, Φ∗ A0 Φ∗ ∈ OP S m (Ω  × IRn ) and the asymptotic formula (7.1.45) is a consequence of Theorem 7.1.11. This completes the proof.  The transformed symbol in (7.1.45) is not very practical from the computational point of view since one needs the explicit form of the matrix Ξ(x , y  ) in (7.1.47) and its inverse. Alternatively, one may use the following asymptotic expansion of the symbol. Lemma 7.1.17. Under the same conditions as in Theorem 7.1.16, the complete symbol class of AΦ also has the asymptotic expansion    1  ∂ α ∂ α iξ ·r  aΦ (x , ξ  ) ∼ −i a(x, ξ) e (7.1.50) α! ∂ξ ∂z | ∂Φ  |z=x ξ= ξ 0≤α

∂x



where x = Φ(x) and r = Φ(z) − Φ(x) −

∂Φ (x)(z − x) . ∂x

(7.1.51)

Moreover, 

∂ α iξ ·r  −i |z=x = e ∂z



0

(−i)

|α|−1



∂ ∂x

α



Φ(x) · ξ

for |α| = 1 , 

for |α| ≥ 2 . (7.1.52)

Proof: In view of Theorem 7.1.16 it suffices to consider properly supported operators A0 and AΦ,0 . Therefore the symbol class to AΦ,0 can be found by using formula (7.1.29), i.e., σAΦ,0 (x , ξ  )







=

e−ix ·ξ AΦ,0 eiξ ·• (x )

=

e−iΦ(x)·ξ A0 eiξ ·Φ(·) (x) for x = Φ−1 (x )





450

7. Introduction to Pseudodifferential Operators

after coordinate transform. With the substitution Φ(z) = Φ(x) +

∂Φ (x)(z − x) + r(x, z) ∂x

we obtain σAΦ,0 (x , ξ  )

=

∂Φ 

e−ix·( ∂x

ξ

) (2π)−n

 

e−i(x−z)·ξ a(x, ξ) ×

IRn Ω ∂Φ  ξ ∂x

=

×eiz·( e−ix·ζ A ◦ Bei•·ζ

) eir(x,z)·ξ dzdξ

where A = a(x, D) and B is the multiplication operator defined by the symbol σB (z, ξ) = eir(x,z)·ξ



with x and ξ  fixed as constant parameters. With formula (7.1.29) we then find

σAΦ,0 (x , ξ  ) ∼ σA◦B (x, ζ) where x = Φ(x) and ζ =

∂Φ   ξ . ∂x

The symbol of A ◦ B is given by (7.1.37), which now implies     1  ∂ α ∂ α ir(x,z)·ξ   a(x, ξ) e −i σAΦ,0 (x , ξ ) ∼ α! ∂ξ ∂z |ξ=ζ |z=x α≥0



 with constant ζ = ∂Φ ∂x ξ , which is the desired To show (7.1.52) we employ induction for we see that    ∂ iξ ·r(x,y) ∂Φ e|z=x = ieiξ ·r(x,z) (z) − ∂zj ∂zj

and ∂ ∂ iξ ·r(x,z) e ∂zk ∂zj |z=x

=

=

formula (7.1.50). α with |α| ≥ 1. In particular   ∂Φ (x) · ξ  =0 ∂xj |z=x

   ∂Φ ∂Φ 2 iξ  ·r(x,z) i e (z) − (x) · ξ  × ∂zj ∂xj   ∂Φ ∂Φ (z) − (x) · ξ  ∂zk ∂xk   ∂2Φ +ieiξ ·r(x,z) (z) · ξ  ∂zj ∂zk |z=x i

∂2Φ (x) · ξ  , ∂xj ∂xk

7.1 Basic Theory of Pseudodifferential Operators

451

which is (7.1.52) for 1 ≤ |α| ≤ 2. For |α| > 2, we observe that every differentiation yields terms which contain at least one factor of the form  ∂Φ  ∂Φ (z) − (x) · ξ  which vanishes for z = x, except the last term which ∂z ∂x is of the form  ∂ α i Φ(x) · ξ  , ∂x which proves (7.1.52).  We comment that in this proof we did not use the explicit representation of AΦ,0 in the form (AΦ,0 v)(x) = (2π)−n 

 

IRn Ω 

×v(y) det

 ∂Φ      a x, ξ  eih+i(x −y )·ξ × ∂x

∂Φ −1  ∂Φ    det dy dξ ∂y ∂x

where h(x , y  , ξ  ) =

 ∂Φ ∂x

 (x − y  ) · ξ  with x = Φ−1 (x ) and y = Φ−1 (y  ) .

Now it follows from Theorem 7.1.11 that   1  ∂ α  ∂ α    a (x , y , ξ ) − i aΦ (x , ξ  ) ∼ Φ α! ∂ξ  ∂y  |y =x |α|≥0

where aΦ (x , y  , ξ  )

=

 ∂Φ    ∂Φ −1  ∂Φ  det . a x, ξ  eir det ∂x ∂y ∂x

 ∂Φ   ∂Φ    Since det a x, ξ is independent of y  , interchanging differentia∂x ∂x tion yields the formula  1 ∂Φ  ∂ α    ∂Φ    a x, det ξ × α! ∂x ∂ξ  ∂x 0≤|α|  ∂Φ  −1  ∂ α  ih(x ,y ,ξ )  (y ) . e det × −i  ∂y ∂y |y =x

aΦ (x , ξ  ) ∼

(7.1.53)

Needless to say that both formulae (7.1.50) and (7.1.53) are equivalent, how ∂Φ −1 ever, in (7.1.53) one needs to differentiate the Jacobian det which ∂y makes (7.1.53) less desirable than (7.1.50) from the practical point of view.

452

7. Introduction to Pseudodifferential Operators

7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn Elliptic pseudodifferential operators form a special class of Lm (Ω) which are essentially invertible. This section we devote to some basic properties of this class of operators on Ω. In particular, we will discuss the connections between the parametrix and the Green’s operator for elliptic boundary value problems. The development here will also be useful for the treatment of boundary integral operators later on. Definition 7.2.1. A symbol a(x, ξ) in S m (Ω × IRn ) is elliptic of order m if there exists a symbol b(x, ξ) in S −m (Ω × IRn ) such that ab − 1 ∈ S −1 (Ω × IRn ) .

(7.2.1)

This definition leads at once to the following criterion. Lemma 7.2.1. A symbol a ∈ S m (Ω × IRn ) is elliptic if and only if for any compact set K  Ω there are constants c(K) > 0 and R(K) > 0 such that |a(x, ξ)| ≥ c(K)ξm for all x ∈ K and |ξ| ≥ R(K) .

(7.2.2)

We leave the proof to the reader. For a pseudodifferential operator A ∈ Lm (Ω) we say that A is elliptic of order m, if one of the representatives of the complete symbol σA is elliptic of order m. For classical pseudodifferential operators A ∈ Lm c (Ω), the ellipticity of order m can easily be characterized by the homogeneous principal symbol a0m and the ellipticity condition a0m (x, ξ) = 0 for all x ∈ Ω and |ξ| = 1 .

(7.2.3)

As an example we consider the second order linear differential operator P as in (5.1.1) for the scalar case p = 1 which has the complete symbol a(x, ξ) =

n 

ajk (x)ξj ξk − i

j,k=1

n  

n  ∂ajk

k=1

j=1

bk (x) −

∂xj

 (x) ξk + c(x) .

(7.2.4)

In particular, Condition (7.2.3) is fulfilled for a strongly elliptic P satisfying (5.4.1); then a02 (x, ξ) :=

n 

ajk (x)ξj ξk = 0 for |ξ| = 0 and x ∈ Ω ,

(7.2.5)

j,k=1

and P is elliptic of order m = 2. Definition 7.2.2. A parametrix Q0 for the operator A ∈ Lm (Ω) is a properly supported operator which is a two–sided inverse for A modulo L−∞ (Ω): A ◦ Q0 − I

=

C1 ∈ L−∞ (Ω) (7.2.6)

and Q0 ◦ A − I

=

C2 ∈ L

−∞

(Ω)

7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn

453

Theorem 7.2.2. (H¨ormander[186], Taylor [416, Chap.III, Theorem 1.3]) A ∈ Lm (Ω) is elliptic of order m if and only if there exists a parametrix Q0 ∈ L−m (Ω) satisfying (7.2.6). Proof: (i) Let A be a given operator in Lm (Ω) which is elliptic of order m. Then to the complete symbol class of A there is a representative a(x, ξ) satisfying (7.2.2). If Q would be known then (7.1.37) with (7.2.6) would imply    1  ∂ α   σQ◦A = 1 ∼ q−m−j (x, ξ)a(x, ξ) + q−m− (x, ξ) × α! ∂ξ 0≤j 0≤ 1≤|α|  ∂ α a(x, ξ) . (7.2.7) × −i ∂x So, to a we choose a function χ ∈ C ∞ (Ω × IRn ) with χ(x, ξ) = 1 for all x ∈ Ω and |ξ| ≥ C ≥ 1 where C is chosen appropriately large, and χ = 0 in some neighbourhood of the zeros of a. Then define q−m (x, ξ) := χ(x, ξ)a(x, ξ)−1

(7.2.8)

and, in view of (7.2.7), recursively for j = 1, 2, . . ., q−m−j (x, ξ) := (7.2.9)     1  ∂ α α ∂ − q−m−j+|α| (x, ξ) −i a(x, ξ) q−m (x, ξ) . α! ∂ξ ∂x 1≤|α|≤j

From this construction one can easily show that q−m−j ∈ S −m−j (Ω × IRn ). The sequence {q−m−j } can be used to define an operator Q ∈ OP S −m (Ω × IRn ) with

∞

q−m−j as the asymptotic expansion of σQ , see Theorem 7.1.3.

j=0

With Q available, we use the decomposition of Q = Q0 + R, Theorem 7.1.9, where Q0 ∈ OP S −m (Ω × IRn ) is properly supported and still has the same symbol as Q. Then we have with Theorem 7.1.14 that Q0 ◦ A ∈ L0 (Ω) and by construction in view of (7.1.37) σQ0 ◦A ∼ 1. Hence, Q0 ◦A−I = R2 ∈ L−∞ (Ω). In the next step it remains to show that A ◦ Q0 − I ∈ L−∞ (Ω), too. The choice of q−m also shows that Q0 is an elliptic operator of order (−1) ∈ OP S m which is also −m. Therefore, to Q0 there exists an operator Q0 (−1) properly supported and satisfies Q0 ◦ Q0 − I = R1 ∈ L−∞ (Ω), where R1 is properly supported, due to Proposition 7.1.6. Then (−1)

Q0

◦ Q0 ◦ A ◦ Q 0 − I

Q0

◦ (I + R2 ) ◦ Q0 − I

=

(−1) Q0

◦ Q0 − I + Q0

= (−1)

(−1)

=

R1 +

(−1) Q0

(−1)

◦ R2 ◦ Q 0

◦ R2 ◦ Q 0 . (−1)

are properly supported, Q0 ◦ R2 ◦ Q0 again is a Since both, Q0 and Q0 (−1) smoothing operator due to Theorem 7.1.14. Hence, Q0 ◦ Q0 ◦ A ◦ Q0 − I =: S1 ∈ L−∞ (Ω).

454

7. Introduction to Pseudodifferential Operators

On the other hand, A ◦ Q0 − I

(−1)

(−1)

◦ Q0 ◦ A ◦ Q0 + (I − Q0

=

Q0

=

S1 − R 1 ◦ A ◦ Q 0

◦ Q0 ) ◦ A ◦ Q 0 − I

and R1 ◦ A ◦ Q0 is a smoothing operator since Q0 and R1 are both properly supported and R1 ∈ L−∞ (Ω). This shows with the previous steps that A ◦ Q0 − I ∈ L−∞ (Ω). Hence, Q0 satisfies (7.2.6) and is a parametrix. (ii) If Q is a parametrix satisfying (7.2.6), then with (7.1.37) we find σA◦Q − aq ∼

   1  ∂ α ∂ α σA (x, ξ) −i q(x, ξ) . α! ∂ξ ∂x

|α|≥1

Hence, σA◦Q −aq ∈ S −1 (Ω×IRn ), i.e. (7.2.1) with b = q. Moreover, σA◦Q −1 ∈  S −∞ (Ω × IRn ) because of (7.2.6). 7.2.1 Systems of Pseudodifferential Operators The previous approach for a scalar elliptic operator can be extended to general elliptic systems. Let us consider the p × p system of pseudodifferential operators A = ((Ajk ))p×p with symbols ajk (x; ξ) ∈ S sj +tk (Ω × IRn ) where we assume that there exist two p–tuples of numbers sj , tk ∈ IR ; j, k = 1, . . . , p. As a special example we consider the Agmon–Douglis–Nirenberg elliptic system of partial differential equations p s j +tk 

β ajk β (x)D uk (x) = fj (x) for j = 1, . . . , p .

(7.2.10)

k=1 |β|=0

Without loss of generality, assume sj ≤ 0. The symbol matrix associated with (7.2.10) is given by 

sj +tk

|β| β jk ajk β (x)i ξ = a (x; ξ)

(7.2.11)

|β|=0

and its principal part is now defined by  ajk sj +tk (x; ξ) :=

|β| β ajk β (x)i ξ

(7.2.12)

|β|=sj +tk jk where ajk sj +tk (x; ξ) is set equal to zero if the order of a (x; ξ) is less than sj + tk . Then

7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn

 ξ  sj +tk jk ajk0 asj +tk x; sj +tk (x; ξ) = |ξ| |ξ|

455

(7.2.13)

is a representative of the corresponding homogeneous principal symbol. For the more general system of operators Ajk ∈ Lsj +tk (Ω) we can associate in the same manner as for the differential operators in (7.2.10) representatives (7.2.11) by first using the principal symbol class (7.1.31) of Ajk and then taking the homogeneous representatives (7.2.13) and neglecting those ajk sj +tk from (7.1.31) of order less than sj + tk . Definition 7.2.3. Let the characteristic determinant H(x, ξ) for the system ((Ajk )) be defined by H(x, ξ) := det



ajk0 sj +tk (x; ξ)

p×p

.

(7.2.14)

Then the system is elliptic in the sense of Agmon–Douglis–Nirenberg 2 if H(x, ξ) = 0 for all x ∈ Ω and ξ ∈ IRn \ {0} . (7.2.15) With this definition of ellipticity, Definition 7.2.2 of a parametrix Q0 = ((Qjk )) for the operator A = ((Ajk ))p×p with Ajk ∈ Lsj +tk (Ω) as well as Theorem 7.2.2 with Qjk ∈ Lsj +tk (Ω) remain valid. The Stokes system (2.3.1) is a simple example of an elliptic system in the sense of Agmon–Douglis–Nirenberg. If we identify the pressure in (2.3.1) with un+1 for n = 2 or 3 then the Stokes system takes the form (7.2.10) with p = n + 1, for n = 3: ⎛

−μΔ 0 0 j +tj 4 s  ⎜ 0 −μΔ 0 β ⎜ ajk β D uk = ⎝ 0 0 −μΔ

k=1 |β|=0

∂ ∂x1

∂ ∂x2

∂ ∂x3

∂ ⎞⎛ ⎞ u1 ∂x1 ∂ ⎟⎜ ⎟ u ∂x2 ⎟ ⎜ 2 ⎟ ∂ ⎠⎝ ⎠ u3 ∂x3

0

=f

u4

where sj = 0 for j = 1, n , sn+1 = −1 and tk = 2 for k = 1, n , tn+1 = 1. The jk for |β| = 2 and j, k = 1, n; coefficients are given by ajk β = −μδ 24 34 41 42 43 a14 (1,0,0) = a(0,1,0) = a(0,0,1) = a(1,0,0) = a(0,1,0) = a(0,0,1) = 1 for n = 3 ; 23 31 32 a13 (1,0) = a(0.1) = a(1,0) = a(0,1) = 1

for n = 2

and all other coefficients are zero. Then the symbol matrix (7.2.12) for (2.3.1) reads ⎛ ⎞ μ|ξ|2 0 0 iξ1 ⎜ 0 μ|ξ|2 0 iξ2 ⎟ ⎟ a(x, ξ) = ⎜ ⎝ 0 0 μ|ξ|2 iξ3 ⎠ iξ1 iξ2 iξ3 0 2

Originally called Douglis–Nirenberg elliptic.

456

7. Introduction to Pseudodifferential Operators

For n = 2, the third column and third row are to be discarded. The characteristic determinant becames H(x, ξ) = det a(x, ξ) = μn−1 |ξ|2n for n = 2, 3 . Hence, the Stokes system is elliptic in the sense of Agmon–Douglis–Nirenberg. Theorem 7.2.3. (see also Chazarain and Piriou [64, Chap.4, Theorem 7.7]) A = ((Ajk ))p×p is elliptic in the sense of Agmon–Douglis–Nirenberg if and only if there exists a properly supported parametrix Q0 = ((Qjk ))p×p with Qjk ∈ L−tj −sk (Ω). Proof: Assume that A is elliptic. By the definition of ellipticity, the homogeneous principal symbol is then of the form sj r tr * ajk0 sj +tk (x; ξ) = ((|ξ| δj ))((asj +tk (x; Θ) ))((|ξ| δrk ))

*= where Θ

ξ |ξ|

and Einstein’s rule of summation is used, and where

* = det ((ajk (x; Θ) * )) for every x ∈ Ω and Θ * ∈ IRn , |Θ| * = 1. H(x; Θ) sj +tk This implies that for every fixed x ∈ Ω there exists R(x) > 0 such that det((ajk (x; ξ) )) = 0 for all |ξ| ≥ R(x). By setting a(x; ξ) := ((ajk (x; ξ) ))p×p and using an appropriate cut–off function χ ∈ C ∞ (Ω × IRn ) with χ(x, ξ) = 1 for all x ∈ Ω and |ξ| ≥ 2R(x) and with χ(x, ξ) = 0 in some neighbourhood of the zeros of det a(x, ξ), we define q−m (x, ξ) by (7.2.8) as a matrix–valued function. Recursively, then q−m−j (x, ξ) can be obtained by (7.2.9) providing us with asymptotic symbol expansions of all the matrix elements of the operator Q which we can find due to Theorem 7.1.3. The properly supported parts of Q define the desired Q0 satisfying the second equation in (7.2.6), i.e., Q0 is a left parametrix. To show that Q0 is also a right parametrix, i.e. to satisfy the first equation in (7.2.6), the arguments are the same as in the scalar case in the proof of Theorem 7.2.2. Conversely, if Q is a given parametrix for A with Qjk ∈ L−tj −sk (Ω) being properly supported then its homogeneous principal symbol has the form jk0 ((q−t (x; ξ) )) j −sk

=

jk0 * )) ((|ξ|−tj −sk q−t (x; Θ) j −sk

=

r0 * ))((|ξ|−sk δrk )) . ((|ξ|−tj δj ))q−t (x; Θ) j −sk

This implies with (7.2.6) sj +tk * ))jk ajk0 ((qt−1 (x; Θ) sj +tk (x; ξ) = |ξ| j +sk jk0 where qt−1 denotes the inverse of ((q−t )). The latter exists since j +sk j −sk jk0 * ))det ((ajk0 (x; Θ) * )) = 1 . (x; Θ) det ((q−t 0 j −sk

7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn

457

Hence, A is elliptic in the sense of Agmon–Douglis–Nirenberg. Because of  (7.2.6) we also have Ajk ∈ Lsj +tk (Ω). In Fulling and Kennedy [130] one finds the construction of the parametrix of elliptic differential operators on manifolds. 7.2.2 Parametrix and Fundamental Solution To illustrate the idea of the parametrix we consider again the linear second order scalar elliptic differential equation (5.1.1) for p = 1, whose differential operator can be seen as a classical elliptic pseudodifferential operator of order m = 2. Let us first assume that P has constant coefficients and, moreover, that the symbol satisfies a(ξ) =

n 

ajk ξj ξk − i

j,k=1

n 

bk ξk + c = 0 for all ξ ∈ IRn .

(7.2.16)

k=1

Then the solution of (5.1.1) for f ∈ C0∞ (Ω) can be written as   1 F f (x) . u(x) = N f (x) := Fξ−1 y→ξ →x a(ξ)

(7.2.17)

The operator N ∈ L−2 (Ω) defines the volume potential for the operator P , the generalized Newton potential or free space Green’s operator. When f (y) is replaced by δ(y − x) then we obtain the fundamental solution E(x, y) for P explicitly:  1 i(x−y)·ξ −n e dξ . (7.2.18) E(x, y) = (2π) a(ξ) IRn

If condition (7.2.16) is replaced by a(ξ) = 0 for ξ = 0 (as for c = 0 in (7.2.16)) then, as in (7.2.8), we take q−2 (ξ) = χ(ξ)

1 a(ξ)

where the cut–off function χ ∈ C ∞ (IRn ) satisfies χ(ξ) = 1 for |ξ| ≥ 1 and χ(ξ) = 0 for |ξ| ≤ We define the parametrix by Qf := Fξ−1 →x (q−2 (ξ)Fy→ξ f )

1 . 2

(7.2.19)

458

7. Introduction to Pseudodifferential Operators

with Q ∈ L−2 (IRn ); and we have u(x) = Qf (x) + Rf (x) with the remainder Rf (x) = Fξ−1 →x



1 − χ(ξ)

 1 f*(ξ) . a(ξ)

(7.2.20)

Since f ∈ C0∞ (Ω) we have f*∈ C ∞ from the the Paley–Wiener–Schwartz 1 having a orem 3.1.3 (iii). Hence, with 1 − χ(ξ) = 0 for |ξ| ≥ 1 and a(ξ) singularity at ξ = 0 and having quadratic growth, due to Lemma 3.2.1,  only 1 * 1 − χ(ξ) a(ξ) f (ξ) defines a distribution in E  . Hence, Rf ∈ C ∞ and R ∈ L−∞ (Ω) by applying Theorem 3.1.3 (ii). (Note that R ∈ OP S −∞ (Ω × IRn )). Thus, Q + R ∈ L−2 (Ω) defining again N := Q + R, the Newton potential operator, which can now still be written as   1 −1 Fy→ξ f (y) (x) . (7.2.21) N f (x) = Fξ→x a(ξ) As in the previous case, we still can define the fundamental solution E(x, y) = N δ(x − y)

(7.2.22)

where δ(x − y) is the Dirac functional with singularity at y. Note that δ ∈ E  (IRn ) and N : E  (IRn ) → D (IRn ) is well defined. E(x, y) will satisfy (3.6.4). We now return to the more general scalar elliptic equation (5.1.1) with C ∞ –coefficients (but p = 1). Here, the symbol a(x, ξ) in (7.2.4) depends on both, x and ξ. The construction of the symbols (7.2.8) and (7.2.9) leads to an asymptotic expansion and via (7.1.18), i.e., by q(x, ξ) :=

∞ 

 Ξ

j=0

ξ tj

 q−2−j (x, ξ)

we define the symbol q where Ξ ∈ C ∞ is a cut–off function satisfying Ξ(y) = 0 for |y| ≤ 12 and Ξ(y) = 1 for |y| ≥ 1. Then q ∈ S −2 (Ω × IRn ) and   1 ei(x−y)ξ q(x, ξ)f (y)dydξ (7.2.23) Q(x, −iD)f (x) = (2π)n IRn Ω

is a (perhaps not properly supported) parametrix Q ∈ L−2 (Ω). With Q = Q0 + R, the equations (7.2.6) yield P ◦Q Q◦P

= =

P ◦ (Q0 + R) (Q0 + R) ◦ P

= =

I + C1 + P ◦ R I + C2 + R ◦ P

=: =:

I + R1 , I + R2 .

(7.2.24)

7.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn

459

Since P is properly supported, P ◦ R and R ◦ P are still smoothing operators; so are R1 and R2 . The equations (7.2.24) read for the differential equation:  (7.2.25) P ◦ (Qf )(x) = f (x) + r1 (x, y)f (y)dy Ω

and

 Q(x, −iD) ◦ P u(x)

=

r2 (x, y)u(y)dy

u(x) +

(7.2.26)

Ω

where r1 (x, y) and r2 (x, y) are the C ∞ (IRn )–Schwartz kernels of the smoothing operators R1 and R2 , respectively. We note that both smoothing operators R1 and R2 can be constructed since in (7.2.24) the left-hand sides are known. Theorem 7.2.4. If R2 extends to a continuous operator C ∞ (Ω) → C ∞ (Ω) and ker(Q ◦ P ) = {0} in C ∞ (Ω) then the Newton potential operator N is given by (7.2.27) u(x) = (N f ) := (I + R2 )−1 ◦ Q(x, −iD)f (x) . where f ∈ C0∞ (Ω). Moreover, in this case, the fundamental solution exists and is given by  (7.2.28) E(x, y) = (N δy )(x) = (I + R2 )−1 ◦ Q(x, −iD)δy (x) where δy (·) := δ(· − y). Our procedure for constructing the fundamental solution, in principle, can be extended to general elliptic systems. Similar to the second order case we will arrive at an expression as (7.2.28). The difficulty is to show the invertibility as well as the continuous extendibility of I + R2 . We comment that this is, in general, not necessarily possible. Hence, for general elliptic linear differential operators with variable coefficients it may not always be possible to find a fundamental solution. We shall provide a list of references concerning the construction of fundamental solutions at the end of the section. In any case, even if the fundamental solution does not exist, for elliptic operators one can always construct a parametrix. From the practical point of view, however, it is more desirable to construct only a finite number of terms in (7.2.9) rather than the complete symbol. This leads us to the idea of Levi functions. 7.2.3 Levi Functions for Scalar Elliptic Equations From the homogeneous constant coefficient case we have an explicit formula to construct the fundamental solution via the Fourier transformation. However, in the case of variable coefficients, fundamental solutions can not be constructed explicitly, in general. This leads us to the idea of freezing coefficients in the principal part of the differential operator and, by following the

460

7. Introduction to Pseudodifferential Operators

idea of the parametrix construction, one constructs the Levi functions which dates back to the work of E.E. Levi [266] and Hilbert [179]. These can be used as approximations of the fundamental solution for the variable coefficient equations. In general, one can show that Levi functions always exist even if there is no fundamental solution. Pomp in [351] developed an iterative scheme for constructing Levi functions of arbitrary order for general elliptic systems, from the distributional point of view. Here we combine his approach with the concept of pseudodifferential operators. To illustrate the idea, we use the scalar elliptic equation of the form (3.6.1) with p = 1 and the order 2κ (instead of 2m). For fixed y ∈ Ω, we define  P0 (y)v(x) = aα (y)Dxα v(x) (7.2.29) |α|=2κ

and write the original equation in the form P u(x) = P0 (y)u(x) − T (y)u(x) = f (x) ,

(7.2.30)

where T (y)u(x)

= =

(7.2.31) P0 (y) − P u(x)    α aα (y) − aα (x) Dx u(x) − aα (x)Dxα u(x) .



|α|=2κ

|α| 0 and γ2 ≥ 0 such that G˚ arding’s inequality holds in the form Re (w, Λσ ΘΛ−σ Aw)p

=1

≥ γ1 w2p

=1

H t (Ω)

H (t −s )/2 (Ω)

− γ2 w2p

=1

(7.2.63) H t −1 (Ω)

 &p t for w ∈ =1 Hcomp (Ω) with supp w ⊂ K where Λσ = Λsj δj p×p (see (4.1.44)) and where γ1 > 0 and γ2 ≥ 0 depend on Θ, A, Ω, the compact K  Ω and on the indices tj and sk . The in (7.2.63) a linear compact operator &p defines &p last tterm −s  (Ω) → =1 Hcomp (Ω) which is given by C : =1 Hcomp (v, Cw)p

=1

H (t −s )/2 (Ω)

= γ2 (v, w)p=1 H t −1 (Ω) .

With this operator C, the G˚ arding inequality (7.2.63) becomes   σ Re w, Λ ΘΛ−σ A + C w p H (t −s )/2 (Ω) ≥ γ1 w2p H t (Ω) . (7.2.64) =1

Proof:

3

=1

For the proof let us consider the operator A1 := ΘΛ−σ AΛ−τ

where τ = (t1 , . . . , tp ) and σ = (s1 , . . . , sp ), which is now a pseudodifferential operator of order zero and let us prove (7.2.63) for A1 first. The strong ellipticity of A implies that the operator Re A1 = 12 (A1 + A∗1 ) 3

The authors are grateful to one of the reviewers who suggested the following proof.

470

7. Introduction to Pseudodifferential Operators

has a positive definite hermitian principal symbol matrix a0Re A1 (x, ξ) whose  1 positive definite hermitian square root a0Re A1 (x, ξ) 2 ∈ S 0 (Ω × IRn ) defines via   1 Bu(x) := (2π)n/2 eix·ξ a0Re A1 (x, ξ) 2 u *(ξ)dξ = B0 u(x) + Ru(x) IRn

a strongly elliptic pseudodifferential operator of order zero, where B0 denotes its properly supported part and R ∈ L−∞ (Ω) due to Theorem 7.1.9. Hence, B admits a properly supported parametrix Q0 due to Theorem 7.2.3. For  ∈ L2 (IRn ), formula u ∈ L2 (Ω) with supp u ⊂ K extended by zero to u (7.2.6) yields u  = Q0 B u + C2 u  = Q0 B 0 u  + (Q0 R + C2 ) u,

(7.2.65)

where C2 ∈ L−∞ , and Q0 R ∈ L−∞ because of Theorem 7.1.14. Since u  has compact support and Q0 B0 is properly supported due to Proposition 7.1.6,  ⊂ K1 . Hence, there exists a compact set K1  Ω and supp Q0 B0 u supp (Q0 R + C2 ) ⊂ K ∪ K1  Ω. Now, (7.2.65) yields  uL2 (IRn ) = uL2 (K) ≤ c1 B0 u L2 (IRn ) + c2  uH −1 (IRn ) with some c1 > 0 since B0 u  has compact support in Ω. Hence, there exists γ0 > 0 such that (B0∗ B0 u , u )L2 (IRn ) ≥ γ0  u2L2 (IRn ) − c2  u2H −1 (IRn ) , . Since Re A1 = B ∗ B + C = B0∗ B0 + B0∗ R + R∗ B0 + R∗ R + C with some C ∈ L−1 c (Ω) due to Theorem 7.1.14, we get Re (A1 u, u) = Re (A1 u , u ) = (B0 u , B0 u ) + (R u, B 0 u ) + (B0 u , R u) + (R u, R u) + (C u , u ) and u2H −1 (IRn ) − c4  uL2 (Ω)  uH −1 (IRn ) Re (A1 u, u)L2 (Ω) ≥ γ0 u2L2 (Ω) − c3  γ0 ≥ u2L2 (Ω) − c5  u2H −1 (Ω) (7.2.66) 2 with an appropriate constant c5 . This, in fact, is already the proposed G˚ arding inequality for the special case αjk = 0. Now, for the general case, we employ the isometries Λα from H s (IRn ) to ◦

H s−α (IRn ) and replace u by ηΛτ w, where η ∈ C0∞ (K 2 ) and K2  Ω with ◦

K  K 2 , η|K ≡ 1 and 0 ≤ η ≤ 1. For w ∈ H τ (Ω) with supp w ⊂ K  Ω now apply (7.2.66) and obtain

7.3 Review on Fundamental Solutions

471

Re (ηΛτ w, ΘΛ−σ AΛ−τ ηΛτ w)L2 (IRn ) ≥ γ0 ηΛτ w2L2 (Ω) − c3 ηΛτ w2H −1 (Ω) ≥ γ0 Λτ w2L2 (Ω) − c3 w2H τ −1 (Ω) − 4γ0 [η, Λτ ]w2L2 (Ω) ≥ γ0 w2H τ (Ω) − c6 w2H τ −1 (Ω)

(7.2.67)

with γ0 > 0 since [η, λτ ] ∈ Lτc−1 due to Corollary 7.1.15. The left–hand side can be reformulated as (ηΛτ w, A1 ηΛτ w) = Re (w, Λτ ΘΛ−σ Aw)L2   + [η, Λτ ]w, A1 ηΛτ w + Λτ w, A1 [η, Λτ ]w . So, with Corollary 7.1.15, p Re (w, Λσ ΘΛ−σ Aw) 

H (tj −sj )/2 (Ω)

= Re (w, Λτ ΘΛ−σ Aw)L2 (Ω)

j=1

≥

γ0 Λτ w2L2 (Ω)

− c6 w2H τ −1 (Ω) − c7 Λτ wL2 (Ω) Λτ −1 wL2 (Ω)

≥ γ0 w2H τ (Ω) − c8 w2H τ −1 (Ω) as proposed.



Remark 7.2.2: In the special case when αjk = 2α is constant, where 2α is the same order for all Ajk , we may choose α = sj = tk , and if the system is strongly elliptic then G˚ arding’s inequality (7.2.64) reduces to the familiar form α Re ((ΘA+C)w, w)L2 (Ω) ≥ γ1 w2H α (Ω) with γ1 > 0 , for all w ∈ Hcomp (Ω) .

(For differential operators Ajk see Miranda [307, p.252]).

7.3 Review on Fundamental Solutions As is well known, the fundamental solution plays a decisive role in the method of boundary integral equations. It would be desirable to have a general constructive method to find an explicit calculation of fundamental solutions. However, in general it is not easy to obtain the fundamental solution explicitly whereas Levi functions can always be constructed. The latter will allow us to construct at least local fundamental solutions. The existence of a fundamental solution is closely related to the existence of solutions of P u = f in Ω . (7.3.1) More precisely, we have the following theorem (Komech [235], Miranda [307]).

472

7. Introduction to Pseudodifferential Operators

Theorem 7.3.1. A necessary and sufficient condition for the existence of a fundamental solution for P is that the equation (7.3.1) admits at least one solution u ∈ E  for every f ∈ E  . Consequently, in all the cases when variational solutions exist, we also have a fundamental solution. To our knowledge it seems that the most comprehensive survey on fundamental solutions of elliptic equations is due to Miranda in [307]. Here we quote his useful remarks: “The first research on fundamental matrices looks at particular systems and is due to C. Somigliana [396], E.E. Levi [265], G. Giraud [144, 145]. For elliptic systems in the sense of Lopatinskii this study then was taken up in general by this author [273], first for equations with constant coefficients and then, with Levi’s method in a sufficiently restricted domain, for equations with variable coefficients. For elliptic systems in the sense of Petrowskii the case of constant coefficients was treated by F. John [219] as a particular case of systems with analytic coefficients and by C.B. Morrey [315, 316] in the case r = 1. Y.B. Lopatinskii [272] proved, with Levi’s method, the existence in the small of fundamental matrices for systems with variable coefficients, while the existence in certain cases of principal fundamental matrices was established by A. Avantaggiati [18]. For elliptic systems in the sense of Douglis and Nirenberg the construction of the fundamental matrix, in the case of constant coefficients, was done by these authors [103] with a procedure due to F. Bureau [57]. Also for these systems, V.V. Grusin [161] and A. Avantaggiati [18] proved the existence of principal fundamental matrices in certain cases. For the case of variable coefficients, some indication relating to the existence in the small is given in §10.6 of the volume [187] by H¨ormander. Finally, for certain systems of particular type see D. Greco [154] and L.L. Parasjuk [343]. As for the case of a single equation, the problem of existence of a fundamental matrix can be related on one hand with that of the existence for arbitrary f of at least one solution of the equation Mu = f and on the other hand with that of the validity of the unique continuation property4 . Each one of these questions has separately been an object for study, but the relationship between the results obtained had to be deepened later. Regarding the first question we note that from theorems of existence in the small of a fundamental matrix it follows that in every sufficiently restricted domain an elliptic system always admits at least one solution.” In the recent publication [351], Pomp also discusses currently available existence results for fundamental solutions. 7.3.1 Local Fundamental Solutions In this section we present Levi’s method for constructing local fundamental solutions. 4

Mu = P u in our notation as in (7.3.1).

7.3 Review on Fundamental Solutions

473

Let us assume that a fundamental solution E(x, y) exists. Then with the Levi function LN (x, y) constructed in (7.2.40) for a scalar operator P or L(N ) (x, y) = LN (x, y) by (7.2.58) for an Agmon–Douglis–Nirenberg system we consider W (x, y) := E(x, y) − LN (x, y) for x, y, ∈ Ω .

(7.3.2)

Then for fixed y ∈ Ω,

 Px W (x, y) = δ(x − y) − δ(x − y) − T (y)NN +1 (x, y)

(7.3.3)

due to (7.2.42) or (7.2.61). The function TN +1 (x, y) is the Schwartz kernel of a pseudodifferential operator in L−N −2 (Ω) and, moreover, TN +1 ∈ C N +1−n (Ω × Ω). For N ≥ n − 1 , TN +1 is at least continuous. Now, W (x, y) can be represented in the form (7.2.47) or (7.2.59), namely  W (x, y) = LN (x, z)Φ(z, y)dz (7.3.4) Ω

with Φ(x, y) the solution of the integral equation  TN +1 (x, y) = Φ(x, y) − TN +1 (x, z)Φ(z, y)dz .

(7.3.5)

Ω

If the existence of E(x, y) is assumed then (7.3.5) has a solution Φ(x, y) and E(x, y) is given by  E(x, y) = LN (x, y) + LN (x, z)Φ(z, y)dz . (7.3.6) Ω

Conversely, if (7.3.5) has a solution Φ(x, y) which is at least continuous for (x, y) ∈ Ω × Ω then E(x, y) in (7.3.6) is the desired fundamental solution. Note that Fredholm’s alternative for the integral equation (7.3.5) can be applied if the integral operator in (7.3.5) defines a compact mapping on the solution space for Φ(x, y), e.g., on the space of continuous functions. In view of the properties of TN +1 , this can be guaranteed if Ω is replaced by a compact region Ω   Ω with a sufficiently smooth boundary ∂Ω  . In this case, the whole construction procedure of the Levi function should be carried out on Ω  . If then for (7.3.5) uniqueness of the solution holds then the corresponding uniquely determined function Φ(x, y) generates a local fundamental solution by (7.3.6) associated with Ω  . If |Ω  | is sufficiently small then the integral operator’s norm in (7.3.5) is less than 1 and Banach’s fixed point theorem in the form of Neumann’s series will provide the local existence of Φ(x, y) and, hence, that of E(x, y). If (7.3.5) has eigensolutions, the representation of E(x, y) in (7.3.6) and the integral equation are to be appropriately modified. We refer to the details

474

7. Introduction to Pseudodifferential Operators

in Miranda [307] and in Ljubiˇc [271]. For bounded regions in IR2 and elliptic systems in the sense of Petrovski, Fichera also gave a complete constructive existence proof in [118]. For real analytic coefficients in Agmon–Douglis–Nirenberg elliptic systems, John presents the construction of a local fundamental solution in [219]. In [425] Vekua constructs, for second order systems with the Laplacian Δ as principal part and analytic coefficients and also for higher order scalar equations in IR2 , the fundamental solution. For a scalar elliptic differential ormander [188, Vol II] operator having C ∞ –coefficients, Theorem 13.3.3 by H¨ provides a local fundamental solution. 7.3.2 Fundamental Solutions in IRn for Operators with Constant Coefficients In the previous chapters we have introduced fundamental solutions for particular elliptic partial differential equations which are defined in the whole space IRn . Fundamental solutions in the whole space IRn are often referred to as principal fundamental solutions. In particular, we have principal fundamental solutions for the Laplacian (1.1.2), the Helmholtz equation (2.1.4), the Lam´e system (2.3.10) and the biharmonic equation (2.4.7). Notice that all these partial differential equations have constant coefficients. Scalar equations with constant coefficients For scalar differential equations with constant coefficients, in particular the elliptic ones, the principal fundamental solution always exists due to Malgrange and Ehrenpreis (see also Folland [125], Wagner [429]). Theorem 7.3.2. (Ehrenpreis [111], Malgrange [280]) For the scalar partial differential operator P with symbol i|α| cα ξ α , the fundamental solution |α|≤m

exists and is given by the complex contour integral   a(η + λξ)  dλ (2π)−n/2 λm eiλξ·(x−y) Fη−1 E(x, y) = →(x−y) a(η + λξ) 2πiλ . am (ξ) λ∈C∧|λ|=1

where am (ξ) = |α|=m

(7.3.7) im cα ξ α is the principal symbol and ξ ∈ C is fixed such

that am (ξ) = 0 (see Wagner [429]). A different formula by K¨onig for this case can also be found in [231]. Treves constructs in [418] fundamental solutions for several differential polynomials. Wagner in [428] illustrates how to apply the method of H¨ ormander’s stairs to construct fundamental solutions in this case. In [341, 342], Ortner presents a collection of some fundamental solutions for constant coefficient operators. For second order systems in IR2 with constant coefficients see also Clements [73].

7.3 Review on Fundamental Solutions

475

John in [219, (3.53)] obtains more explicit expressions for fundamental solutions depending on n odd or even. For n odd he obtains the general representation formula  : eiλ(x−y)·ξ i (n−1)/2 E(x, y) = dλdωξ (Δ ) sign[ξ · (x − y)] y 4(2πi)n a(λξ) |ξ|=1

|λ|=M

(7.3.8) where M is chosen large enough such that all zeros of a(λξ) are contained in the circle |λ| = M . He also simplifies this formula for subcases of odd n. For n even and n < m, he obtains the explicit expression [219, (3.64)]  : eiλ(x−y)·ξ n−1 i λ log |(x − y) · ξ| dλdωξ . E(x, y) = n+1 (2π) a(λξ) |ξ|=1,ξ∈IRn

|λ|=M

(7.3.9) For even n ≥ m, he presents E(x, y) only for the case of the homogeneous equation with a(ξ) = am (ξ):   m 1 1 n/2 E(x, y) = − dωξ . (x−y)·ξ log |(x−y)·ξ| (Δ ) y n m!(2πi) am (−iξ) |ξ|=1,ξ∈IRn

(7.3.10) In special cases, these integrations can be carried out explicitly. For instance, if P is the iterated Laplacian in IRn , P u = Δq u with q ∈ IN

(7.3.11)

then a fundamental solution is given in [219, (3.5)], (−1)q Γ ( n2 − q) |x − y|2q−n for n odd and for even n > 2q ; 22q π n/2 Γ (q) (7.3.12) and for even n ≤ 2q, one has [219, (3.6)], E(x, y) =

n

E(x, y) =

−(−1) 2 |x − y|2q−n log |x − y| . 2q−1 n/2 2 π (q − 1)!(q − n2 )!

(7.3.13)

For the Helmholtz operator in IRn , P u = Δu + k 2 u one has E(x, y) =

 k 1 4 2π|x − y|

n−2 2

H n−2 (k|x − y|) 2

where Hν is the Bessel function of the second kind.

(7.3.14)

476

7. Introduction to Pseudodifferential Operators

Petrovski elliptic systems with constant coefficients Fichera in [121] has carried out Fritz John’s construction of the fundamental solution for even order, m = 2m , homogeneous elliptic systems in the sense of Petrovski which have the form  Pu = aαβ Dα+β u in IRn (7.3.15) |α|=|β|=m

where u = (u1 , . . . , up ) and aα,β = a∗β,α . The symbol matrix P is then given by  (7.3.16) σαβ (ξ) = (−1)m aαβ ξ α+β ; and ellipticity in the sense of Petrovski means:   Q(ξ) := det (−1)m aαβ ξ α+β = 0 for all ξ ∈ IRn with ξ = 0 . (7.3.17)  Let L(ξ) be defined by the inverse of the symbol matrix times Q, i.e., −1    L(ξ) = Q(ξ) (−1)m aαβ ξ α+β

(7.3.18)

which consists of the cofactors to the symbols σαβ (ξ) and, hence, of homogeneous polynomials. Due to Fichera [121], the fundamental solution then is given explicitly by  D)S(x, y) E(x, y) = L(x, (7.3.19) where for n odd, S(x, y) =

1 4(2πi)n−1 (m − 1)!

(Δy )



n−1 2

|(x − y) · ξ| Q(ξ)

m−1

dωξ

(7.3.20)

|ξ|=1

and for n even 

n −1 (Δy ) 2 S(x, y) = n 4(2πi) m!

|(x − y) · ξ|m log |(x − y) · ξ| dωξ . (7.3.21) Q(ξ)

|ξ|=1

Agmon–Douglis–Nirenberg elliptic systems with constant coefficients In [219] Fritz John represents explicitly the fundamental solution for the principal part of an elliptic system (7.2.10), i.e., Pj u =

p 



β ajk β D uk for j = 1, . . . , p ,

k=1 |β|=sj +tk

having the symbol matrix (7.2.13), i.e.,

(7.3.22)

7.3 Review on Fundamental Solutions

σjk (ξ) := ajk0 sj +tk (ξ) =



|β| β ajk0 β i ξ .

477

(7.3.23)

|β|=sj +tk

(sj + tk ) ≥ 0 and let P q (ξ) denote the components Let mk := maxj=1,...,p of the inverse matrix to σjk (ξ) . Then the fundamental solution is given by the matrix components for n odd: Eq (x, y) =

1 4(2πi)n−1 (mq

×(Δy )

n−1 2



× (7.3.24) − 1)!  mq −1  (x − y) · η sign (x − y) · η P q (−iη)dωη ,

|η|=1,η∈IRn

and for n even, −1 × (7.3.25) mq !(2πi)n   mq n ×(Δy ) 2 (x − y) · η log |(x − y) · η|P q (−iη)dωη .

Eq (x, y) =

|η|=1,η∈IRn

7.3.3 Existing Fundamental Solutions in Applications For various existing problems in applications, explicit fundamental solutions are available. Here we collect a list of a few relevant references which may be useful but is by no means complete. For harmonic elastodynamics, fundamental solutions were given in [191]. For the Yukawa equation, for Δm − sΔm−1 and the Oseen equation one finds fundamental solutions in [198]. In the book by Natroshvili [321] one finds the construction of the fundamental solution of anisotropic elasticity in IR3 (see also Natroshvili et al [322, 323]). The book by Kythe [255] contains several examples ranging from classical potential theory, elastostatics, elastodynamics, Stokes flows, piezoelectricity to some nonlinear equations such as the p–Laplacian and the Einstein–Yang–Mills equations. The proceedings by Benitez (ed.) [29] contain the topics: methodology for obtaining fundamental solutions, numerical evaluation of fundamental solutions and some applications for not–well–known fundamental solutions. In the engineering community one may find many further interesting examples of fundamental solutions. To name a few, we list some of these publications: Balaˇs et al [23], Bonnet [37], Brebbia et al [44], Gaul et al [140], Hartmann [173], Kohr and Pop [233], Kupradze et al [253] and Pozrikidis [353] — without claiming completeness.

8. Pseudodifferential Operators as Integral Operators

All of the integral operators with nonintegrable kernels are given in terms of computable Hadamard’s partie finie, i.e. finite part integrals [168], which can be applied to problems in applications (Guiggiani [163], Schwab et al [380]). In this chapter, we discuss the interpretation of pseudodifferential operators as integral operators. In particular, we show that every classical pseudodifferential operator is an integral operator with integrable or nonintegrable kernel plus a differential operator of the same order as that of the pseudodifferential operator in case of a nonnegative integer order. In addition, we also give necessary and sufficient conditions for integral operators to be classical pseudodifferential operators in the domain. Symbols and admissible kernels are closely related based on the asymptotic expansions of the symbols and corresponding pseudohomogeneous expansions of the kernels as examined by Seeley [385]. The main theorems in this context are Theorems 8.1.1, 8.1.6 and 8.1.7 below.

8.1 Pseudohomogeneous Kernels In order to introduce appropriate kernel functions for integral operators we need the concept of pseudohomogeneous functions as defined in Definition 3.2.1, (3.2.1). In this context, we first recapture the definition of pseudohomogeneous functions. Definition 8.1.1. A function kq (x, z) is a C ∞ (Ω×IRn \{0}) p s e u d o h o m o geneous function (w.r. to z) of degree q ∈ IR: if q ∈  IN0 ; if q ∈ IN0 , (8.1.1) where pq (x, z) is a homogeneous polynomial in z of degree q having C ∞ – coefficients and where the function fq (x, z) satisfies kq (x, tz) kq (x, z)

= =

tq kq (x, z) for every t > 0 and z = 0 fq (x, z) + log |z|pq (x, z)

fq (x, tz) = tq fq (x, z) for every t > 0 and z = 0 . In short, we denote the class of pseudohomogeneous functions of degree q ∈ IR by Ψ hfq . Note that for q ∈ IN0 , in this case kq is positively homogeneous of degree q with respect to z as well as fq in the case q ∈ IN0 . © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_8

480

8. Pseudodifferential Operators as Integral Operators

A kernel function k(x, x − y) with (x, y) ∈ Ω × Ω , x = y, is said to have a pseudohomogeneous expansion of degree q if there exist kq+j ∈ Ψ hfq+j for j ∈ IN0 such that k(x, x − y) −

J 

kq+j (x, x − y) ∈ C q+J−δ (Ω × Ω)

(8.1.2)

j=0

for some δ with 0 < δ < 1. In what follows, for simplicity, we call this class of kernel functions pseudohomogeneous kernels of degree q. The class of all kernel functions with pseudohomogeneous expansions of degree q ∈ IR will be denoted by Ψ hkq (Ω). As examples, the fundamental solutions of the Helmholtz equation are of this form: Ek (x, y) ∈ Ψ hk0 in (2.1.18) for n = 2 and Ek (x, y) ∈ Ψ hk−1 for n = 3 in (2.1.19). We note that for q > −n, the integral operator  kq (x, x − y)u(y)dy (8.1.3) Ω

for u ∈ C0∞ (Ω) is well defined as an improper integral. For q ≤ −n, however, kq (x, x − y) is non integrable except that u(y) and its derivatives up to the order κ := [−n − q] vanish at y = x. For fixed x ∈ Ω, hence, (8.1.3) defines a homogeneous distribution on C0∞ (Ω\{x}). In order to extend this distribution to all of C0∞ (Ω), we use the Hadamard finite part concept which is the most natural one for integral operators. Definition 8.1.2. Let Ω0 with Ω 0  Ω be a convex domain with x ∈ Ω0 * the radial distance from x to y ∈ ∂Ω0 depending on the angular and R(Θ) * = x−y on the unit sphere. coordinate Θ |x−y| The Hadamard finite part integral operator is defined by  p.f. kq (x, x − y)u(y)dy := (8.1.4) 

Ω

   1  (y − x)α Dα u(x) dy + kq (x, x − y) u(y) − dα (x)Dα u(x) α! |α|≤κ

Ω

|α|≤κ

where κ := [−n − q] and  := −n − q − κ. For 0 <  < 1,  1 1 * Θ * α kq (x, −Θ)dω * R−j− (Θ) dα (x) = − (j + ) α! +

1 α!



 |Θ|=1

kq (x, x − y)(y − x)α dy Ω\Ω0

(8.1.5)

8.1 Pseudohomogeneous Kernels

481

for all multi–indices α with |α| = κ − j and j = 0, . . . , κ. If  = 0, then   1 1 * Θ * α kq (x, −Θ)dω * log R(Θ) + kq (x, x − y)(y − x)α dy dα (x) = α! α!  |Θ|=1

Ω\Ω0

(8.1.6) for |α| = κ, i.e. j = 0; and the dα (x) for j = 1, . . . , κ are given by (8.1.5) where  = 0. * = 1 will Note that in the one–dimensional case n = 1, integrals over |Θ| always be understood as  * Ψ (x, Θ)dω = Ψ (x, 1) + Ψ (x, −1) (8.1.7)  |Θ|=1

according to Definition 3.2.2. Here, Ω0 with x ∈ Ω0 and Ω 0 ⊂ Ω is any star–shaped domain with * = (y − x)/|y − x|, where dω(Θ) * is the surface element on respect to x, and Θ * the unit sphere and R(Θ) describes the boundary of Ω0 in polar coordinates. Note that this definition resembles the Definition 3.2.2 of Hadamard’s finite part integral. In the following we are going to show that every classical pseudodifferential operator can be written as an integral operator in the form of Hadamard’s finite part integral operators with pseudohomogeneous kernels together with a differential operator and vice versa. Our presentation follows closely the article by Seeley [385] (see also [384]), the book by Pomp [351] and Schwab et al [380]. In view of the definition of Lm c (Ω) and Theorem 7.1.2, this means that for A ∈ Lm c (Ω) we have the general representation   (Au)(x) = cα (x)Dα u(x) + p.f. k(x, x − y)u(y)dy (8.1.8) |α|≤2N

where cα (x)

=

Ω

  1 (y − x)α ψ(y)ei(x−y)·ξ dya(x, ξ)dξ α! n  IR Ω  − p.f. k(x, x − y)(y − x)α ψ(y)dy (8.1.9)

(2π)−n

Ω

with k(x, x − y) given by (7.1.12). Note that for n + m ∈ IN0 and m + n ≥ 0, the coefficients aα (x) must vanish since the finite part integral is a classical pseudodifferential operator of order m whereas the orders of aα (x)Dα (x) are |α| ∈ IN0 .

482

8. Pseudodifferential Operators as Integral Operators

8.1.1 Integral Operators as Pseudodifferential Operators of Negative Order We begin with the case A ∈ Lm c (Ω) for m < 0 since, in this case, the corresponding integral operators do not involve finite part integrals. Theorem 8.1.1. (Seeley [385, p. 209]) Let m < 0. Then A ∈ Lm c (Ω) if and only if  (Au)(x) = k(x, x − y)u(y)dy for all u ∈ C0∞ (Ω) (8.1.10) Ω

with the Schwartz kernel k satisfying k ∈ Ψ hk−m−n (Ω). Remark 8.1.1: Note that for the elliptic differential equations, the parametrix Q0 (7.2.6) as well as the free space Green’s operator (7.2.11) are of the form (8.1.10) with m < 0. The proof needs some detailed relations concerning the properties of pseudohomogeneous kernels. We, therefore, first present these details and return to the proof of Theorem 8.1.1 later on. Lemma 8.1.2. Let k−n ∈ Ψ hf−n , i.e. k−n (x, z) be a homogeneous function of degree −n and let ψ(r) be a C ∞ –cut–off function, 0 ≤ ψ(r) ≤ 1 with ψ(r) = 0 for r ≥ 2 and ψ(r) = 1 for 0 ≤ r ≤ 1. Then  k−n (x, z)ψ(|z|)e−iz·ξ dz bψ (x, ξ) := p.f. =

IRn n/2

(2π)

 Fz→ξ k−n (x, z)ψ(|z|)

has the following properties: bψ (x, ξ) = aψ (x, ξ) + c(x){a0 + a1 log |ξ|} with aψ ∈ S 0 (Ω × IRn ) and c(x) :=

1 ωn

  |Θ|=1

(8.1.11)

* k−n (x, Θ)dω, for n ≥ 2 where,

for n ≥ 2, a0 = (2−1 π −n/2 )c0 , a1 = (2−1 π −n/2 )c−1 with the constants (n) (n) c0 , c−1 given in Gelfand and Shilov [143, p.364], i.e., (n)

c0

(n)

=

−cE

(n) c−1

=

2.

Moreover,

(n)

with the Euler–Mascheroni constant cE .

 ξ  ξ = a0ψ (x, ξ0 ) with ξ0 := lim aψ x, t t→∞ |ξ| |ξ|

(8.1.12)

8.1 Pseudohomogeneous Kernels

483

and  a0ψ (x, ξ0 ) =



* − c(x) k−n (x, Θ)

 |Θ|=1



log

1 * · ξ0 | |Θ

−

* · ξ0  iπ Θ dω (8.1.13) * · ξ0 | 2 |Θ

 In the case n = 1, one has c(x) = 12 k−1 (x, 1) + k−1 (x, −1) , 1 ∞ cos t dt a0 = 2α0 , a1 = −2 where α0 = (cos t − 1) dt t + t ; and 0

 1

1



a (x, ξ0 ) = − 2 k−1 (x, 1) − k−1 (x, −1) iπ sign ξ0 . 0

(8.1.14)

Proof: From the definition of bψ it follows with the Paley–Wiener Theorem 3.1.3 that bψ ∈ C ∞ (Ω × IRn ). Hence, we only need to consider the behaviour of bψ (x, ξ) for |ξ| ≥ q. There, with the Fourier transform of distributions, we write bψ (x, ξ)

(2π)n/2 Fz→ξ k−n (x, z)

=

 −(2π)n/2 Fz→ξ {k−n (x, z) 1 − ψ(|z|) } I1 (x, ξ) − I2 (x, ξ) .

=

We first consider I2 (x, ξ) for |ξ| ≥ 1 and the multiindex γ with |γ| ≥ 1:    ∂ γ γ ξ I2 (x, ξ) = e−iz·ξ − i {k−n (x, z) 1 − ψ(|z|) }dz . ∂z |z|≥1

Since

   ∂ γ  {k−n (x, z) 1 − ψ(|z|) } ≤ cγ (x)|z|−n−|γ| ,  −i ∂z

we have |ξ γ I2 (x, ξ)| ≤ cγ (x) for x ∈ Ω and any γ with |γ| ≥ 1 . Hence,   ξ   I2 ∈ S −∞ (Ω × IRn ) and I2 x, t  ≤ t−|γ| cγ (x) for t ≥ 1 . (8.1.15) |ξ| Now we consider the case n ≥ 2. For I1 (x, ξ) we first modify the homogeneous kernel function as o

k −n (x, z) := k−n (x, z) − c(x)|z|−n , which satisfies the Tricomi condition  o * * = 0. k −n (x, Θ)dω( Θ)  |Θ|=1

(8.1.16)

(8.1.17)

484

8. Pseudodifferential Operators as Integral Operators

Then I1 (x, ξ)

o

=

(2π)n/2 Fz→ξ k −n (x, z) + c(x)(2π)n/2 Fz→ξ |z|−n

=

a0ψ (x, ξ) + c(x){a0 + a1 log |ξ|}

where, after introducing polar coordinates, we arrive at ∞

 a0ψ (x, ξ)

=

p.f.

o

 0 r −1 * −iΘ·ξ * k −n (x, Θ)e r drdω(Θ)

0  |θ|=1

 =

o

* k −n (x, Θ)

 |θ|=1

 1 

 e−iΘ·ξ0 r − 1 r−1 dr

0

∞ +

  * e−iΘ·ξ0 r r−1 dr dω(Θ)

1

with (8.1.17). From the Calder´ on–Zygmund formula (see also Mikhlin and Pr¨ ossdorf[305, p. 249])  ∞ 

e

 0r −iΘ·ξ

dr + −1 r

0

∞



e−iΘ·ξ0 r

1

* dr  * · ξ0 | − iπ Θ · ξ0 + α0 = − log |Θ * · ξ0 | r 2 |Θ (8.1.18)

With some constant α0 given explicitly in [305] we find  a0ψ (x, ξ)

  * o * log |Θ * · ξ0 | + iπ Θ · ξ0 + α0 dω . k −n (x, Θ) * · ξ0 | 2 |Θ

=−  |Θ|=1

For a given vector ξ = |ξ|ξ0 ∈ IRn , the integrand is weakly singular along * = 1. Hence, a0 (x, ξ) is uniformly * · ξ0 = 0 on the unit sphere |Θ| the circle Θ ψ bounded for every x ∈ K  Ω and all ξ ∈ IRn \ {0} and is homogeneous of degree zero. For n = 1, the proof follows in the same manner, except the Tricomi condition (8.1.17) now is replaced by 0 0 (x, 1) + k−1 (x, −1) = 0 . k−1

(8.1.19)

Here, instead of the Calder´on–Zygmund formula (8.1.18) we use its one– dimensional version, i.e., ∞ p.f. 0

e−iξz

dz iπ ξ = − log |ξ| − + α0 . z 2 |ξ|

(8.1.20)

8.1 Pseudohomogeneous Kernels

485

Collecting the above results, we find by using (8.1.17), bψ (x, ξ) = a0ψ (x, ξ) − I2 (x, ξ) + c(x){a0 + a1 log |ξ|} . Since lim aψ (x, tξ0 ) = a0ψ (x, ξ0 ) − lim I2 (x, tξ0 ) = a0ψ (x, ξ0 )

t→∞

t→∞

from (8.1.15) we obtain the desired property (8.1.12).



The next lemma, due to Seeley [385], is concerned with the differentiation and primitives of pseudohomogeneous functions. Lemma 8.1.3. Let m ∈ IR , k ∈ IN0 and m − k > −n.  ∂ α f (z) ∈ Ψ hfm−k for all |α| = k. If m ∈ IN (i) If f (z) ∈ Ψ hfm then −i ∂z and if fm , the homogeneous part of f , satisfies the compatibility conditions  * Θ * α dω = 0 for all |α| = m for n ≥ 2 fm (Θ) (8.1.21) |z|=1

*= where Θ

z |z| ,

and f (1) + (−1)m fm (−1) = 0 for n = 1 ,

then

∂fm ∂zj

will satisfy

 |z|=1

and

∂fm * β Θ dω = 0 for all |β| = m − 1 if n ≥ 2 ∂zj

(8.1.22)

  fm (1) + (−1)m−1 fm (−1) = 0 for n = 1 .

(ii) Let n ≥ 2 and h(α) (z) ∈ Ψ hfm−k for all α with |α| = k with the additional property that there exists some function F ∈ C ∞ (IRn \ {0}) and α  ∂ h(α) (z) = −i F (z) for all α with |α| = k . ∂z Then there exists a function f ∈ Ψ hfm with the property that F (z) − f (z) = pk−1 (z) is a polynomial of degree less than k. If m = 0 then f ∈ Ψ hfm is uniquely determined. In the one–dimensional case n = 1, the space IR \ {0} is not connected. Therefore we obtain only the weaker result

486

8. Pseudodifferential Operators as Integral Operators

F (z) − f (z) = p± k−1 (z) with two polynomials for z > 0 and z < 0, respectively, which in general might be different, provided m ∈ IN0 or m ∈ IN0 and m − k ≥ 0. In the case m ∈ IN0 and m − k < 0, the function h(k) (z) must also satisfy the additional assumption h(k) (1) − (−1)k h(k) (−1) = 0 . Proof: (i) The function f ∈ Ψ hfm is of the form f = fm (z) with fm (tz) = tm fm (z) for t > 0 and z = 0, provided m ∈ IN0 . Then differentiation with |α| = 1 α (α) yields for hm−1 (z) := ∂∂zfαm (z) the relation (α)

tm hm−1 (z) = tm

∂ α fm ∂ α fm (α) (z) = t (tz) = thm−1 (tz) . α ∂z ∂z α

(α)

Hence, hm−1 ∈ Ψ hfm−1 and also m − 1 ∈ IN0 . For m ∈ IN0 , the function f is of the form f (z) = fm (z) + pm (z) log |z| where fm (z) is a positively homogeneous function and pm (z) is a homogeneous polynomial of degree m. Then ∂ ∂ 1 ∂ f (z) = fm (z) + zk pm (z) · 2 + log |z| pm (z) . ∂zk ∂zk |z| ∂zk The first two terms now are positively homogeneous of degree m − 1 whereas ∂z∂k pm (z) is a homogeneous polynomial of degree m − 1. By repeating the same argument k times, the assertion follows. If, in addition, (8.1.21) is fulfilled then we distinguish the cases n ≥ 2 and n = 1. We begin with the case n ≥ 2 and consider the integral     ∂fm β  ∂fm * * β z  dω = r2m−2 (Θ)Θ dω I(r) := ∂zj ∂zj |z|=r  |Θ|=1

where β is an arbitrarily chosen multiindex with |β| = m − 1 and dω 2 denotes the  n surface element of the unit sphere. Integrating r I(r) = 2 =1 z I(r) from r = 0 to , we obtain with the divergence theorem:

8.1 Pseudohomogeneous Kernels



 r

n+1

I(r)dr =

0

 =−

∂ fm (z) ∂zj

V ()



=−

fm (z)

∂ ∂zj

∂fm β z ∂zj

V ()

' z

β

n 

zβ

V ()

n 

n 

( z2

'

 fm (z)z

dv +



dv + 2m+n

=1

β

n 

( z2

zj n−2 dω

=1

|z|=

( z2

dv

=1

( z2

=1

'

'

487

* Θ *β Θ *j dω . fm (Θ)

 |Θ|=1

V () denotes the volume of the ball with radius . The last integrals on d the right–hand side vanish due to assumption (8.1.21). Then taking d on both sides at  = 1, we find  n     ∂z β 2 β * * * * * I(1) = − fm (Θ) (Θ) Θ + Θ Θj dω . ∂zj =1

 |Θ|=1

∂β * *2 *β Θ *j is homogeneous of order (Θ)Θ and Θ Each of the polynomials ∂z j |β| + 1 = m. Again, with (8.1.21) for each of these terms, we obtain  ∂f * β Θ dω = 0 for any |β| = m − 1 . I(1) = ∂zj  |Θ|=1

Now we consider the case n = 1. Then  fm (1)

+

 fm (−1)(−1)m−1

1 =

 {f  (1) + fm (−1)(−1)m−1 }dr

0

1 =

 fm (r)r−m+1 dr +

0

1

 fm (−r)r−m+1 (−1)m−1 dr ,

0

 since fm (r) is positively homogeneous of degree (m − 1). Integration by parts gives with (8.1.21):

 (1) fm

+

 fm (−1)(−1)m−1

1 =−

fm (r)(1 − m)r−m dr

0

1 + (−1)

m−1

fm (−r)(1 − m)r−m dr

0

= −{fm (1) + fm (−1)(−1)m } = 0 , as proposed.

488

8. Pseudodifferential Operators as Integral Operators

(ii) For this part of the theorem we confine ourselves only to the case k = 1. Then the general case can be treated by standard induction. We will consider the three different subcases m < 0 , m > 0 and m = 0, separately. For m < 0, let ∂F (j) = ihm−1 (z) ∈ Ψ hfm−1 for j = 1, . . . , n . ∂zj (j)

Since we assume that F (z) is a given primitive of the hm−1 the indefinite line integral F (z) = i

z  n

(j)

hm−1 (ζ)dζj + c0 for z ∈ IRn \ {0} ,

∞ j=1

for n ≥ 2 is independent of the path of integration avoiding the origin. All that remains to show is the positive homogeneity of f (z) :=

z  n

(j)

hm−1 (ζ)dζj

∞ j=1

which is well defined because of * lim |ζ|m−1 = 0 lim hm−1 (ζ) = hm−1 (Θ) (j)

(j)

ζ→∞

ζ→∞

uniformly and m − 1 < −1. A simple change of variables ζj = tζj with (j) any t > 0 shows with the homogeneity of hm−1 for z = 0 that 1  (j) h (z)zj m j=1 m−1 n

f (z) =

which is obviously homogeneous of degree m. This is Euler’s formula for positively homogeneous functions of degree m. For n = 1, the integration paths are from (sign z)∞ to z and are different for z > 0 and z < 0. Hence, in general one obtains two different constants c± 0 accordingly. For m > 0 and m ∈ IN, we proceed in the same manner as before by defining z  n n 1  (j) (j) h (z)zj . f (z) := hm−1 (ζ)dζj = m j=1 m−1 j=1 0

This representation also shows that f is uniquely determined.

8.1 Pseudohomogeneous Kernels

489

Again, in the case n = 1, one may obtain different contants c± 0 corresponding to z > 0 or z < 0, respectively. Now, let us consider the case m ∈ IN and n ≥ 2. Here, (j)

(j)

h(j) (z) = hm−1 (z) + pm−1 (z) log |z| = −i

∂F (z) ∂zj

is pseudohomogeneous of degree m − 1. Hence, ∂F (j) (j) (tz) = itm−1 hm−1 (z) + itm−1 pm−1 (z) {log |z| + log t} ∂zj for every t > 0 and z = 0. For F (z), we find the path–independent line integral n z  ∂F (ζ)dζj + c F (z) = ∂z j j=1 0

where c is any constant. By changing the variable of integration ζ = τ z and integrating along the straight lines where τ ∈ [0, 1], we find F (z)

=

i

n 1 

  (j) (j) τ m−1 hm−1 (z) + pm−1 (z) log |z| zj dτ

j=1τ =0 n  

1

+i

(j)

τ m−1 log τ dτ pm−1 (z)zj + c ,

j=1 0

i.e., the explicit representation  n  1 (j) i  (j) (j) hm−1 (z) + pm−1 (z) log |z| − pm−1 (z) zj + c F (z) = m j=1 m with any constant c. Therefore,  n  i  (j) 1 (j) hm−1 (z) − pm−1 (z) zj , fm (z) = m j=1 m i  (j) p (z)zj . m j=1 m−1 n

pm (z)

=

Both functions, fm (z) and the polynomial pm (z) are homogeneous of degree m and F = fm + pm log |z| + c. Finally, for n ≥ 2, let us consider the case m = 0. Here the desired form of f is

490

8. Pseudodifferential Operators as Integral Operators

f (z) = c0 + g0 (z) + p0 log |z| where g0 and p0 both are to be found. The yet unknown function g0 (z) is positively homogeneous of degree zero, hence, the desired function n  ∂g0 j=1

∂zj

(z)zj is positively homogeneous

of degree zero again, due to the first part of our theorem. Now we require that the condition   n ∂g0 Θj dω = 0 (8.1.23) ∂zj j=1 |z|=1

is satisfied. This implies with zj ∂g0 ∂F = + p0 2 = h(j) (z) ∂zj ∂zj |z|

(8.1.24)

that p0 must be chosen as to satisfy  p0 |z|=1

 n n   zj Θ dω = j |z|2 j=1 j=1

h(j) (z)Θj dω .

(8.1.25)

|z|=1

With the constant p0 , we now define the function  n z n z    ∂g0 ζj h(j) (ζ) − p0 2 dζj g0 (z) = dζj := ∂zj |ζ| j=1 j=1 Θ

(8.1.26)

Θ

where Θ is a point on the unit sphere which fulfills n  

 h(j) (Θ) − p0 Θj Θj = 0 .

j=1

This is always possible because of (8.1.24). Since h(j) = ∂ log |z| ∂zj ,

∂F ∂zj

and

zj |z|2

=

the value of the line integral in (8.1.26) is independent of the path as long as IRn \ {0} is simply connected. It will be independent of the path in every case if for a closed curve C on the unit sphere, C : z = Θ(τ ) for τ ∈ [0, T ] with Θ(0) = Θ(T ) = 0, the integral n :  

h(j) (Θ(τ )) − p0 Θ(τ )

j=1 C

vanishes. Since h(j) (z) =

∂F ∂zj

 dΘ dτ

j

dτ

with F ∈ C ∞ (IRn \ {0}) and

8.1 Pseudohomogeneous Kernels

491

n n  d  2 ∂Θj =0 Θj (τ ) = Θj (τ ) dτ j=1 ∂τ j=1

for any curve on |Θ(τ )| = 1, we find, indeed, n :  ∂g0 dζj = 0 . ∂zj j=1 C

To show that g0 is positively homogeneous of degree zero, we first show with (8.1.24) and (8.1.25) tΘ n Θ

j=1

∂g0 (ζ)dζj ∂zj

=

n tΘ  j=1

=

h(j) (τ Θ) − p0

Θ

log t

n 

τ Θj  Θj dτ τ2

 h(j) (Θ)Θj − p0 = 0

j=1

? for every t > 0. Then, with ζj = ζj |z|,

g0 (z)

=

|z|Θ n  n z   ∂g0 ∂g0 dζj + dζj ∂z ∂zj j j=1 j=1 |z|Θ

Θ

=

0+

n  j=1

=

n  j=1

z/|z| 

z/|z| 

∂g0  (|z|ζ)|z|d ζj ∂zj

Θ

∂g0   (ζ)dζj = g0 ∂zj



z |z|



Θ

where we have used the homogeneity of

∂g0 ∂zj

due to (8.1.24).

We note that we may add to g0 an arbitrarily chosen constant which defines a homogeneous function of degree zero. Hence, g0 is not unique. In the remaining case n = 1 and m = 0, for k = 1 we have for h(1) ∈ Ψ hf−1 the explicit representation h(1) (z) = h(1) (sign z) · Hence,

1 . |z|

f (z) = (sign z) · h(1) (sign z) log |z| + c± 0 ,

492

8. Pseudodifferential Operators as Integral Operators

and only with h(1) (1) − (−1)h(1) (−1) = 0 we find

f (z) = h(1) (1) log |z| + c± 0 ∈ Ψ hf0 

as proposed.

In the following lemma we show how one can derive the homogeneous symbol from the integral operator with a kernel given as a pseudohomogeneous function. Lemma 8.1.4. Let kκ (x, z) ∈ Ψ hfκ with κ > −n. Then the integral operator Km defined by  (Km u)(x) = kκ (x, x − y)u(y)dy for x ∈ Ω , u ∈ C0∞ (Ω) (8.1.27) Ω

belongs to Lm c (Ω) where m = −κ − n. In addition, if ψ(z) is a C ∞ cut–off function with ψ(z) = 1 for |z| ≤ ε , ε > 0, and ψ(z) = 0 for |z| ≥ R then  (8.1.28) aψ (x, ξ) := e−ix·ξ kκ (x, x − y)ψ(x − y)eiy·ξ dy Ω m (Ω × IRn ) and the limit is a symbol in Sc

am (x, ξ) := lim t−m aψ (x, tξ) t→+∞

(8.1.29)

exists for x ∈ Ω and ξ = 0 and defines the positively homogeneous representative of degree m of the complete symbol class σKm ∼ am (x, ξ). Proof: With ψ(z), first write    (Km u)(x) = kκ (x, x−y)ψ(x−y)u(y)dy+ kκ (x, x−y) 1−ψ(x−y) u(y)dy . Ω

Ω

 Since the kernel {kκ (x, x − y) 1 − ψ(x − y) } of the second integral operator is a C ∞ (Ω × Ω) function, the latter defines a smoothing operator due to Theorem 7.1.10 (iv). Hence, it suffices to consider only the first operator  (Kψ u)(x) := kκ (x, x − y)ψ(x − y)u(y)dy . Ω

Note that kκ (x, z)ψ(z) is weakly singular and has compact support with respect to z, hence, the symbol aψ (x, ξ) given by (8.1.28) is just its Fourier transformation and, therefore, due to the Paley–Wiener–Schwartz theorem

8.1 Pseudohomogeneous Kernels

493

m (Theorem 3.1.3) aψ ∈ C ∞ (Ω × IRn ). It remains to show that aψ ∈ Sc (Ω × n IR ), that am (x, ξ) is well defined by (8.1.29) and, finally, that aψ ∼ am (for |ξ| ≥ 1). We first show the existence of the limit am . To do so, choose a multi–index γ with |γ| > −m and consider  ξ γ aψ (x, ξ) − t−m aψ (x, tξ)    z  = ξγ dz e−iz·ξ kκ (x, z) ψ(z) − ψ t n IR  z  γ +ξ dz log t e−iz·ξ pκ (x, z)ψ t IRn    z   ∂ γ  = dz e−iz·ξ − i kκ (x, z) ψ(z) − ψ ∂z t ε≤|z|



+

e−iz·ξ

 0≤β≤γ |β|≤κ

IRn

 ε≤|z|≤R

e−iz·ξ



−i

 ∂ β pκ (x, z) × ∂z

 ∂ γ−β  z  dz log t ψ × cβγ − i ∂z t

provided t ≥ R, by using the Leibniz rule. Then we obtain    I2λγ (t) + I3βγ (t) ξ γ aψ (x, ξ) − t−m aψ (x, tξ) = I1 (t) + 0≤β≤γ |β|≤κ

0≤λ −m, for fixed x ∈ Ω, the integral I1 (t) converges uniformly as t → ∞ for all |z| ≥ ε:

494

8. Pseudodifferential Operators as Integral Operators



e−iz·ξ kκ−|γ| (x, z) {ψ(z) − 1} dz . (γ)

lim I1 (t) =

t→∞

|z|≥ε

A simple manipulation shows that the second term in I2λγ (t) can be estimated by   z     (λ) dz  kκ−|λ| (x, z)t|λ|−|γ| ψγ−λ cλγ t ε≤|z|≤tR

tR

rκ−|λ| t|λ|−|γ| rn−1 (1 + | log r|)dr = O(t−1 log t)

≤

c(x, λ, γ, ε, R)

→

0 as t → ∞ since |γ| > κ + n .

r=ε

For the third term we find similarly |I3βγ (t)| = O(t−1 log t) → 0 as t → ∞ . Hence, the limit exists; and for ξ = 0 we find am (x, ξ)

:= =

lim t−m aψ (x, tξ) (8.1.30)  (γ) aψ (x, ξ) − ξ −γ e−iz·ξ kκ−|γ| (x, z) {ψ(z) − 1} dz

t→∞

− ξ −γ

|z|≥ε





0≤λ 0 and ξ = 0 .

t→+∞

Finally, we show that aψ (x, ξ) − χ(ξ)am (x, ξ) ∈ S −∞ (Ω × IRn ) for any C –cut–off function χ with χ(ξ) = 0 for |ξ| ≤ 12 and χ(ξ) = 1 for all |ξ| ≥ 1.   Ω be any compact subset; consider any multiindices α, β, γ. Let Ω Then, for |ξ| ≥ 1 from (8.1.30), ∞

8.1 Pseudohomogeneous Kernels

495

 ∂ β  ∂ α  aψ (x, y) − am (x, ξ) ξγ ∂x ∂ξ  ∂ β   ∂ α  (γ) e−iz·ξ kκ−|γ| (x, z) {ψ(z) − 1}dz = ∂ξ ∂x |z|≥ε    ∂ α  ∂ β  (λ) e−iz·ξ kκ−|λ| (x, z) ψγ−λ (z)dz cλγ + ∂ξ ∂x 0≤λ κ + n + |α| the estimates  ∂ β  ∂ α     α, β, γ) . aψ (x, ξ) − am (x, ξ) |ξ γ | ≤ C(Ω,  ∂x ∂ξ  x∈Ω,|ξ|≥1 sup

 m (Ω×IRn ). Hence, aψ (x, ξ)−χ(ξ)am (x, ξ) ∈ S −∞ (Ω×IRn ) and aψ (x, ξ) ∈ Sc  Remark 8.1.2: For κ ∈ IN0 , consider the whole class of integral operators with the kernels of the form kκ (x, z) = fκ (x, z) + pκ (x, z) log |z| + qκ (x, z)

(8.1.31)

where qκ (x, z) = |γ|=κ cγ (x)z γ is a homogeneous polynomial of degree κ. Then all kκ of the form (8.1.31) belong to Ψ hfκ . For given fκ and pκ , and for any choice of qκ (x, z) in (8.1.31), kκ has the same positively homogeneous principal symbol   ξ a0m (x, ξ) = am x, |ξ|m |ξ| since the integral operator defined by the kernel function qκ is a smoothing operator. For fixed x and κ ∈ IN0 , the symbol of kκ (x, z) = fκ (x, z) is a positively homogeneous function given by the Fourier transform of fκ which also defines a homogeneous distribution of degree m = −κ − n with a singularity at ξ = 0 for m < 0.

496

8. Pseudodifferential Operators as Integral Operators

Let us recall that a distribution u is said to be homogeneous of degree λ ∈ C if tλ+n u, ϕ(xt) = u, ϕ for all ϕ ∈ C0∞ (IRn ) and t > 0 .

(8.1.32)

For the case κ ∈ IN0 , and pseudohomogeneous kernels of the form (8.1.31), we again consider the Fourier transform of the kernel. The latter defines the density of a distribution whose Fourier transform can be written as  6κ (x, ξ) = (pκ x, ·) log | · | (x, ξ) + Fz→ξ fκ (x, z) . k We first consider f κ (x, •)(ξ)

=

(2π)

−n/2



fκ (x, z)ψ(z)e−iz·ξ dz

|z|≤R

+ |ξ|−2k (2π)−n/2



(8.1.33)

  e−iz·ξ (−Δkz ) fκ (x, z)(1 − ψ(z)) dz

|z|≥ε

which exists for ξ = 0 since both integrals exist, provided that for the second one 2k > κ + n. Similar to the previous analysis, one can easily show that f*κ (x, ξ) = O(|ξ|m ) as |ξ| → 0. Moreover, with the identity •  (z) = tn ϕ(tz) ϕ * for t > 0 t we obtain 

  ξ dξ t

=

fκ (x, z)ϕ(z)dz *

=

f*κ (x, ξ)ϕ 

IRn

= tn+m IRn

 fκ (x, z)ϕ(tz)dz *

tn



IRn

tm+n

f*κ (x, ξ)ϕ(ξ)dξ

(8.1.34)

IRn

for all ϕ ∈ C0∞ (IRn \{0}) and t > 0 , m = −κ−n. According to the definition of homogeneity of generalized functions in H¨ormander [188, Vol.I, Definition 3.2.2], Equation (8.1.34) implies that f*κ (x, ξ) is a positively homogeneous function of degree m for ξ = 0. To extend f*κ (x, ξ) as a distribution on C0∞ (IRn ), we use the canonical extension through the finite part integral

8.1 Pseudohomogeneous Kernels



f*κ (x, ξ)ϕ(ξ)dξ

p.f. IRn



497

(8.1.35)

  1 ∂αϕ  ξ α α (0) dξ f*κ (x, ξ) ϕ(ξ) − α! ∂ξ

=

|α|≤κ

|ξ| 0 , ξ = 0 . Then k−m−n (x, z) := (2π)−n p.f.

 eiz·ξ am (x, ξ)dξ

(8.1.39)

IRn

exists and defines a positively pseudohomogeneous kernel function of degree −m − n. Proof: Let χ(ξ) be a C ∞ cut–off function with χ(ξ) = 1 for |ξ| ≥ 1 and χ(ξ) = 0 for |ξ| ≤ 12 . Then each of the following integrals,  −n eiz·ξ am (x, ξ)χ(ξ)dξ k(x, z) = (2π) IRn   −n + (2π) p.f. eiz·ξ 1 − χ(ξ) am (x, ξ)dξ , IRn

exists since the first has integrable integrand because of m < −n and the second one is defined by the Fourier transform of a distribution with compact

8.1 Pseudohomogeneous Kernels

499

support, therefore it is a C ∞ (Ω × IRn )–function according to the Paley– Wiener–Schwartz theorem, Theorem 3.1.3. Hence, for x fixed, k(x, z) is a Fourier transformed positively homogeneous distribution. We now show that k ∈ C ∞ (Ω × IRn \ {0}). For z = 0 one finds with (7.1.11)    ∂ α k(x, z) = (2π)−n |z|−2k eiz·ξ (Δξ )k ξ α am (x, ξ)χ(ξ) dξ −i ∂z +(2π)−n



IRn ∩{|ξ|≥ 12 }



∂ α −i ∂z

 eiz·ξ 1 − χ(ξ) am (x, ξ)dξ .

(8.1.40)

IRn ∩{|ξ|≤1}

The first integral on the right–hand side is continuous if |α| < −n − m + 2k for any chosen k ∈ IN0 whereas the second integral is C ∞ (Ω × IRn ) for any α ≥ 0. Hence, k(x, z) ∈ C ∞ (Ω × IRn \ {0}). Since, in general, the Fourier transform of a positively homogeneous distribution is not necessarily positively homogeneous, we consider first the derivatives  ∂ α k(x, z) k (α) (x, z) := − i ∂z given by (8.1.40) with |α| > −m − n and k chosen with 2k > |α| + n + m. For these α one will see that k (α) is positively homogeneous of degree −m − n − |α|. For this purpose we use (7.1.11) and begin with the limit lim tm+n+|α| k (α) (x, tz)

t→0+

= (2π)

−n

lim t

m+n+|α|



t→0+

|tz|

  eitz·ξ ξ α 1 − χ(ξ) am (x, ξ)dξ

+ = (2π)−n lim

t→0+

 eitz·ξ (Δξ )k ξ α am (x, ξ)χ(ξ) dξ

IRn ∩{|ξ|≥ 12 }







−2k

IRn ∩{|ξ|≤1}



|z|−2k 

 α  eiz·ξ (Δξ )k ξ  am (x, ξ  )χ

|ξ  |≥t 12

+

e

iz·ξ  α

ξ



 1−χ

ξ t





ξ t



dξ 

 am (x, ξ  )dξ  ,

|ξ  |≤t

where the transformation ξ  = tξ and the positive homogeneity of am (x, ξ) have been employed. Since |α| > −m−n, the integrand of the second integral on the right–hand side is continuous, uniformly bounded and tends to zero as t → 0+. The first term on the right–hand side can be rewritten as follows:

500

8. Pseudodifferential Operators as Integral Operators

lim tm+n+|α| k (α) (x, tz)

t→0+

=

(2π)−n |z|−2k



 α  eiz·ξ (Δξ )k ξ  am (x, ξ  )χ(ξ  ) dξ 

1  2 ≤|ξ |



+(2π)−n lim |z|−2k t→0+

    ξ     χ(ξ  ) dξ  eiz·ξ (Δξ )k ξ α am (x, ξ  ) χ t

1  2 t≤|ξ |≤1

=

(2π)

−n

|z|

−2k

+(2π)−n



 eiz·ξ (Δξ )k ξ α a(x, ξ)χ(ξ)dξ

1 2 ≤|ξ|



eiz·ξ ξ α a(x, ξ)(1 − χ(ξ))dξ

|ξ|≤1

=

k

(α)

(x, z) ,

since the second term has a continuous, uniformly bounded integrand which tends to the corresponding integrand in (8.1.40). Thus, lim tm+n+|α| k (α) (x, tz) = k (α) (x, z) .

t→0+

(8.1.41)

Equation (8.1.41) implies that k (α) (x, z) ∈ Ψ hfκ−|α| is positively homogeneous with κ = −m − n. Hence, for n ≥ 2 it follows from Lemma 8.1.3 part (ii) that k(x, z)−p|α|−1 (z) ∈ Ψ hf−m−n where p|α|−1 (x, z) is some polynomial of degree less than |α|. In what follows, for n ≥ 2 we shall now present the explicit form of the kernel k(x, z) = f−m−n (x, z) + p−m−n (x, z) log |z| which shows that p|α|−1 = 0. From Definition 8.1.2 applied to (8.1.39) we find with Ω0 = {ξ ∈ IRn | |ξ| < R} for κ = −m − n > 0,      1 k(x, z) = (2π)−n (iz)α ξ α dξ am (x, ξ) eiξ·z − (8.1.42) α! −

 |α| 0 reads after proper change of variables,

8.1 Pseudohomogeneous Kernels

k(x, tz)

(2π)−n

=

−

⎧ ⎪ ⎨ ⎪ ⎩

 |α|n →0

and

where ξ0 = ξ  /|ξ  | and qm+1−j is given by (9.4.7).  we write To show (9.4.6) for Qv,

9.4 Pseudodifferential Operators with Symbols of Rational Type

am−j (, D)wη ()    −n = (2π) ei ·ξ v*(ξ  ) |ξ  |≤1



+(2π)−n



577

 ein ξn am−j ( , n ), (ξ  , ξn ) g*(ηξn )dξn dξ 

|ξn |≤c0



  ei ·ξ v*(ξ  )a0m−j ( , n ), (ξ  , ξn ) ein ξn g*(ηξn )dξn dξ  

|ξ  |≤1 |ξn |≥c0

+(2π)



−n



   ei ·ξ v*(ξ  )a0m−j ( , n ), (ξ  , ξn ) ein ξn g*(ηξn )dξn dξ 

|ξ  |≥1 IR

=:

I1 + I 2 + I 3 .

Since in I1 the domain of integration is compact and the integrand is C ∞ , we get      ei ·ξ v*(ξ  ) ein ξn am−j ( , n ), (ξ  , ξn ) dξn dξ  . lim I1 = (2π)−n 0 m−j+1 and the integrand is holomorphic in Im z ≤ 0 ∧ |z| ≥ c0 provided 0 ≤ η < −n . Then we write      I2 = (2π)−n ei ·ξ v*(ξ  ) a0m−j ( , n ), (ξ  , z) ein z g*(ηz)dzdξ  |ξ  |≤1

c\[−c0 ,c0 ]

and the integrand depends continuously on η on the compact domain of integration. Taking the limit yields lim I2

0 m+n+1, i.e., a C ∞ –Schwartz kernel. This completes the proof of Theorem 9.4.3.   Corollary 9.4.4. Let A ∈ Lm c (Ω) satisfy all the conditions of Theorem 9.4.3. Then  c v() := AΦ ,0 (v ⊗ δ)( , n ) for  ∈ Ur , n > 0 and v ∈ D(Ur ) Q r (9.4.22)   c maps C ∞ (Ur ) to C ∞ Ur × has a continuous extension to n ≥ 0 and Q 0 (0, ε) . The trace Qc v( ) =

lim

0 0 is chosen  sufficiently large so that all the poles of am−j ( , 0), (ξ  , z) in the upper half–plane are enclosed in the interior of cc ⊂ C. The proof is identical to that of Theorem 9.4.3 except n ≥ η > 0 and Im z ≥ 0. r with Lemma 9.4.5. Let Γτ be the family of planes n = τ = const in U  τ ∈ Iτ . Then the limits of Q(τ )v() given by (9.4.22) in (9.4.5) and (9.4.22) for n → τ ±, respectively, exist for every τ ∈ Iτ and depend continuously on τ.  Proof: The poles of a0m−j ( , τ + n ) ; (ξ  , z) , i.e., the zeros of  ℘d(j) ( + τ n) ; (ξ  , z) in (9.4.13) depend continuously on τ . Therefore, the contour integrals defined in (9.4.14) also depend on τ continuously. Hence,  m+1−j v)(τ )() in the family given by (9.4.20) coneach of the potentials (Q verges to the corresponding limit qm+j−1 (τ,  , D)v for each Γτ . Each of these limits then depends continuously on τ . As in (9.4.21) the remaining operator R has a kernel that now depends continuously on the parameter τ , it follows that the limit operators Q(τ ) and Qc (τ ) exist and depend on τ continuously.  Remark 9.4.1: As a consequence of Lemma 9.4.5, the existence of the limits Q and Qc implies that the extension conditions are fulfilled for each τ ∈ Iτ . The latter implies that therefore the canonical extension conditions (9.3.31) are satisfied. This means that together with the special parity conditions (9.4.3) all the assumptions of Theorem 9.3.8 are satisfied for our class of operators A having symbols of rational type. Hence, together with Lemma 9.4.1, Theorem 9.3.9 holds as well. To sum up these consequences we now have the following theorem which, again, is the Plemelj–Sochozki jump relation for the class of operators having symbols of rational type.  Theorem 9.4.6. Let A ∈ Lm c (Ω) have a symbol of rational type. Let AΦr,0 ∈ m  Lc (Ur ) be the properly supported part of AΦr in the parametric domain. Then  the jump of AΦr,0 v ⊗ δ(n ) across n = 0 is given by

9.4 Pseudodifferential Operators with Symbols of Rational Type

581



lim

0 m − j + 1 in (9.4.31). To estimate the remaining integral we want to show :  0 −1 am−j − am−jH )( , 0), (ξ  , z) dz (2π) |z|=R

=

(2π)



−1

ξn ≤−R

−(2π)

−1



   a− m−jR ( , 0), (ξ , ξn ) dξn    a+ m−jR ( , 0) , (ξ , ξn ) dξn .

R≤ξn

The latter then can be estimated by the use of (9.4.30). To this end, employ (9.4.10), introduce polar coordinates about z = 0 and write   (a0m−j − am−jH ) ( , 0), (ξ  , z) dz c+ R



 =

lim

n →0+

lim

η→0+ c+ R

  (a0m−j − am−jH ) ( , n ), (ξ  , z) ein z g*(ηz)dz

9.4 Pseudodifferential Operators with Symbols of Rational Type

583

with 0 < η < n where g*(ηz) is defined by (9.4.10). Here, c+ R is the contour iϑ with 0 ≤ ϑ < π}. Because of the estimate (9.4.11), defined by c+ R = {Re for every η > 0, with an application of the Cauchy integral theorem applied to the integrand which is a holomorphic function in the upper half–plane Im z ≥ 0 for |z| ≥ R ≥ c0 , we obtain   (a0m−j − am−jH ) ( , 0), (ξ  , z) dz c+ R

= − lim

n →0+

 =



 lim

η→0+ |ξn |≥R

a− m−jR

  (a0m−j − am−jH ) ( , n ), ξ ein ξn g*(ηξn )dξn 

( , o)(ξ , ξn ) dξn −







ξn ≤−R

   a+ m−jR ( , 0), (ξ , ξn ) dξn .

ξn ≥R

iδ with π ≤ δ ≤ 2π}, we obtain Similarly, with c− R := {z = Re   (a0m−j − am−jH ) ( , 0), (ξ  , z) dz c− R

 =

   a− m−jR ( , 0), (ξ , ξn ) dξn −

ξn ≤−R



   a+ m−jR ( , 0) , (ξ , ξn ) dξn .

R≤ξn

Collecting terms, we find [qm+1−j ]( , ξ  )

=



1 ∂ γ am−j   ( , 0), (0, 1) ξ γ  γ γ! (∂ξ ) |γ|=m−j+1   −1 + (2π)−1 a− dξ − (2π) m−jR n

i

ξn ≤−R

a+ m−jR dξn

R≤ξn

for every R ≥ c0 and every fixed  ∈ Ur and ξ  ∈ IRn−1 \ {0}. By taking the limit as R → ∞, in view of the estimate (9.4.30) we obtain the desired result.  Remark 9.4.2: In Theorem 9.4.6, as we mentioned already in Remark 9.4.1, the invariance property of the finite part integral is crucial for the behaviour of potential operators. In particular this type of potential operators includes all the classical boundary integral operators as investigated by R. Kieser in [225]. This implies that numerical computations for this class of operators can be performed by the use of classical coordinate transformations. We now turn to investigating the mapping properties of the Poisson op defined by (9.4.5). erator Q

584

9. Pseudodifferential and Boundary Integral Operators

 defined by (9.4.5) satisfies the estiTheorem 9.4.7. The linear operator Q mate  ϕQv

1

H s−m− 2 (IRn ∩{n ≤0})

≤ cvH s (K)

(9.4.33)

r ), and any compact set K  Ur and every for any s ∈ IR where ϕ ∈ C0∞ (U ∞ v ∈ C0 (K). The constant c depends on K, ϕ and s.  c reads The corresponding estimate for Q  c v ϕQ

1

H s−m− 2 (IRn ∩{n ≥0})

≤ cvH s (K) .

(9.4.34)

To prove this theorem we first collect some preliminary results. Lemma 9.4.8. Let K  IRn−1 be a compact set. Then the distributions of the form u() = v( ) ⊗ δ(n ) with  = ( , n ) and v ∈ C0∞ (K) will satisfy the estimate u

1

H s− 2 (K×IR)

≤ cvH s (K)

(9.4.35)

provided s < 0. Proof: Using duality, the trace theorem and the trace operator (4.2.32), we find for ϕ ∈ C0∞ (IRn ):     = sup |u, ϕIRn | = sup  v( )(γ0 ϕ)d  u s− 12 H

(K×IR)

ϕ 1 −s ≤1

ϕ 1 −s ≤1

2

≤



c · vH s (K) ·

2

sup ϕ 1 −s ≤1

K

γ0 ϕH −s (K) ≤ c · vH s (K)

2

for any s ∈ IR with s < 0.



We remark that, as a consequence of Lemma 9.4.8 we already obtained the desired estimate (9.4.33) provided s < 0. In order to extend this estimate to arbitrary s ≥ 0, we first analyze the function qm+1−j in more detail. Lemma 9.4.9. Let α, β ∈ IN0 and K  Ur be any compact subset. Then qm+1−j (, ξ  ) given in (9.4.14) satisfies the estimates |βn Dαn qm+1−j (, ξ  )| ≤ cξ  m+1+α−β−j

(9.4.36)

for n ∈ [−ε, 0) ,  ∈ K and ξ  ∈ IRn−1 where c = c(j, α, β, K) is a constant. Proof: From (9.4.14) we obtain, by using the transformation z = z  |ξ  |, z  ∈ c, |ξ  | ≥ 1 and with integration by parts with respect to z  ,

9.4 Pseudodifferential Operators with Symbols of Rational Type

585

βn Dαn qm+1−j (, ξ  )   α  1 (iz)γ ein z Dα−γ = βn a0m−j , (ξ  , z) dz n γ 2π |ξ  |c

=

1 2π



|ξ  |c

=

0≤γ≤α

 α  i β    βn ein z Dzβ (iz)γ Dα−γ a0m−j , (ξ  , z) dz n γ n

0≤γ≤α

  α 1  × |ξ  |−β+γ+m−j+1 γ 2π 0≤γ≤α       × (i)β ein |ξ |z Dzβ (iz  )γ Dα−γ a0m−j , (ξ0 , z  ) dz  . n c

Then we obtain the estimates   β α n D qm+1−j (, ξ  ) ≤ c|ξ  |m+1−j+α−β for |ξ  | ≥ 1 n whereas for |ξ  | ≤ 1 the left–hand side is bounded since it is smooth on |ξ  | ≤ 1 and has compact support with respect to  and n ≤ 0. This yields (9.4.36).  Lemma 9.4.9 indicates that q(, ξ  ) for n ≤ 0 has properties almost like a symbol although the inequality (9.4.36) is not strong enough for n → 0. However, for applying the techniques of Fourier transform we need a C ∞ extension of qm+1−j to n ≥ 0. To this end we use the Seeley extension theorem proved in [383] by putting ⎧  ⎪ qm+1−j ( , n ), ξ  for n ≤ 0 and ⎪ ⎨ ∞    pm+1−j (, ξ ) := ⎪ for n > 0 λp qm+1−j ( , −2p n ), ξ  ⎪ ⎩ p=1

(9.4.37) where the sequence of real weights λp =

1,∞ , =p

1 − 2− 1 − 2p−

for p ∈ IN

(9.4.38)

satisfies the infinitely many identities ∞ 

λp 2p = (−1) for all  ∈ IN

(9.4.39)

p=1

and these series converge absolutely. With the relations (9.4.39) one can easily show that all derivatives of qm+1−j (, ξ) are continuously extended across

586

9. Pseudodifferential and Boundary Integral Operators

n = 0 since the term–wise differentiated series (9.4.37) still converges absolutely and uniformly. Since qm+1−j has compact support with respect to  and n ≤ 0, also pm+1−j has compact support in . The extended functions pm+1−j define operators    −n+1 Pm+1−j v() := (2π) ei ·ξ pm+1−j (, ξ  )* v (ξ  )dξ  (9.4.40) 

IRn−1



for  ∈ Ur × (−ε, ε) and Pm+1−j v()

 m+1−j v() for n ≤ 0 Q

=

(9.4.41)

 m+1−j is given by (9.4.20) via (9.4.16). In order to analyze the mapwhere Q ping properties of Pm+1−j , we now consider the Fourier transformed p*m+1−j of pj , i.e.  p*m+1−j (ζ, ξ  ) := (2π)−n/2 e−iζ· pm+1−j (, ξ  )d . (9.4.42) IRn

Lemma 9.4.10. For any q, r ∈ IN, there exist constants c(q, r, j) such that p*m+1−j satisfies the estimates  |* pm+1−j (ζ, ξ  )| ≤ c(1 + |ξ  |)m−j (1 + |ζ  |)−r 1 +

|ζn | −q . 1 + |ξ  |

(9.4.43)

Proof: We begin with the case |ξ  | ≥ 1. Then from the definition of pm+1−j , we have     −n/2 p*m+1−j (ζ, ξ ) = (2π) e−i ·ζ ×  ∈IRn−1

×



0 e

−in ζn

(2π)

−1

+ n =0

e−in ζn (2π)−1

 ein z a0m−j ( , n ), (ξ  , z) dzdn

z∈|ξ  |c

n =−∞

∞





z∈|ξ  |c

∞ 

λp ei2

p

n z 0 am−j

 ( , −2p n ), (ξ  , z) dzdn d .



p=1

By the homogeneity of a0m−j and ξ  = |ξ  |ξ0 and by scaling n = |ξ  |n , ζn = |ξ  |−1 ζn and z = |ξ  |z  , one obtains      m−j −n/2−1 p*m+1−j (ζ, ξ ) = |ξ | (2π) e−i ·ζ {. . . }d ,  ∈IRn−1

where

9.4 Pseudodifferential Operators with Symbols of Rational Type

{. . . } =



0



e−in ζn

z  ∈c

 n =−∞

∞ +  n =0



e−in ζn

∞  p=1

:

:

   n ein z a0m−j ( ,  , (ξ0 , z  ) dz  d n |ξ |

e−in 2

λp

587

p 

z

a0m−j



 , −2p

z  ∈c

 n     . , (ξ , z ) dz d  n 0 |ξ  |

In the integrands above, we see that the second integral contains the function   2p n λp f  , −  , ξ0 , |ξ | p=1

∞ 

which is the C ∞ Seeley extension of :         n n   −1 f  ,  , ξ0 := (2π) ein z a0m−j  ,  , (ξ0 , z  ) dz  |ξ | |ξ | z  ∈c

from n ≤ 0 to n > 0. The function f has compact support with respect  to ( , n ) for every fixed moreover, is uniformly bounded. Therefore,   ξnand, its Seeley extension F  , |ξ | has the same properties and its function series converges absolutely and uniformly. Since  n     m−j −n/2 p*m+1−j (ζ, ξ ) = |ξ |  (2π) e−i·ζ F ( ,  ) , ξ0 d |ξ |  ∈IRn

where  = ( , n ) and ζ = (ζ  , ζn ), the Paley–Wiener–Schwartz theorem 3.1.3 yields the estimates  −N |ξ  |m−j for ζ ∈ IRn and ξ  ∈ IRn−1 , |ξ  | ≥ 1 . |* pm+1−j (ζ, ξ  )| ≤ cN (1 + |ζ|) Here, since N ∈ IN is arbitrary, the latter implies with N = r + q,  −q |ζn |   −r 1+ (1 + |ξ  |)m−j , |* pm+1−j (ζ, ξ )| ≤ cN (1 + |ζ |) 1 + |ξ  | the proposed estimate (9.4.43) for |ξ  | ≥ 1. For |ξ  | ≤ 1 one finds the uniform estimates |* pm+1−j (ζ, ξ  )| ≤ cN (1 + |ζ  |)−r (1 + |ζn |)−q by using the Paley–Wiener–Schwartz theorem 3.1.3 again. Then 1 ≤ 1+|ξ  | ≤ 2 implies the desired estimate (9.4.43) for |ξ  | ≤ 1 as well. This completes the proof.  Lemma 9.4.10, in contrast to Lemma 9.4.9, will give us the following crucial mapping properties.

588

9. Pseudodifferential and Boundary Integral Operators

 m+1−j defined by (9.4.20) satisfies the estiLemma 9.4.11. The operator Q mates  m+1−j v Q

1

H s−m+j− 2 (IRn ∩{n ≤0})

≤

Pm+1−j v

≤

cvH s (IRn−1 ) for v ∈ C0∞ (Ur ) .

(9.4.44)

1

H s−m+j− 2 (IRn )

Proof: Our proof essentially follows the presentation in the book by Chazarain and Piriou [64, Chap.5, Sec.2]. Since   ψH t (IRn− ) = inf Ψ H t (IRn ) with Ψ |IRn = ψ −

where IRn− = IRn ∩ {n ≤ 0} and  m+1−j v)() = (Pm+1−j v)() for n < 0 , (Q the first estimate in (9.4.44) is evident. In order to estimate the norm of Pm+1−j v we use the Fourier transform of Pm+1−j v. The Parseval identity implies Pm+1−j v, ϕ   v, ϕ := Pm+1−j v()ϕ()d = Pm+1−j * IRn

=

(2π)



−n+1



 ξ  )(1 + |ζ|)s−m+j− 12 × p*m+1−j (ζ − ξ,

ζ∈IRn ξ  ∈IRn−1 1

 × v*(ξ  )(1 + |ζ|)−s+m−j+ 2 ϕ(−ζ)dξ * dζ

where ξ = (ξ  , 0) ∈ IRn for any ϕ ∈ C0∞ (IRn ). Consequently, |Pm+1−j v, ϕ| ≤ ϕ

1

H −s+m−j+ 2 (IRn )

where

Vm+1−j vL2 (IRn )

 v(ζ) . Vm+1−j v(ζ) := (1 + |ζ|)s−m+j− 2 Pm+1−j 1

The estimate (9.4.43) of p*m+1−j in Lemma 9.4.10 gives us     1 Vm+1−j vL2 (IRn ) ≤ c (1 + |ξ  |)s |* v (ξ  )| (1 + |ζ|)s−m+j− 2 × IRn

IRn−1

 ×(1 + |ξ  |)m−j−s (1 + |ζ  − ξ  |)−r 1 + =: =

|ζn | 1 + |ξ  |

−q 

dξ 

2 dζ

Wm+1−j vL2 (IRn )   (1 + |ξ  |)s |* v (ξ  )|χm+1−j (·, ξ  )dξ  L2 (IRn ) IRn−1

 12

9.4 Pseudodifferential Operators with Symbols of Rational Type

589

where χm+1−j (ζ, ξ  ) is defined by the kernel function {. . . } above. We note that χm+1−j (ζ, ξ  ) contains one term of convolutional type |ζ  − ξ  | in IRn−1 which motivates us to estimate χm+1−j by a convolutional kernel with the help of Peetre’s inequality, namely 

1 + |ξ  | 1 + |ζ  |



≤ (1 + |ξ  − ζ  |)|| for any  ∈ IR .

(9.4.45)

In χm+1−j (ζ, ξ  ) we use (9.4.45) with  = m − j − s and, also, with  = 1 and obtain 1

χm+1−j (ζ, ξ  ) ≤ (1 + |ζ|)s−m+j− 2 (1 + |ζ  |)m−j−s ×  −q |ζn | × (1 + |ξ  − ζ  |)|m−j−s|−r 1 + . (1 + |ζ  |)(1 + |ξ  − ζ  |) ? The last factor in χm+1−j contains a term of the form |ζn | (1 + |ζ  |) which ? motivates us to introduce the new variable ζn = ζn (1 + |ζ  |). Then    2 Wm+1−j vL2 = |(Wm+1−j v ζ  , (1 + |ζ  |)ζn ) |2 (1 + |ζ  |)dζ  dζn IR IRn−1





≤ IR IRn−1

×

 

(1 + |ξ  |)s |* v (ξ  )| ×

IRn−1

2 1 χm+1−j (ζ  , ζn (1 + |ζ  |), ξ  ) dξ  (1 + |ζ  |) 2 dζ  dζn 

and  1 χm+1−j (ζ  , ζn (1 + |ζ  |), ξ  ) (1 + |ζ  |) 2  s−m+j− 12 9 1 ≤ (1 + |ζ  |)m−j−s+ 2 1 + |ζ  |2 + (1 + |ζ  |)2 |ζn |2 × ' ×(1 + |ζ  − ξ  |)|m−j−s|−r Since √1 (1 2

|ζn | 1+ 1 + |ζ  − ξ  |

(−q .

√ + |ζ  |)(1 + |ζn |) ≤ (1 + . . .) ≤ (1 + |ζ  |)(1 + |ζn |) ,

we finally find  1 χm+1−j (ζ  , ζn (1 + |ζ  |), ξ  ) (1 + |ζ  |) 2 −q  1 |ζn | ≤ c(1 + |ζ  − ξ  |)|m−j−s|−r 1 + (1 + |ζn |)s−m+j− 2 .   1 + |ζ − ξ | Hence,

590

9. Pseudodifferential and Boundary Integral Operators

Wm+1−j v2L2 (IRn )

 D  D ≤c D (1 + |ξ  |)s |* v (ξ  )|(1 + | • −ξ  |)|m−j−s|−r × IR



× 1+

|ζn | 1 + | • −ξ  |

−q

IRn−1

D2 D dξ  D 2

L (IRn−1 )

× (1 + |ζn |)2(s−m+j)−1 dζn .(9.4.46)

For every fixed ζn ∈ IR, the inner integral is a convolution integral with respect to ξ  whose L2 –norm can be estimated as:  . . . L2 (IRn−1 )

≤

c(1 + | • |)s |* v (•)| L2 (IRn−1 ) × '   |m−j−s|−r × (1 + |η |) 1+ IRn−1

≤

cvH s (IRn−1 ) (1 + |ζn |)−q



|ζn | 1 + |η  |

(9.4.47)

(−q dη 

(1 + |η  |)|m−j−s|−r+q dη  .

IRn−1

This estimate holds for any choice of r and q ∈ IN provided the integral  (1 + |η  |)|m−j−s|−r+q dη  = crq < ∞ , IRn−1

which is true for |m − j − s| − r + q < −n + 1. Inserting (9.4.47) into (9.4.46) implies  Wm+1−j v2L2 (IRn ) ≤ cc2rq v2H s (IRn−1 ) (1 + |ζn |)2(s−m+j)−1−2q dζn . IR

Choose q > max{s − m + j − 1 , 1} and then r > |m − j − s| + n − 1 + q, which guarantees the existence of the integrals in (9.4.46). In turn, we obtain the desired estimate (9.4.44).  Proof of Theorem 9.4.7: By using the pseudohomogeneous expansion of the Schwartz kernel to AΦr ,0 , we write AΦr ,0 =

L 

Am−jΦr ,0 + AΦr R

(9.4.48)

j=0

 and the Schwartz kernel kR of AΦ R is in the where Am−jΦr ,0 ∈ Lm−j (Ω), r c   × Ω)  with some δ  ∈ (0, 1). Correspondingly, we have class C −m−n+L−δ (Ω     m+1−j v() = (2π)−n+1 ei ·ξ v*(ξ  ) qm+1−j (, ξ  )dξ  Q IRn−1

= Am−jΦr ,0 (v ⊗ δ)() for n = 0

(9.4.49)

9.4 Pseudodifferential Operators with Symbols of Rational Type

591

and  AΦr R (v ⊗ δ)() =

 kR ,  − (η  , 0) v(η  )dη 

IRn−1

 Hence, we may decompose Q  in the form for L ≥ m + 1 and  ∈ Ω.  Qv() =

L 

 m+1−j v() + AΦ R ( ⊗ δ)() , Q r

j=0

 Therefore, with Lemma 9.4.8 we can estimate where AΦr ,R ∈ Lm−L−1 (Ω). c the last term on the right–hand side as ϕAΦr ,R (v ⊗ δ)

1

H s−m− 2 (IRn )

≤ cv ⊗ δ

1

H m−L+ 2 (IRn−1 ×IR)

≤ cvH m−L+1 (K)

provided L > s + 1. Here, c = c(s, L, K, ϕ). We note that we need to consider ϕAΦr ,R instead of AΦr R since the middle estimate is only valid if  varies in a compact subset  For the estimates of ϕQv,  we use Lemma 9.4.9 and obtain of Q.  ϕQv|

1 H s−m− 2

(IRn ∩{n ≤0})

≤

L 

cj vH s−j (IRn−1 ) + cvH s−L+1 (K)

j=0

≤ c vH s (K) if we choose L > max{s + 1, 1}. The proof of estimate (9.4.34) follows in exactly the same manner. This completes the proof of Theorem 9.4.7.   defined by (9.4.5) is a continuous Corollary 9.4.12. The linear operator Q ∞ ∞ ¯n  ¯ n ∩U r ) where mapping from C0 (Ur ) into C (IR− ∩ Ur ) and also into C ∞ (IR + n n ¯ and IR ¯ denote the lower and upper closed half space, respectively. IR − + The proof is an immediate consequence of Theorem 9.4.7 since s can be chosen arbitrarily large in (9.4.33) and (9.4.34). In order to extend Theorem 9.4.7 to sectional traces, let us introduce the following definition.  Definition 9.4.2. For k ∈ IN0 , let V ∈ C k (−ε, ε) , D (Ur ) , n → V (n ) ∈ D (Ur ) be a family of distributions which are k timescontinuously differentiable on n ∈ (−ε, ε). Then for the distribution v ∈ D (Ur ×(−ε, ε) defined as   v, ϕ = V (n ) , ϕ(•, n )dn for ϕ ∈ D Ur × (−ε, ε) , IR

592

9. Pseudodifferential and Boundary Integral Operators

the distributions γj v :=

 d j  V (0) ∈ D (Ur ) for 0 ≤ j ≤ k dn

are called the sectional traces of orders j on the plane n = 0. Theorem 9.4.13. (Chazarain and Piriou [64, Theorem 5.2.4. i)])   Let A ∈ Lm c (Ω × Ω) have a symbol of rational type and v ∈ D (Γ ). Then   Qv(x) := A(v ⊗ δΓ ) (x) for x ∈ Ω and  \Ω  c v(x) := A(v ⊗ δΓ ) (x) for x ∈ Ω Q  and γkc Q  c v, of any order k ∈ IN0 have, respectively, the sectional traces γk Qv ∞  on Γ . Moreover, for ϕ ∈ C0 (Ur ) and for any s ∈ IR and any k ∈ IN0 , we have the estimate D  ∂ k D  D s−m− 1 Dϕ ≤ cvH s (K) (9.4.50) Qv 2 (IRn ∩{n ≤0}) H ∂n for every v ∈ H s (K), where K  Ur is any compact set. The constant c k   c holds depends on K, ϕ, s and k. The corresponding estimate for ∂∂n Q as well, and we have D  ∂ k D  c v D s−m− 1 Dϕ ≤ cvH s (K) . Q 2 (IRn ∩{n ≥0}) H ∂n

(9.4.51)

Proof: As before, we fix one of the local charts (Or , Ur , χr ) in the tubular  of Γ and consider the representation of A in the parametric neighbourhood Ω  domain Ur = Ur × (−ε, ε) in the form (9.4.48). Then one of the terms as in (9.4.49), we decompose, m+1−j v() + Rv()  for n ∈ (−ε, ε] , Am−j,Φr,0 (v ⊗ δ)() = B where m+1−j v() := B











ei ·ξ v*(ξ  )pm+1−j (, ξ  )dξ 

|ξ  |≥1

and m+1−j v() := R

 |ξ  |≤1

ei ·ξ v*(ξ  )pm+1−j (, ξ  )dξ  ,

(9.4.52)

9.5 Surface Potentials on the Boundary Manifold Γ

593

 is a smoothing operator with where pm+1−j (, ξ  ) is given in (9.4.37) and R  ∞  a C Schwartz kernel. Since pm+1−j (, ξ ) is in C∞ Ur × IRn−1 , the right– hand side in (9.4.52) defines a distribution in C ∞ (−ε, ε) , D (Ur ) , and the sectional traces   ∂ k     γk Bm+1−j v = lim ei ·ξ v*(ξ  ) pm+1−j (, ξ  ) dξ  0 0 are to be exchanged  with n ≥ 0 and n < 0.

9.5 Surface Potentials on the Boundary Manifold Γ In the previous section we discussed surface potentials and their limits for n → 0 only for the half–space. There we introduced the extension conditions (9.3.7), the canonical extension and Tricomi conditions (9.3.31), (9.3.32), the special parity conditions (9.3.27) and the transmission conditions according to H¨ormander (9.3.40) as well as the operators with symbols of rational type. Up to now, our discussion was confined to the half–space situation only. To extend these ideas to surface potentials for a general boundary manifold Γ , we need to generalize these concepts to manifolds. Let us recall that  for A ∈ Lm ) and the c (Ω) and any local chart (Or , Ur , χr )of an atlas A(Γ  r with corresponding local transformation Φ : O → U × (−ε, ε) =: U r r r         ( , n ) = Φr x + n n(x ) and (n , 0) = Φr (x ) and  = χr (x ) for x ∈ Γ , and its inverse x = Ψr () (see (9.2.1) and (9.2.2)), the surface potential in the half space is defined by   r v() = AΦr,0 v ⊗ δ(ηn ) () (9.5.1) Q for any v ∈ C0∞ (Ur ). Hence, it is natural to consider the mapping defined by  Γ u(x) := A(u ⊗ δΓ )(x) Q

(9.5.2)

594

9. Pseudodifferential and Boundary Integral Operators

r \ Γ and v = χr∗ u where u ∈ C ∞ (Or ). Then for x = x + n n(x ) with x ∈ O 0  ∗ ∗  Γ u = Φ∗ Q  Q r r χr∗ u = Φr AΦr,0 Φr∗ (u ⊗ δΓ ) = Φr AΦr,0 χr∗ u ⊗ δ(n ) . (9.5.3)  Γ u(x) as x → x ∈ Or ⊂ Γ , we need the To consider the limits of Q following definition of the extension conditions.  Γ satisfies the extension conditions Definition 9.5.1. The operator Q − + for n → 0 (n → 0 ) if and only if for every local chart of an atlas A(Γ ) the associated operators  r = Φr∗ Q  Γ χ∗r Q satisfy the extension conditions (9.3.7) for m+1 ∈ IN0 and ν = 0, . . . , [m+1]; and, for m + 1 ∈ IN0 , the extension conditions (9.3.7) for ν = 1, . . . , m + 1.  Γ is said to satisfy the Tricomi condiFor m + 1 ∈ IN0 , the operator Q  tions iff all the operators Qr satisfy the Tricomi conditions (9.3.8). Now we are using a partition of unity {ϕr }r∈I to an atlas A(Γ ) as in Section 9.1 and the family {ψr }r∈I with ψr ∈ C0∞ (Or ) and ψr (x ) = 1 for x ∈ supp ϕr . Then we are in the position to establish the following theorem for the surface potential.  Theorem 9.5.1. Let A ∈ Lm c (Ω ∪ Ω) with m ∈ IR. Then the limit QΓ u(x ) :=

lim

0>n →0

  Γ u(x) where x = x + n n(x ) ∈ Ω Q

(9.5.4)

always exists if m < −1. For m > −1 it exists in the case m + 1 ∈ IN0 iff  Γ satisfies the extension conditions. For m + 1 ∈ IN0 , the limit exists iff Q  Γ satisfies the extension conditions and the Tricomi conditions. If the limit Q exists then QΓ ∈ Lm+1 (Γ ) is a pseudodifferential operator of order m + 1 on c  Γ u(x), correspondingly. Γ . These results hold also for QΓ,c u(x ) = lim Q 0n →0

exists iff the extension conditions (9.3.7) for m + 1 ∈ IN0 and (9.3.7) and (9.3.8) for m + 1 ∈ IN0 are satisfied due to Theorem 9.3.2. Furthermore, Qr is a pseudodifferential operator Qr ∈ Lm+1 (Ur ). c t ∩ Ω we have r ∩ O For x ∈ O   Γ u(x) = A(u ⊗ δΓ )(x) = Φ∗ Φr∗ AΦ∗ Φr∗ (u ⊗ δΓ ) (x) Q r r  = Φ∗r AΦr (χr∗ u) ⊗ δ(ηn ) (x)   r (χr∗ u) (x) = Φ∗r Q   t (χt∗ u) (x) . = Φ∗t Q

9.5 Surface Potentials on the Boundary Manifold Γ

595

For n → 0, i.e., x → x ∈ Or , the limits exist due to the extension conditions and, hence, are equal. This means (QΓ u)(x ) = (χ∗r Qr χr∗ u)(x ) = (χ∗t Qt χt∗ u)(x ) for arbitrary charts with x ∈ Or ∩ Ot = Ort = ∅ and u ∈ C0∞ (Ort ). Consequently, in terms of the partition of unity, by   1 − ψr (x ) χ∗r Qr χr∗ ϕr u , (9.5.5) ψr (x )χ∗r Qr χr∗ ϕr u + QΓ u := r∈I

r∈I

(Γ ) is well defined. the pseudodifferential operator QΓ ∈ Lm+1 c For QΓ,c , the proof follows in the same manner.



In order to express QΓ in (9.5.5) in terms of finite part integral operators,  Γ in (9.5.2) it follows that Equawe observe that from the definition of Q tion (9.5.3) can be written in terms of the Schwartz kernels of A and AΦr,0 , respectively, i.e.,   kA (x, x − y  )u(y  )dsy QΓ u(x) = y  ∈Γ



= η  ∈Ur



=

  8 kA Ψr () , Ψr () − Tr (η  ) u Tr (η  ) γr (η  )dη    r \ Γ (9.5.6) kAΦr,0 ,  − (η  , 0) u Tr (η  ) dη  for x ∈ O

η  ∈Ur

and any u ∈ C0∞ (Or ), where x = ψr () , y  = Tr = ψr (η  , 0) and γr is given by (3.4.15). Consequently,   8 (9.5.7) kAΦr,0 ,  − (η  , 0) = kA Ψr () , Ψr () − Ψr (η  , 0) γr (η  ) r \ (η  , 0). Since kA ∈ ψhk−m−n , for n = 0 and, by continuity, for all  ∈ U  Lemma 8.2.1 implies kAΦr,0 ∈ ψhk −m−n in Ur and for n = 0, the kernel  function kAΦr,0 (η  , 0) ; ( − η  , 0) belongs to the class ψhk−(m+1)−(n−1) on Ur . Therefore the corresponding finite part integral operators define on Ur pseudodifferential operators in Lm+1 (Ur ). c For these operators we can show that the following lemma holds.  Lemma 9.5.2. Let A ∈ Lm c (Ω ∪ Ω) and suppose that the finite part integral operator  8  kA Tr ( ) ; Tr ( ) − Tr (η  ) v(η  ) γr (η  )dη  (9.5.8) Qf,r v( ) = p.f. Ur

596

9. Pseudodifferential and Boundary Integral Operators

for  ∈ Ur is invariant under change of coordinates. Then the finite part integral on Γ given by  p.f. Γ

:=

kA (x , x − y  )u(y  )dsy 

ψr (x ) p.f.

r∈I



8  kA x ; x − Tr (η  ) (χr∗ ϕr u)(η  ) γr (η  )dη 

Ur

+





1 − ψr (x )



r∈I



kA (x , x − y  )ϕr u(y  )dsy

(9.5.9)

Or

defines a pseudodifferential operator on Γ belonging to the class Lm+1 (Γ ). c Let Λ := χr ◦ Tt : Ut → Ur denote the coordinate transform generated by the local charts (Ort , Ur , χr ) and (Ort , Ut , χt ) and let η  ∈ Ut , η  = Λ(η  ) ∈ Ur . We shall now show the relations χ∗r Qf,r χr∗ = χ∗t Qf,t χt∗ on D(Ort ) . (9.5.10)  With vr ( ) := χr∗ u we have vt ( ) = vr Λ( ) since χ∗r vr = u = χ∗t vt . From (9.2.4) we have for the surface element √ √ ds(η  ) = γr dη  and ds(η  ) = γt dη  . With

 ∂xk ∂xk ∂ = g m  , n = n = n ,   ∂j ∂m ∂j k,m

from (3.4.5) together with  ∂  ji  g ∂ j j

=

 ∂xk ∂xk g m  g ji ∂m ∂j

k,m,j

=

 ∂xk ∂  ∂ i m i m g = g  ∂m ∂xk ∂ m m k,m

we obtain det hence,

 ∂  e ∂j

 det

(g  )−1

 ∂ 2  ∂j

=

det

=

g  . g

 ∂  i

∂m

(g  )−1 ,

Then, for n = 0, with (3.4.24) and n = n = n , we obtain

9.5 Surface Potentials on the Boundary Manifold Γ

 det

597

∂Λ 2 γt = ∂η  γr

and, consequently, since the finite part integral operator Qf,r in (9.5.8) is invariant under change of coordinates by assumption, we find  (Qf,r vr ) Λ( )       √  ∂Λ  kA Tr Λ( ) , Tr Λ( ) − Tr Λ(η  ) γr det × = p.f. ∂η  Ut  ×vr Λ(η  ) dη     √ = p.f. kA Tt ( ) , Tt ( ) − Tt (η  ) vt (η  ) γt dη  Ut

=

Qf,t vt ( ) .

(9.5.11)

This shows that Qf,t and Qf,r satisfy the compatibility condition (9.5.10) and Theorem 9.1.1 implies that the family χ∗r Qf,r χr∗ defines by (9.5.9) a (Γ ).  pseudodifferential operator in Lm+1 c Now we return to the operator QΓ defined by (9.5.5). In terms of the locals charts, as an immediate consequence of Theorem 9.3.2 and Theorem 9.5.1, the representation (9.5.5) of QΓ leads to the following lemma.  Γ satisfy the extension conditions (see Definition 9.5.1). Lemma 9.5.3. Let Q  If m + 1 ∈ IN0 let QΓ also satisfy the Tricomi conditions. Then QΓ admits the representation in the form QΓ u(x )      ∗ = ψr (x )χr p.f. kAΦr,0 ( , 0) ; ( − η  , 0) (χr∗ ϕr u)(η  )dη  r∈I



Ur

− Tr (χr∗ ϕr u) (x )    ∗  + 1 − ψr (x ) χr kAΦr,0 ( , 0) ; ( − η  , 0) χr∗ (ϕr u)(η  )dη  , r∈I 

Ur 

x = Tr ( ) ∈ Γ

(9.5.12)

where Tr is the tangential differential operator in (9.3.20) associated with any chart (Or , Ur , χr ) in the atlas A(Γ ), and where kAΦr,0 is given by (9.5.7). If m+1 ∈ IN0 then (9.5.12) holds with T = 0 and, moreover, is invariantly defined by  QΓ u(x ) = p.f.

kA (x , x − y  )u(y  )dsy .

(9.5.13)

Γ

For the exterior limits with 0 < n → 0, the same results hold for AΓ,c if −Tr in (9.5.12) is replaced by +Tr .

598

9. Pseudodifferential and Boundary Integral Operators

We note that the finite part integral in (9.5.12) for m + 1 ∈ IN0 is not generally invariant under the coordinate transformations generated by changes of the local charts. However, with the appropriate parity conditions for the Schwartz kernel of A, the finite part integrals in (9.5.12) will become invariant.  Definition 9.5.2. The operator A ∈ Lm c (Ω ∪ Ω) with m + 1 ∈ IN0 is said  iff the members of the to satisfy the special parity conditions in Ω ∪ Ω pseudohomogeneous kernel expansion of A satisfy kκ+j (x, y − x) = (−1)m−j kκ+j (x, x − y) for 0 ≤ j ≤ m + 1

(9.5.14)

 , y ∈ IRn . and x ∈ Ω ∪ Ω We remark that due to Theorem 8.2.9 with σ0 = 0, the special parity conditions (9.5.14) are invariant under changes of coordinates. Hence, this  r of the definition implies the special parity conditions (9.3.27) for every Q  form (9.5.1) and  ∈ Ur in any chart of the atlas. Therefore, by using Theorem 9.3.6, we have the following theorem.  Theorem 9.5.4. Let A ∈ Lm c (Ω) with m + 1 ∈ IN0 satisfy the special parity  conditions (9.5.14) and let QΓ satisfy the extension conditions at Γ . Then QΓ as in (9.5.12) can also be defined invariantly by  QΓ u(x ) = p.f. kA (x , x − y  )u(y  )dsy − T u(x ) , (9.5.15) Γ

so is QΓ,c , i.e., QΓ,c u(x )

 =

p.f.

kA (x , x − y  )u(y  )dsy + Tc u(x )

(9.5.16)

Γ

for x ∈ Γ . Proof: The invariance of the finite part integral operator is now a consequence of Theorem 9.3.6 since the special parity conditions guarantee that the finite part integrals in (9.5.12) are invariant under the change of coordinates generated by two compatible charts of the atlas. Hence, Lemma 9.5.3 implies that QΓ and QΓc in (9.5.15) and (9.5.3) are defined invariantly, respectively.  In order to introduce the concept of canonical extension conditions for Γ , let Γτ with τ ∈ Iτ denote the family of parallel surfaces associated with the local chart (Or , Ur , χr ) by Γτ : x = Ψr ( , τ ) = x + τ n(x ) where x = Tr ( ) ∈ Γ Then we may define

and  ∈ Ur .

9.5 Surface Potentials on the Boundary Manifold Γ

 Γ u(x) := A(u ⊗ δΓ )(x) = Q τ τ



599

 kA x, (x + τ n(x ) − y  , 0) u(y  )dsy .

y  ∈Γτ

(9.5.17) r and  = 0 we have For u ∈ C0∞ (Ur ) and x = x = (τ − n )n(x ) ∈ O  Γ u(x) = Φ∗ Q  Q r r (τ )χr∗ u . τ r . Hence, in order to obtain the limits of  r (τ ) is defined by (9.5.1) in U where Q  Γ u(x) in (9.5.17) as n → 0; i.e., x → x + τ u(x ) ∈ Γτ for each τ ∈ Iτ , we Q τ now carry over the definition of the canonical extension conditions from the family of planes in the parametric domain to the family of parallel surfaces  Γτ in Ω. Definition 9.5.3. We say that the canonical extension conditions are satisfied for A at Γ iff for each chart (Or , Ur , χr ) of the atlas A(Γ ) the  Schwartz kernel of the associated operator AΦr,0 ∈ Lm c (Ur ) satisfies the r . canonical extension conditions (9.3.31) where  ∈ U  Clearly, for operators A ∈ Lm c (Ω ∪ Ω) with m + 1 ∈ IN0 satisfying the canonical extension conditions in the vicinity of Γ and, in addition, the canonical Tricomi conditions (9.3.32) in every local chart, Theorem 9.5.4 holds with more simplified tangential differential operators T and Tc , respectively, in view of Theorem 9.3.8. Now we are in the position to formulate the following fundamental theorem for the surface potential on Γ in the case m + 1 ∈ IN0 . For m + 1 ∈ IN0 , the corresponding result is already formulated in Theorem 9.5.1, where the extension conditions are sufficient without further restrictions.  Theorem 9.5.5. Let A ∈ Lm c (Ω ∪ Ω) with m + 1 ∈ IN0 satisfy the canonical extension conditions (cf. Definition 9.5.3) and the special parity conditions  Γ u from the interior Ω as well as from the (9.5.14). Then the limits of Q n exterior IR \ Ω exist. We have   Γ u x + n n(x ) = QΓ u(x ) Q 0>n →0  kA (x , x − y  )u(y  )dsy − T u(x ) for x ∈ Γ = p.f. lim

y  ∈Γ

and lim

0n →0

 S ◦ A(u ⊗ δΓ ) x + n n(x )  (S ◦ kA )(x , x − y  )u(y  )dsy − T u(x ) for x ∈ Γ = p.f. y  ∈Γ

due to Theorem 9.5.5 where  T u = Tc u = 12 (γc − γ) S ◦ A(u ⊗ δΓ ) . However, since S is a tangential operator. we have γS = S|Γ γ and γc S = S|Γ γc . Consequently, with Theorem 9.5.5 applied to A, T u = Tc u = S|Γ 12 (γc − γ)A(u ⊗ δΓ ) = S|Γ ◦ T u = S|Γ ◦ Tc u . 

9.5 Surface Potentials on the Boundary Manifold Γ

601

 Γ has continuous limits under the extension conditions. For surSo far Q  face potentials Qv() on the plane IRn−1 we have shown that for opera tors with symbols of rational type, the potentials Qv() for n = 0 and ∞ ∞ v ∈ C0 (Ur ), can be extended to C –functions up to n → 0± , respectively, and they define continuous mappings on Sobolev spaces. To obtain  Γ when Γ is a general surface, we now confine corresponding results for Q  ourselves to this subclass of operators A ∈ Lm c (Ω ∪ Ω) having symbols of rational type. Then Lemma 9.4.2 implies that all the operators AΦr,0 generated by local charts of an atlas A(Γ ) have symbols of rational type, too. As a consequence, the following theorem can be established and resembles all the previous properties.  Theorem 9.5.7. Let A ∈ Lm c (Ω ∪ Ω) with m ∈ Z have a symbol a(x, ξ) whose j −th asymptotic terms am−j (x, ξ) are of rational type for j = 0, . . . , m with respect to any local chart. Then all the assumptions of Theorem 9.5.5 are fulfilled and Theorem 9.5.5 applies to this class of operators. By the same arguments, Corollary 9.5.6 remains valid also for this class of operators, provided aj−m are of rational type for j = 0, . . . , m + 1. For this special class of operators, the mapping properties of surface potentials on the plane IRn−1 in Sobolev spaces have been given in Theorem 9.4.7 and Corollary 9.4.12 implies that these surface potentials define continuous  ∩ IRn ) and to C ∞ (U  ∩ IRn ), respectively. mappings from C0∞ (Ur ) to C ∞ (U − + For operators other than of rational type, the latter mapping properties are not true, in general. For this reason, we now only consider the mapping conditions for surface potentials on Γ for the class of operators with symbols of rational type.  Theorem 9.5.8. Let A ∈ Lm c (Ω ∪ Ω) with m ∈ Z have a symbol of ratio nal type. Then the operators QΓ given by (9.5.2) with x ∈ Ω or x ∈ Ω c , respectively, define the following linear mappings which are continuous: Γ Q

:

H s (Γ )

→

H s (Γ )

→

C ∞ (Γ )

→

C ∞ (Γ )

→

E  (Γ )

→

E  (Γ )

→

1

H s−m− 2 (Ω) and 1  ∩ Ω c ) for s ∈ IR , H s−m− 2 (Ω

(9.5.19)

C ∞ (Ω) and  \ Ω) ; C ∞ (Ω

(9.5.20)

E  (Ω) and  \ Ω) . E  (Ω

(9.5.21)

and, hence, Γ Q

:

Γ Q

:

Proof: With the partition of unity {ϕr }r∈I to an atlas A(Γ ) and the corresponding family of functions ψr ∈ C0∞ (Or ) with ψr (x) = 1 for x ∈ supp ϕr ,  Γ u(x) for u ∈ C ∞ in the form we can write Q

602

9. Pseudodifferential and Boundary Integral Operators

 Γ u(x) = Q



 r χr∗ ϕr u)(x) ψr (x)(Φ∗r Q

r∈I

+

 



1 − ψr (x) kA (x, x − y  )ϕr (y  )u(y  )dsy .

r∈I y  ∈Γ

 Since in the last sum the kernels 1 − ψr (x) kA (x, x − y  )ϕr (y  ) are C ∞ – r  and y  ∈ Γ , the mapping results follow from those of Q functions of x ∈ Ω in Theorem 9.4.7 and its Corollary 9.4.12.  In closing this section, we remark that the operators QΓ and QΓ,c when Γ , belong to Lm+1 (Γ ). This implies in ever they exist as the limits of Q c particular that QΓ and QΓ,c are continuous mappings on the Sobolev spaces, QΓ , QΓ,c : H s (Γ ) → H s−m−1 (Γ ) for every s ∈ IR . and, consequently, are also continuous mappings on C ∞ (Γ ).

9.6 Volume Potentials With the mapping properties available for the pseudodifferential operators we first consider the volume potentials Af and their mapping properties.  s (Ω ∪ Ω)  and A ∈ Lm (Ω ∪ Ω),  we denote by Af (x) for x ∈ Ω ∪ Ω  For f ∈ H c s−m  the volume potential. Then clearly Af ∈ Hloc (Ω ∪ Ω). Moreover, then (4.2.32) implies that γ0 Af 

1

H s−m− 2 (Γ )

+ γ0c Af 

1

H s−m− 2 (Γ )

≤ cf H s (Ω∪Ω) for s > m + 

1 , 2

(9.6.1)

where c is independent of f . For the remaining indices s and in what follows, we now confine ourselves to the subclass of operators with symbols of rational type. Theorem 9.6.1. (Boutet de Monvel [39])  Let A ∈ Lm c (Ω ∪ Ω). Then the following mappings are continuous:  s (Ω) −→ H s−m (Ω) for every s ∈ IR , A : H 1 A : H s (Ω) −→ H s−m (Ω) for s > − . 2

(9.6.2) (9.6.3) ◦

 \ Ω) if Ω is replaced by K where K is The corresponding results hold in (Ω  \ Ω. any compact set with K ⊂ Ω

9.6 Volume Potentials

603

Note that for s → ∞ the estimate (9.6.3) implies that A maps C ∞ (Ω) continuously into C ∞ (Ω). Proof: ◦

◦

 be a compact set with Ω  K where K is the open interior i) Let K  Ω ∪ Ω  and  s (Ω). Then Af ∈ H s−m (Ω ∪ Ω) of K, and let f ∈ H loc Af H s−m (K) ≤ cf H s (Ω) with some constant c depending on K, A and s. For any w ∈ C0∞ (Ω) we then find |(Af, w)L2 (Ω) | = |(Af, w)L2 (K) | ≤ c f H s (Ω) wH m−s (Ω) . This yields by duality (Af )H s−m (Ω)   = sup |(Af, w)L2 (Ω) | for w ∈ C0∞ (Ω) with wH m−s (Ω) ≤ 1 and

Af H s−m (Ω) ≤ c f H s (Ω)

 s (Ω). which implies (9.6.2) since C0∞ (Ω) is dense in H ∞ ii) Now let f ∈ C0 (K) and define  f (x) for x ∈ Ω, 0 f (x) := 0 for x ∈  Ω, where K and w are as above. Then (Af 0 , w)L2 (K) = (f 0 , A∗ w)L2 (K) = (f 0 , A∗ w)L2 (Ω) and the restriction  ∗

rΩ A w(x) :=

A∗ w(x) 0

for x ∈ Ω, for x ∈  Ω

 −s (Ω), provided −s < 1 . Hence, satisfies rΩ A∗ w ∈ H 2 |(Af 0 , w)L2 (Ω) | = |(Af 0 , w)L2 (K) | = |(f 0 , rΩ A∗ w)L2 (K) | ≤ cf 0 H s (Ω) A∗ wH −s (Ω) ≤ c f 0 H σ (Ω) wH m−s (Ω) .

604

9. Pseudodifferential and Boundary Integral Operators

Again, taking the supremum for wH m−s (Ω) ≤ 1 we find by duality the estimate 1 Af H m−s (Ω) ≤ cf H s (Ω) , if s > − , 2 i.e., (9.6.3) since C ∞ (Ω) is dense in H s (Ω).



For the subclass of operators having symbols of rational type, the traces of the volume potentials on the surface Γ exist for some scale of Sobolev spaces as in the standard trace theorem. It should be noted that the operators γ0 Af belong to the class of Boutet de Monvel’s trace operators. He also developed the complete calculus for treating compositions of trace and Poisson operators (Boutet de Monvel [39, 40, 41], Grubb [159], Schulze [379] and Chazarain and Piriou [64, p.287]).  Theorem 9.6.2. Let A ∈ Lm c (Ω ∪ Ω) have a symbol of rational type. Then 1 s  (Ω) and s > m + we have the estimate for f ∈ H 2

γ0 Af 

1

H s−m− 2 (Γ )

≤ cf H s (Ω) .

(9.6.4)

≤ cf H s (Ω)

(9.6.5)

If f ∈ H s (Ω) then γ0 Af 

1

H s−m− 2 (Γ )

provided s > max{− 12 , m + 12 }. The constant c is independent of f . ◦

 \ Ω) if Ω is replaced by K, where K The corresponding results hold in (Ω ◦  \ Ω and Ω ⊂ K. is any compact set with K ⊂ Ω  s (Ω) then Af ∈ H s−m (Ω) due to (9.6.2), and the Trace Proof: If f ∈ H Theorem 4.2.1 implies (9.6.4) for s − m > 12 . For f ∈ H s (Ω) we obtain Af ∈ H s−m (Ω) from (9.6.3) provided aditionally s > − 12 .   s (Ω  \ Ω), where f |Ω = 0, then one has more generally If, however, f ∈ H Theorem 9.6.3. (Chazarain and Piriou [64, Theorem 5.2.2])  s  Let A ∈ Lm c (Ω ∪ Ω) have a symbol of rational type. Then, for f ∈ H (Ω \ Ω) and σ ∈ IR, the sectional trace γ0 Af exists and γ0 Af 

1

H s−m− 2 (Γ )

≤ cf H s (Ω\Ω) . 

(9.6.6)

≤ cf H s (Ω) ,

(9.6.7)

 s (Ω), we have For f ∈ H γ0c Af 

1

H s−m− 2 (Γ )

where the constants c are independent of f .

9.7 Strong Ellipticity and Fredholm Properties

605

 \ Ω) with Proof: For the proof of (9.6.6), let v ∈ C ∞ (Γ ) and f ∈ C0∞ (Ω K := supp f . Then   ∗ vγ0 Af ds = (v ⊗ δΓ , Af )L2 (Ω∪Ω)  = A (v ⊗ δΓ ), f L2 (Ω∪Ω)  , Γ ∗  where A∗ ∈ Lm c (Ω ∪ Ω) has a symbol of rational type as well, and A (v ⊗ δΓ ) is a Poisson operator or surface potential operating on v. Since f = 0 in Ω, we obtain   f) 2  | | A∗ (v ⊗ δΓ ), f L2 (Ω\Ω) | = |(Qv,  L (Ω\Ω)

 c vH −s (K) , Q ≤ cf H s (Ω\Ω)   c is the operator defined by (9.4.5) with respect to A∗ and for 0 < where Q n → 0. From Theorem 9.4.7 with s replaced by m + 12 − s, we obtain     vγ0 Af ds ≤ cf   s  v m+ 1 −s H (Ω\Ω) 2 H

(Γ )

Γ

and γ0 Af 

1

H s−m− 2 (Γ )

=

 

sup v

H

m+ 1 −s 2 (Γ )

≤1



 vγ0 Af ds ≤ cf H s (Ω\Ω) 

Γ

as proposed.  \ Ω is replaced by Ω.  The proof of (9.6.7) follows in the same manner if Ω

9.7 Strong Ellipticity and Fredholm Properties For completeness, in this section we now give an overview and collect some basic results for boundary integral equations with respect to their mapping and solvability properties, the concept of strong ellipticity, variational formulation and Fredholmness. In Section 7.2.5, the concept of strong ellipticity for systems of pseudodifferential operators in IRn has been introduced. Theorem 7.2.7 is the consequence of strong ellipticity in the form of G˚ arding’s inequalities associated with corresponding bilinear forms on compact subdomains. However, inequality (7.2.63) is restricted to Sobolev spaces on compact subregions. On the other hand, if G˚ arding’s inequality is satisfied for operators defined on a compact boundary manifold Γ , it holds on the whole Sobolev spaces on Γ . In this case, the boundary integral equations can be reformulated in the form of variational equations on Γ in terms of duality pairings on boundary Sobolev spaces associated with the scalar products in G˚ arding’s inequality.

606

9. Pseudodifferential and Boundary Integral Operators

With the formulation in variational form together with G˚ arding’s inequality Theorem 5.3.10 may be applied and provides the Fredholm alternative for the solvability of these variational formulations of the corresponding integral equations. In Chapter 5 we already have shown that G˚ arding’s inequality is satisfied for boundary integral equations which are intimately associated with strongly elliptic boundary value problems for partial differential equations satisfying G˚ arding’s inequality in the domain. There G˚ arding’s inequalities for these boundary integral equations are consequences of the G˚ arding inequalities for the corresponding partial differential equations. We emphasize that the G˚ arding inequalities for the corresponding boundary integral equations are formulated in terms of the corresponding energy spaces. For more general pseudodifferential operators on Γ , we now require strong ellipticity on Γ as in Section 9.1.1 which will lead to G˚ arding inequalities for a whole family of variational formulations in various Sobolev spaces including the corresponding energy spaces. For the benefit of the reader let us recall the definition of strong ellipticity for a system ((Ajk ))p×p of pseudodifferential s +t operators on Γ with Ajk ∈ Lcj k (Γ ). We say that A is strongly elliptic,if there exists a constant β0 > 0 and a C ∞ –matrix valued function Θ(x) = Θj (x) p×p such that Re ζ  Θ(x)a0 (x, ξ  )ζ ≥ β0 |ζ|2

(9.7.1)

−1   p 0  for all x ∈ Γ and ξ ∈ IR with |ξ | = 1 and for all ζ ∈ C . Here, a (x, ξ ) =  0  ajk (x, ξ ) is the principal part symbol matrix of A. For strongly elliptic operators, Theorem 9.1.4 is valid. The corresponding variational formulation reads: &p &p For given f ∈ =1 H −s (Γ ) find u ∈ =1 H t (Γ ) such that p a(v, u) := v, ΛσΓ ΘΛ−σ Γ Au

=1

H (t −s )/2 (Γ ) p = v, ΛσΓ ΘΛ−σ Γ f

=1

H (t −s )/2 (Γ )

(9.7.2)

&p s s for all v ∈ =1 H t (Γ ) where ΛσΓ = ((ΛΓj δj ))p×p and ΛΓj = (−ΔΓ + 1)sj /2 with the Laplace–Beltrami operator ΔΓ . As we notice, in Equation &p(9.7.2) the bilinear form is formulated with respect to the Sobolev space =1 H (t −s )/2 (Γ )–inner product. In practice, however, one prefers the L2 –scalar product instead. For this purpose, we employ ΛτΓ−σ with τ = (t1 , . . . , tp ) as a preconditioner. Then the bilinear form becomes −σ τ v, ΛτΓ ΘΛ−σ Γ AuL2 (Γ ) = v, ΛΓ ΘΛΓ f L2 (Γ ) .

(9.7.3)

Alternatively, we may rewrite (9.7.3) in the form a(v, u) = Bv, AuL2 (Γ ) = Bv, f L2 (Γ ) with Bv := Λ−σ Θ∗ Λτ v . (9.7.4)

9.7 Strong Ellipticity and Fredholm Properties

607

Special examples of this formulation have been introduced in Section 5.6; in particular the equations (5.6.27) and (5.6.28) for the variational formulations of boundary integral equations of the second kind where Θ = ((δjk ))p×p . In (5.6.27), we may identify ΛΓ with V −1 and in (5.6.28) ΛΓ with D since their principal symbols are the same. The bilinear form in (9.7.3) or (9.7.4) satisfies a G˚ arding inequality in the form Re {a(w, w) + w, ΛτΓ−σ CwL2 (Γ ) } ≥ β0 w2p with some constant β0 > 0 for all w ∈

=1

&p =1

H t (Γ )

(9.7.5)

H t (Γ ) where

ΛτΓ−σ C = ((ΛtΓ −s Cj ))p×p contains the compact operators ΛtΓ −s Cj : H tj (Γ ) → H −t (Γ ) due to Theorem 9.1.4. Here, from G˚ arding’s inequality (9.7.5) we may consider &p t H (Γ ) as the energy space for the boundary integral operator A and =1 the variational equations (9.7.3) or (9.7.4). However, in order to analyze these equations in other Sobolev spaces as &p t +ε well, we need a variational formulation for the mapping A : H (Γ ) → =1 &p −sk +ε H (Γ ) for any ε ∈ IR. To this end, we introduce the bilink=1 ear & variational equations of the same boundary equation Au = f for p u ∈ =1 H t +ε (Γ ), i.e. (9.7.6) aε (v, u) := v, Λτ +ε ΘΛ−σ+ε AuL2 (Γ ) = v, fε L2 (Γ ) &p where fε = Λτ +ε ΘΛ−σ+ε f with f ∈ =1 H −s +ε (Γ ); or equivalently, aε (v, u) = Bε v, AuL2 (Γ ) = Bε v, f L2 (Γ ) where

(9.7.7)

Bε v = Λ−σ+ε Θ∗ Λτ +ε v .

For the solvability of (9.7.6) or (9.7.7), we now consider corresponding G˚ arding inequalities for all the bilinear forms aε (u, v), where ε ∈ IR. Lemma 9.7.1. If A = ((Ajk ))p×p is a strongly elliptic system of pseudodifs +t ferential operators Ajk ∈ Lcj k (Γ ) on Γ , then for every ε ∈ IR there exist arding’s inequality holds in the form constants γε > 0 , γ1ε ≥ 0 such that G˚ Re aε (w, w) + β1ε w2p for all w ∈

&p =1

=1

H t −1+ε (Γ )

≥ βε w2p

=1

H t +ε (Γ )

(9.7.8)

H t +ε (Γ ).

Proof: Let Aεjk := Λε Ajk Λ−ε , then Aεjk ∈ Lcj k (Γ ) and the principal symbol matrices Aεjk and Ajk are the same. Hence, Aε = ((Aεjk ))p×p is also strongly elliptic for any ε ∈ IR and Theorem 9.1.4 may be applied to Aε providing G˚ arding’s inequality (9.1.20) with corresponding constants γ0 and γ1 s +t

608

9. Pseudodifferential and Boundary Integral Operators

depending on ε. By setting v = Λε w with w ∈ the fact that Λε is an isomorphism we find

&p =1

H t +ε (Γ ) and employing

Re aε (w, w) = Re w, Λτ +ε ΘΛ−σ+ε AwL2 (Γ ) = Re v, Λτ ΘΛ−σ Aε vL2 (Γ ) ≥ β0 v2p

=1

≥

βε w2p

H t (Γ )

=1

− β1 v2p

H t +ε (Γ )

=1

−

H t −1 (Γ )

β1ε w2p

=1

H t +ε−1 (Γ )

, 

which corresponds to (9.7.8). Note that the compact operators Cε are now defined by the relations

= v, Λτ −σ+2ε Cε wL2 (Γ ) = v, wp=1 H t −1+ε (Γ ) . (9.7.9) As a consequence of G˚ arding’s inequality (9.7.8), for fixed ε ∈ IR the boundary integral operators Aε are Fredholm operators of index zero. More precisely, the following Fredholm theorem holds. &p Theorem 9.7.2. Let H = =1 H t +ε (Γ ) with fixed ε ∈ IR. Then for the variational equation (9.7.7), i.e. v, Cε wp

=1

H (t −s )/2 (Γ )

aε (v, u) = v, fε L2 (Γ ) for all v ∈ H there holds Fredholm’s alternative: i) either (9.7.10) &p has exactly one solution for any fε ∈ any f ∈ =1 H −s +ε or ii) the homogeneous problem

&p

aε (v, u0 ) = 0 for all v ∈ H

=1

(9.7.10)

H −t −ε (Γ ), i.e.,

(9.7.11)

has a finite–dimensional eigenspace of dimension k. In this case, the inhomogeneous variational equation (9.7.10) has solutions if and only if the orthogonality conditions fε (v0(j) ) = v0(j) , fε L2 (Γ ) = 0

(9.7.12)

are satisfied where v0(1) , . . . , v0(k) are a basis of the eigenspace to the adjoint homogeneous variational equation aε (v0 , w) = 0 for all w ∈ H . The dimension of this eigenspace is the same as that of (9.7.11). Clearly, this Theorem is identical to Theorem 5.3.10 which was established already in Section 5.3.3 since aε satisfies the G˚ arding inequality (9.7.8). Now, for the family of variational equations (9.7.7) corresponding to the parameter ε ∈ IR we have the following version of the shift–theorem.

9.7 Strong Ellipticity and Fredholm Properties

609

Theorem 9.7.3. i) Uniqueness: If for any ε0 ∈ IR, the corresponding variational equation (9.7.10) with aε0 is uniquely solvable then each of these equations with ε ∈ IR is also uniquely solvable. p ii) Solvability: If u0 ∈ H=1 H t +ε0 (Γ ) is an eigensolution of &pthe homoget +ε neous equation (9.7.11) with aε0 for some ε0 then u0 ∈ ∩ (Γ ) =1 H ε∈IR

= C ∞ (Γ ). Hence, the eigenspaces of all the equations (9.7.11) with ε ∈ IR are identical. &p iii) Regularity: If u ∈ =1 H t +ε0 (Γ ) is a solution of the variational equation p & H t +ε (Γ ) aε0 (v, u) = fε0 (v) for all v ∈ =1

&p

&p −s +ε (Γ ) and ε ≥ ε0 then u ∈ =1 H t +ε (Γ ). =1 H @ &p t +ε0 (Γ ) = C ∞ (Γ ) Proof: We begin by showing ii) first, i.e. u0 ∈ =1 H with f ∈

ε∈IR

for every eigensolution u0 ∈ H of (9.7.11) &p with any fixed ε = ε0 . Then, by the definition of aε0 we have for u0 ∈ =1 H t +ε0 (Γ ): 0 = aε0 (v, u0 ) = v, Λτ +ε0 ΘΛ−σ+ε0 Au0 L2 (Γ ) for every v ∈ C ∞ (Γ ) . This implies Au0 = 0 in

p &

H −s +ε0 (Γ )

=1

since Λτ +ε0 , Θ and Λ−σ+ε0 are isomorphisms. Then, by duality, we also have v, Λτ +ε0 +1 ΘΛ−σ+ε0 +1 Au0 L2 (Γ ) = 0 for v ∈ C ∞ . Hence,

aε0 +1 (v, u0 ) = 0 for all v ∈ C ∞ ,

which implies with G˚ arding’s inequality 1 β1,ε0 +1 u0 2p H t +ε0 . =1 βε0 +1 &p @ By induction we find u0 ∈ ε≥ε0 =1 H t +ε (Γ ) and with the embedding &p @ H α (Γ ) ⊂ H β (Γ ) for α ≥ β we obtain u0 ∈ ε≥IR =1 H t +ε (Γ ) = C ∞ (Γ ). Consequently, if u0 is an eigenfunction of aε0 , it is in C ∞ (Γ ) and satisfies Au0 = 0 which implies aε (v, u0 ) = 0 for every ε ∈ IR and v ∈ C ∞ (Γ ). This establishes the assertion ii). For the uniqueness in all the spaces &p let us assume that equation (9.7.10) with aε0 has a unique solution u ∈ =1 H t +ε0 (Γ ) and that there is nonuniqueness for the equation with aε1 for some ε1 = ε0 . Then the dimension of the eigenspace to aε1 satisfies k ≥ 1; the eigensolutions are in C ∞ (Γ ) and are eigensolutions to aε for every ε ∈ IR, in particular for ε = ε0 , which contradicts our assumption of uniqueness. This establishes that uniqueness for one fixed ε0 implies uniqueness for all ε ∈ IR, i.e. assertion i). u0 2p

=1

H t +ε0 +1

≤

610

9. Pseudodifferential and Boundary Integral Operators

To show assertion iii), let uε0 ∈

with fε0

&p =1

H t +ε0 (Γ ) be a solution of

aε0 (v, uε0 ) = v, fε0 L2 (Γ ) for all v ∈ C ∞ (Γ ) (9.7.13) &p = Λτ +ε0 ΘΛ−σ+ε0 f and f ∈ =1 H −s +ε (Γ ) where ε > ε0 . Hence, Auε0 = f ∈

p &

H −s +ε (Γ )

(9.7.14)

=1

due to the definition of the variational equation (9.7.6) and since Λσ+ε0 ΘΛ−σ+ε0 is an isomorphism. From equation (9.7.14) and by duality, we then find that uε0 is also a solution of aε0 +1 (v, uε0 ) = v, Λτ +ε0 +1 ΘΛ−σ+ε0 +1 Auε0 L2 (Γ ) = Λ−σ+ε0 +1 Θ∗ Λτ +ε0 +1 v, f L2 (Γ ) = v, fε0 +1 L2 (Γ )

(9.7.15)

for all&v ∈ C ∞ (Γ ) provided ε − ε0 ≥ 1. If temporarily we assume that p arding’s inequality (9.7.8) for aε0 +1 , uε0 ∈ =1 H t +ε0 +1 (Γ ) then, by using G˚ we find uε0 2p

=1

≤

1

H t +ε0 +1 (Γ )



β1ε0 +1 uε0 2p

t +ε0 (Γ ) =1 H

βε0 +1 1  β1ε0 +1 uε0 2p H t +ε0 (Γ ) ≤ =1 βε0 +1

+ Re Λτ +ε0 +1 uε0 , ΘΛ−σ+ε0 +1 L2 (Γ )



 + cε0 f p=1 H −s +ε0 +1 (Γ ) uε0 p=1 H t +ε0 +1 (Γ ) .

This inequality implies the a priori estimate   uε0 p=1 H t +ε0 +1 (Γ ) ≤ c(ε0 ) f p=1 H −s +ε (Γ ) + uε0 p=1 H t +ε0 (Γ ) (9.7.16) with ε ≥ ε0 +1. In order to relax the temporary regularity, assumption on uε0 we now apply a standard density argument. More let fj ∈ C ∞ (Γ ) &p precisely, −σ +ε (Γ ) from (9.7.14). be a sequence of functions approximating f in =1 H also satThen we find uε0 j ∈ C ∞ (Γ ) as corresponding solutions of (9.7.13), &p isfying (9.7.14) with fj instead of f , and uε0j → uε0 in =1 H t +ε0 (Γ ). Because of uε0 j ∈ C ∞ (Γ ), these functions automatically satisfy (9.7.16) with uε0 j instead of uε0 and fj instead of f . Then, for j → ∞, the right–hand side of (9.7.16) converges to the corresponding expression with f and uε0 . Hence, from left–hand side we see that uε0 j is a Cauchy sequence in the space &p thet +ε 0 +1 H ) which converges to the limit uε0 in this space, hence we =1 &(Γ p obtain uε0 ∈ =1 H t +ε0 +1 (Γ ). If ε − ε0 ∈ IN then by induction and after finally many steps we obtain the proposed regularity. If ε − ε0 ∈ IN, first prove the estimate for the integer case and then use interpolation to obtain the same conclusion. 

9.8 Strong Ellipticity of Boundary Problems and Integral Equations

611

9.8 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations In Section 5.6.6 we considered a class of direct boundary integral equations whose variational sesquilinear forms satisfy G˚ arding’s inequalities since the corresponding boundary value problems are strongly elliptic. Grubb presented in [158] a general approach for general, normal elliptic boundary problems satisfying coerciveness or G˚ arding’s inequalities, and constructed associated pseudodifferential operators on the boundary which form there a strongly elliptic system as well. Here we present a short review of the work by Costabel et al in [90], where a more special case of elliptic even order boundary value problems were considered. 9.8.1 The Boundary Value and Transmission Problems Let us consider the boundary value problem (3.9.1) and (3.9.2), i.e., P u = 0 in Ω , Rγu = ϕ on Γ

(9.8.1) (9.8.2)

where P is an elliptic operator (3.9.1) of order 2m with m ∈ IN, and R is a m × 2m matrix R = ((Rjk )) j=1,...,m, and every Rjk is a p × p matrix of k=1,...,2m

tangential differential operators with C ∞ coefficients and of equal orders ord Rjk = μj − k + 1 , 0 ≤ μj ≤ 2m − 1 , j = 1, . . . , m .

(9.8.3)

By Rj we denote the j–th row of (3.9.2). Thus, Rj γu :=

2m  k=1

Rjk γk−1 u =

2m  k=1

Rjk

 ∂ k−1  u Γ , j = 1, . . . , m ∂nk−1

(9.8.4)

 is the restriction to Γ of a differential operator of order μj−1 defined in Ω. For the complementing boundary operators S = ((Sjk )) j=1,...,m, we assume k=1,...,2m

the orders ord Sjk = 2m − μj − k

R

(9.8.5)

and require the fundamental assumption (3.9.4) for M = S , i.e., N = M−1 is a matrix of tangential differential operators. For the boundary value problem (9.8.1) and (9.8.2), we further assume that the Lopatinski–Shapiro conditions (3.9.6) and (3.9.7) are satisfied. Related with the boundary value  ⊂ IRn is a bounded problem we will consider a transmission problem where Ω e c   domain that contains Ω ⊂ Ω and where Ω := Ω ∩ Ω denotes a bounded ∂ k u on exterior. Γe := ∂Ω e \Γ is the exterior boundary. The traces γk u = ∂n Γ are defined by sectional traces for any u ∈ H t (Ω) satisfying the differential equation (9.8.1) in Ω.

612

9. Pseudodifferential and Boundary Integral Operators

Lemma 9.8.1. Let u ∈ H t (Ω) satisfy the differential equation (9.8.1) in Ω, where t ≥ 0. Then u has sectional traces γk u on Γ of any order k ∈ IN0 and γk u

1

H t−k− 2 (Γ )

≤ cuH t (Ω) .

(9.8.6)

Proof: If u ∈ H t (Ω) then with Theorem 4.1.5 we can extend u to u  ∈ t t t     H (Ω ∪ Ω) and the mapping u → u  is continuous from H (Ω) to H (Ω ∪ Ω). Hence,  \ Ω)  t−2m (Ω f := P u ∈H since P u  = 0 in Ω. To the elliptic operator P there exist a properly supported  due to Theorem 7.2.2 which has a symbol of (Ω ∪ Ω) parametrix Q ∈ L−2m c rational type. Relation (7.2.6) implies u  = Qf + R ◦ P u  −∞  with a smoothing operator R ∈ L (Ω ∪ Ω). Then  ∂ k  , ◦ Q ∈ L−2m+k (Ω ∪ Ω) c ∂n and (9.6.6) in Theorem 9.6.3 implies with m replaced by −2m + k and σ = t − 2m the desired estimate:  ∂ k γk u t−k− 21 ≤ γ0 ◦ Qf  t−k− 12 H (Γ ) H (Γ ) ∂n  ∂ k + γ0 R ◦ Pu  t−k− 12 H (Γ ) ∂n ≤ c1 f H t−2m (Ω\Ω) + c2  uH t (Ω∪Ω)   ≤ c3  uH t (Ω∪Ω)  ≤ c4 uH t (Ω) .  Clearly, Lemma 9.8.1 also holds for any u ∈ H t (Ω e ) satisfying P u = 0 in Ω e for the sectional traces γkc u. Then Γ defines a transmission boundary between Ω and Ω e . (See also Section 5.6.3.)  functions to Ω e , i.e., Let Ce∞ denote the restrictions of C0∞ (Ω)  . Ce∞ = {u = ϕ|Ω e where ϕ ∈ C0∞ (Ω)}

(9.8.7)

Now we impose the following Assumptions A.1–A.3: A1: There exist two sesquilinear forms aΩ (u, v) and aΩ e (ue , ve ) on C ∞ (Ω) and Ce∞ , respectively, such that Re {aΩ (u, u) + aΩ e (ue , ue )} m    = Re (Rj γu) (Sj γu) − (Rj γue ) (Sj γue )ds j=1 Γ

+

m   j=1 Γ

e

(Rj γu) (Sj γue ) ds

(9.8.8)

9.8 Strong Ellipticity of Boundary Problems and Integral Equations

613

is fulfilled for every u ∈ C ∞ (Ω) , ue ∈ Ce∞ satisfying P u = 0 in Ω and P ue = 0 in Ω e . Assumption A1 corresponds to the validity of the first Green’s formula on Ω and on Ω e . For the sesquilinear forms aΩ and aΩ e we require continuity and G˚ arding’s inequality as follows. A2: There exits a positive constant c such that |aΩ (u, v)| + |aΩ e (ue , ve )|   ≤ c (uH m (Ω) + ue H m (Ω e ) )(vH m (Ω) + (ve H m (Ω e ) )

(9.8.9)

for all u, v ∈ C ∞ (Ω) and ue , ve ∈ Ce∞ . A3: G˚ arding’s inequality: There exist positive constants β0 and ε, and a constant c such that   Re aΩ (u, u) + aΩ e (ue , ue ) (9.8.10) ≥ β0 (u2H m (Ω) + ue 2H m (Ω e ) ) − c(u2H m−ε (Ω) + ue 2H m−ε (Ω e ) ) for all (u.ue ) ∈ C ∞ (Ω) × Ce∞ satisfying the transmission condition Rγu = Rγue on Γ .

(9.8.11)

Note that under these assumptions the sesquilinear forms   (P u) vdx and (P ue ) v e dx

(9.8.12)

Ωe

Ω

satisfy G˚ arding’s inequalities on the subspaces of H m (Ω) and of H m (Ω e ) with elements fulfilling the boundary conditions Rγu = 0 on Γ

. and Rγue = 0 on Γ ; Rγue = 0 on Γe := ∂Ω e ∩ ∂ Ω (9.8.13)

9.8.2 The Associated Boundary Integral Equations of the First Kind As exemplified in Section 3.9, the solution to the boundary value problem (9.8.1) and (9.8.2) can in Ω be represented in terms of Poisson operators or multilayer potentials via (3.9.18) and (3.9.18), i.e., u(x) = −

2m−1  2m−−1  =0

t=0

m m      Kt Pt++1 N+1,j ϕj + N+1,m+j λj ⊗ δΓ j=1

j=1

(9.8.14) for x ∈ Ω, where λ = (λ1 , . . . , λm ) = Sγu denotes the yet unknown Cauchy data of u on Γ . Invoking Theorem 9.4.3 for the traces of Poisson operators we obtain on Γ , by taking traces of (9.8.14), the relations

614

9. Pseudodifferential and Boundary Integral Operators

γu = −

m 2m−1   j=1

−

2m−−1 

=0

=0

2m−1  

t=0

k=0

m 2m−1   2m−−1  j=1

γ0

γ0

2m−1  

t=0

k=0

∂ k−1 Kt ((Pt++1 N+1,m+j λj ) ⊗ δΓ ) ∂n ∂ k−1 Kt ((Pt++1 N+1,j ϕj ) ⊗ δΓ ) . ∂n

(9.8.15) Applying the tangential differential operator R to these relations one obtains boundary integral equations of the first kind for λ on Γ , namely m 2m−1   j=1

= ϕq +

=0

−Rq γKt Pt++1 N+1,m+j λj

t=0

m 2m−1   j=1

in short

2m−−1 

2m−−1 

Rq γKt Pt++1 N+1,j ϕj ,

t=0

=0 m 

(9.8.16)

Aqj λj = ψq , q = 1, . . . , m ,

(9.8.17)

j=1

where the operators Aqj and right–hand sides ψq are defined by (9.8.16). In view of Theorem 9.4.3, the operators Aqj are classical pseudodifferential μ +μ +1−2m (Γ ) which also can be represented via (8.1.84) operators Aqj ∈ Lcq j and (8.1.85) in terms of boundary integral operators with Hadamard’s finite part integral operators composed with tangential differential operators. As shown by Costabel et al in [90], we have the following theorem. Theorem 9.8.2. ([90, Theorem 3.9]) Under the assumptions A.1–A.3 there exist positive constants β0 and δ, and a constant c0 such that Re

m  

λq Aqj λj ds ≥ β0

j,q=1 Γ

for all λ = (λ1 , . . . , λm ) ∈

m 

λj 2

j=1 m &

1 H μj −m+ 2

(Γ )

− c0

m  j=1

λj 2

1

H μj −m+ 2 −δ (Γ )

(9.8.18) H

−m+μj −m+ 12

(Γ ).

j01

For showing G˚ arding’s inequality (9.8.18) let us consider the following transmission problem (similar to Sections 3.9.2 and 5.5.3). 9.8.3 The Transmission Problem and G˚ arding’s inequality For any given λ ∈ C ∞ (Γ ) let us define the boundary potentials

9.8 Strong Ellipticity of Boundary Problems and Integral Equations

u(x) ue (x)

 := −

m 2m−1   j=1

2m−−1 

  Kt (Pt++1 N+1,m+j λj ) ⊗ δΓ

t=0

=1



Ω Ωe

for x ∈

615

.

(9.8.19)

Then these functions are defined by Poisson operators based on the pseu     ∂ dodifferential operators Kt with the Schwartz kernels E(x, y) in ∂ny  having symbols of rational type, see (3.8.1) and (3.8.2). Then u(x) and Ω ×Ω ue (x) can be extended to C ∞ (Ω) and C ∞ (Ω e ), respectively, due to Theorem 9.5.8. Moreover, u satisfies (9.8.1) in Ω and P ue = 0 in Ω e . With the jumps of their traces across Γ , [γu] = γc ue − γu on Γ we find for u and ue given by (9.8.19), [γu] = −

m 2m−1   j=1

2m−−1 

[γKt ]Pt++1 N+1,m+j λj ;

(9.8.20)

t=0

=1

and with (3.9.35) and (3.9.36) we obtain R[γu] = 0 on Γ . On the other hand, (9.8.15) gives Aλ = −Rγu = −Rγc ue on Γ , and

m  

 λq Aqj λj ds = −

q,j=1 Γ

(Rγu) λds .

Γ

Now, we invoke Assumption A1 in the form of (9.8.8) and obtain Re

m  

λq Aqj λj ds

(9.8.21)

q,j=1 Γ

m     (Rj γue ) (Sj γue )ds = Re aΩ (u, u) + aΩ e (ue , ue ) − Re j=1 Γ

e

where on Γe the representation (9.8.19) is to be inserted. Since the integration in (9.8.19) is performed over Γ whereas the traces in (9.8.21) are taken on Γe , and the Schwartz kernels of the operators in (9.8.19) are in C ∞ (Γ × Γe ), the mappings

616

9. Pseudodifferential and Boundary Integral Operators

λ∈

m ,

1

H −m+μj + 2 −ε (Γ ) → (Sj γue |Γe , Rj γue |Γe ) ∈ L2 (Γe ) × L2 (Γe )

j=1

are continuous. So, Re

m  

  λq Aqj λj ds ≥ Re aΩ (u, u)+aΩ e (ue , ue ) −cλ2 m

q,j=1 Γ

1

H −m+μj + 2 −ε (Γ )

.

j=1

Now, (9.8.10) of Assumption A3 implies Re

m  

  λq Aqj λj ds ≥ β0 u2H m (Ω) + ue 2H m (Ω e )

q,j=1 Γ



−c u2H m−ε (Ω) + ue 2H m−ε (Ω e ) + λ2 m

 1

H −m+μj + 2 −ε (Γ )

(9.8.22) .

j=1

Whereas in (9.8.20) we were using the mapping λ → [γu]|Γ , we now need for the first terms on the right–hand side in (9.8.22) the mapping [γu]|Γ → λ. To this end, in addition to (9.8.15), where the traces are obtained for Ω  x → Γ , we also use the representation of ue for x ∈ Ωe : ue (x) =

m 2m−1   j=1

−

=0

2m−−1   t=0

2m−1  2m−−1   =0

t=0

Γ

 ∂  t  E(x, y) × ∂ny   × Pt++1 N+1,j ϕej + N+1,m+j λej ds

  ∂  t E(x, y) Pt++1 γe uds . ∂ny

(9.8.23)

Γe

As we can see, all the operators on the right–hand side of (9.8.23) are Poisson operators. Then we apply the trace γc for Ω e  x → Γ , subtract (9.8.15) and apply the tangential differential operator S to obtain from (3.9.29) the relation λ = S[γu] = B[γu] − Cλ (9.8.24) where B = ((Bjk )) j=1,...,m, is defined by the tangential operators of S and the k=1,...,2m

traces from Ω e and Ω, respectively, having symbols of rational type. Hence, 2m−μj −k with Theorem 9.4.3 and Corollary 9.4.4, the operators Bjk ∈ Lc (Γ ) are classical pseudodifferential operators on the manifold Γ . The operator C in (9.8.22) is obtained from applying Sγc on Γ to the last terms in (9.8.21), which are surface potentials but with charges γe ue on Γe , which are given by applying γe on Γ to the potential ue in Ω e in (9.8.19), where λ is given on Γ . Since for x ∈ Γ and y ∈ Γe we have E(·, ·) ∈ C ∞ (Γ × Γe ), we conclude that

9.9 Remarks

C :

m ,

1

H μj −m+ 2 −δ (Γ ) →

j=1

m ,

617

1

H μj −m+ 2 (Γ )

j=1

is continuous for every δ > 0. We choose δ = ε. Thus, (9.8.24) yields an estimate λ2 m

1

H μj −m+ 2 (Γ )

j=1

≤ c1 [γu]2m

j=1

+ c2 λ2 m

1

H μj −k+ 2 (Γ )

 ≤ 2c1 γu22m



+ γc ue 22m

1

μj −k+ 2 (Γ ) k=1 H

+ c2 λ2 m

1

H μj −m+ 2 −ε (Γ )

j=1

1

H μj −m+ 2 −ε (Γ )

1

μj −k+ 2 (Γ ) k=1 H

.

(9.8.25)

j=1

Since u and ue are solutions of P u = 0 in Ω and P ue = 0 in Ω e , respectively, we can apply for their sectional traces Lemma 9.8.1, and finally get   λ2 ≤ c3 u2H m (Ω) + ue 2H m (Ω e ) + c2 λ2 m m μ −m+ 1 μ −m+ 1 −ε H

j

2

H

(Γ )

j=1

j

2

(Γ )

j=1

(9.8.26) with c3 > 0. For uH m−ε (Ω) and ue H m−ε (Ω) in (9.8.22) we employ Theorem 9.4.7 for the Poisson operators in (9.8.19) where s = μj − m + 12 − ε and obtain u2H m−ε (Ω) + ue 2H m−ε (Ω e ) ≤ λ2 m

1

H μj −m+ 2 −ε (Γ )

.

(9.8.27)

j=1

Inserting (9.8.26) and (9.8.27) into (9.8.22) we finally obtain Re

m  

λq Aqj λj ds ≥ β1 λ2 m

q,j=1 Γ

1

H μj −m+ 2 (Γ )

j=1

− cλ2 m

1

H μj −m+ 2 −ε (Γ )

,

j=1

the proposed G˚ arding inequality (9.8.18).

9.9 Remarks The treatment of elliptic boundary value problems via their reformulation in terms of integral equations in the domain and on the boundary has a long history going back to G. Green [155], C.F. Gauss [141, 142], C. Neumann [335] and H. Poincar´e [350]. More recently, this approach has lead to the general theory of elliptic boundary value problems in terms of systems of pseudodifferential equations in the domain and on the boundary manifold. So, our approach may be seen as a rather particular and explicit example within the general theory. The general approach extending the theory of pseudodifferential operators see, is due to Boutet de Monvel. He developed a complete

618

9. Pseudodifferential and Boundary Integral Operators

calculus where all the compositions of pseudodifferential operators satisfying the transmission conditions with trace operators, Poisson operators and Newton potential operators are included. He established a class of operators encompassing the elliptic boundary value problems as well as their solution operators. Moreover, his class of operators is an algebra closed under composition ( see e. g., Boutet de Monvel [39, 40, 41], Grubb [159], Eskin [112], Schrohe and Schulze [378, 379], Vishik and Eskin [427]). M. Bach in [19] and Bach, Nazarov and Wendland in [20] used a system of pseudodifferential operators to model and analyse the propagation of cracks. For the general theory of pseudodifferential operators there is a vast amount of literature available, e.g., Agranovich [6], Calder´on and Zygmund [61], Chazarain and Piriou [64], Dieudonn´e [100], Egorov and Shubin [109], Folland [125, 126], Giraud [144], H¨ormander [187], Petersen [346], Seeley [384], Taira [412], Taylor [416], Treves [420], Wloka et al [440], Park [344] to name a few.

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

The treatment of boundary value problems in Chapter 5 was based on variational principles and lead us to corresponding boundary integral equations in weak formulations. The mapping properties of the boundary integral operators were derived from the variational solutions of the corresponding partial differential equations. This approach is restricted to only those boundary integral operators associated with boundary value problems which can be formulated in terms of general variational principles based on G˚ arding’s inequality. On the other hand, the boundary integral operators can also be considered as special classes of pseudodifferential operators. In the previous chapters 7 to 9, we have presented some basic properties of pseudodifferential operators. The purpose of this chapter is to apply the basic tools from previous pseudodifferential operator theory to concrete examples of this class of boundary integral equations for elliptic boundary value problems in applications. In particular, the three–dimensional boundary value problems for the Helmholtz equation in scattering theory, the Lam´e equations of linear elasticity and the Stokes system will serve as model problems. Two–dimensional problems will be pursued in the next chapter. For the specific examples in Chapters 2 and 3 our reduction of boundary value problems to boundary integral equations is based on the availability of the explicit fundamental solution E. As is well known, in applications often the fundamental solution cannot be constructed explicitly (see, e.g., Section 7.3 and for anisotropic elasticity in IR3 Natroshvili [321]). However, one may still be able to modify the setting of the reduction from boundary value problems to integral equations either based on the fundamental solution of the principal part of the differential equations if available (see Mikhailov [295]) or on the Levi functions including some parametrix as in Section 7.2 (see Pomp [351]). We will, in the following, refer to the latter as “generalized fundamental solutions”. These always generate classical pseudodifferential operators  as will be verified below. These pseudodifferential operators belong on Ω ∪ Ω to a subclass of the Boutet de Monvel algebra which has its origin in works of Vishik and Eskin [427] and Boutet de Monvel [39, 40, 41] in the 60’s for regular elliptic pseudodifferential boundary value problems (see also Grubb [159]). Specifically, following the presentation in [159] let us consider a simple © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_10

620

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

model of this algebra, P u := −Δu + qu = f in Ω ⊂ IR3 , γ0 u = ϕ on Γ

(10.0.1)

with the smooth coefficient q = q(x) ≥ q0 > 0 in Ω, and a positive constant q0 . Then the unique solution u can be represented in the form u = V λ − W ϕ + N f − N qu in Ω ,

(10.0.2)

where V and W are the simple and double layer potentials of the Laplacian, i.e. (1.2.1), (1.2.2), respectively, and N is the Newton potential  N f (x) = E(x, y)f (y)dy for x ∈ Ω Ω

in terms of the fundamental solution E(x, y) of the Laplacian given by (1.1.2) for n = 3, and λ, ϕ are the Cauchy data of u in (10.0.1). Then we may rewrite (10.0.2) in the form Au − V λ = N f − W ϕ in Ω

(10.0.3)

Au := Iu + N qu .

(10.0.4)

by setting

Taking the trace of (10.0.3), we obtain the boundary integral equation γ0 N qu − γ0 V λ = γ0 N f − (I + γ0 W )ϕ on Γ .

(10.0.5)

Hence, the boundary value problem is now reduced to the following coupled system of domain and boundary integral equations,          u N , −W f u A , −V = (10.0.6) A := λ γ0 N , −(γ0 W + I) ϕ λ γ0 N q , −γ0 V for the unknowns u in Ω and λ on Γ . The matrix A of operators on the left–hand side of (10.0.6) is a special simple case of operators belonging to the Boutet de Monvel operator algebra. In terms of his terminology, A in (10.0.4) is a pseudodifferential operator in Ω with the symbol {|ξ|2 +q(x)}−1 for |ξ| ≥ 1 which is of rational type and, hence, satisfies the transmission conditions (9.3.40). The simple layer potential V is a Poisson operator, Υ = γ0 N q is called a trace operator and −γ0 V is a pseudodifferential operator on Γ . The matrix of operators in (10.0.6) is called a general Green’s operator on Ω. The operator A in (10.0.6) is invertible and we have

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

' A−1 =

B −1 (γ0 V )−1 γ0 Nq B −1

621

( , −B −1 V (γ0 V )−1  , −(γ0 V )−1 I + γ0 Nq B −1 V (γ0 V )−1 (10.0.7)

where  B = A − V (γ0 V )−1 γ0 Nq and Nq g := N (qg) ;

(10.0.8)

also belongs to the algebra. By following Grubb’s example in [159, 160], we may also formulate the general Green operator associated with the boundary value problem (10.0.1):   PΩ + G , K A= (10.0.9) Υ ,Q where we have u + q u)|Ω PΩ u := (−Δ

(10.0.10)

with u  the extension of u by zero on IR \ Ω ; Υ u = γ0 u on Γ ; the Poisson operator is given by   K = B −1 V (γ0 V )−1 (I + γ0 W ) − W , (10.0.11) 3

and Q is a pseudodifferential operator on the boundary, defined by Υ K = γ0 K = I, the identity in this example. The singular Green’s operator G on Ω is here defined by   G = B −1 − V (γ0 V )−1 γ0 N + N − N . (10.0.12) The above simple example shows that the general Green operators in (10.0.6) and (10.0.9) exhibit all of the essential features in the Boutet de Monvel theory which are formed in terms of compositions of boundary and domain integral operators, their inverses and traces. These also provide the basis of the boundary integral equation approach. Similarly, in general, in terms of generalized fundamental solutions, the reduction to boundary integral equations can still be achieved (see e.g. Mikhailov [295]).To be more specific, let F (x, y) be a generalized fundamen together with the tal solution for the differential equation (3.9.1) in Ω ∪ Ω boundary conditions (3.9.2). If the distribution ϕ in (3.7.10) is replaced by F instead of E then the representation formula (3.9.18) is replaced by the generalized representation formula for the solution of the partial differential equation under consideration, i.e.   T (x, y)u(y)dy = F (x, y)f (y)dy u(x) − Ω

Ω\{x}

−

2m−1  2m−−1  =0

p=0

Kp Pp++1

m  j=1

N+1,j ϕj +

m  j=1



N+1,m+j λj (x) .

(10.0.13)

622

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

Here we tacitly employed the property  F (x, y) = δ(x − y) − T (x, y) P(y)

for generalized fundamental solutions. In particular, for F = E one has T (x, y) = 0 and recovers the representation formula (3.9.29), whereas for a parametrix F , the kernel T (x, y) is C ∞ ; for a Levi function of degree j ≥ 0, we have T (x, y) = Tj+1 (x, y) with T ∈ Ψ hkj−1 and the operator  (see (7.2.42) and (7.2.43)). T ∈ L−j−2 (Ω ∪ Ω) ∼ j+1

The operators Kp now are defined in (3.8.2) with E replaced by F . For ease of reading, we abbreviate Equation (10.0.13) as follows: u(x) − T u(x) = N f (x) + Vλ(x) − Wϕ(x) for x ∈ Ω . ∼ ∼

(10.0.14)

In the same manner as in Section 3.9, we apply the boundary operators Rγ and Sγ to equation (10.0.14) and obtain (in addition to (10.0.14)) the two sets of boundary integral equations corresponding to (3.9.30): Tu ϕ − Rγ ∼

Tu λ − Sγ ∼

= =

Rγ N ∼ f + RγVλ − RγWϕ , Sγ N ∼ f + SγVλ − SγWϕ .

(10.0.15) (10.0.16)

For solving the boundary value problem (3.9.1), (3.9.2) with given f in Ω and ϕ on Γ , we now have at least two choices. Coupled domain and boundary integral equations Equations (10.0.14) together with (10.0.15) define a coupled system of domain and boundary integral equations, (  ' (  ' N T , −V , −W I −∼ f u ∼ , (10.0.17) = T , RγV −Rγ N , I + RγW Rγ ∼ ϕ λ ∼ whereas from (10.0.14) and (10.0.16) we arrive at the system '

T, I −∼ T, −Sγ ∼

(  ' N, u ∼ = Sγ N λ ∼,

−V I − SγV

−W −SγW

(  f . ϕ

(10.0.18)

We seek the solution u ∈ H s+m (Ω) with the given and unknown Cauchy data ϕ and λ in the spaces Rγu

=

ϕ∈

m ,

1

H m+s−μj − 2 (Γ ) and

j=1

Sγu

=

λ∈

m , j=1

1

H −m+s+μj + 2 (Γ ) ,

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

623

respectively. R and S are given rectangular matrices of tangential differential  having the orders order(Rjk ) = μj −k+1 operators along Γ and extended to Ω where 0 ≤ μj ≤ 2m − 1 and order(Sjk ) = 2m − μj − k and satisfying the fundamental assumptions, see (3.9.4) and (3.9.5). In order to eliminate u in the domain, the Fredholm integral operator I− T ∼ of the second kind must be invertible, which can always be guaranteed by an T becomes a compact integral operator appropriate choice of F ; and since ∼ on various solution spaces on Ω. On the other hand, the boundary integral equations (10.0.15) and (10.0.16) for the unknown Cauchy datum λ on Γ are boundary integral equations of the first and second kind, respectively, as in Section 3.9. To set up a variational formulation for the system (10.0.17) or (10.0.18) , one needs the mapping properties of the corresponding operators which can be obtained by the use of the mapping properties of the corresponding pseudodifferential operators in the domain, their composition with differential operators and with traces on the boundary. We may classify these operators in the following categories: Mapping properties −j−2 −2m  T 1) The pseudodifferential operators N ∼ ∈ Lc (Ω ∪ Ω) and ∼ ∈ Lc (Ω ∪  map distributions defined on Ω into itself. They have the mapping Ω) properties

: H −m+s (Ω) : H m+s (Ω)

if s > m − 12 , if s > −m − 12 , (10.0.19) (see Theorem 9.6.1) where the last imbedding is compact since j ≥ 0. 2) The operators V and W are surface potentials defining Poisson operators which map distributions with support on Γ into distributions defined on  They have the following mapping properties (see Theorem 9.5.8): Ω ∪ Ω. N ∼ T ∼

V W

:

:

→ H m+s (Ω) → H m+s+j+2 (Ω) → H m+s (Ω)

m , j=1 m ,

1

H −m+s+μj + 2 (Γ ) → H m+s (Ω) , 1

H m+s−μj − 2 (Γ ) → H m+s (Ω) .

(10.0.20)

(10.0.21)

j=1

3) The trace operators defined on Γ by traces of pseudodifferential operators  (see (3.9.3) and Theorem 9.6.2): on Ω ∪ Ω Let s0 := μm + 12 − m and s0 := max{s0 , m − 12 }. Then

624

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

 −m+s (Ω) → Rγ N ∼ : H Rγ N ∼ : H

−m+s

(Ω) →

m , j=1 m ,

1

H m+s−μj − 2 (Γ ) for s > s0 , (10.0.22) H

m+s−μj − 12

Correspondingly, with s1 := ν1 + have  −m+s (Ω) → Sγ N ∼ : H Sγ N : H −m+s (Ω) → ∼  m+s (Ω) → T : H Rγ ∼ T : H m+s (Ω) → Rγ ∼  m+s (Ω) → T : H Sγ ∼ T : H m+s (Ω) → Sγ ∼

(Γ ) for s > s0 ,

j=1

m , j=1 m ,

1 2

− m and s1 := max{s1 , m − 12 } we 1

H −m+s+μj + 2 (Γ ) for s > s1 , 1

H −m+s+μj + 2 (Γ ) for s > s1 ;

(10.0.23)

j=1

m , j=1 m ,

1

H m+s−μj − 2 (Γ ) compactly for s > s1 , 1

H m+s−μj − 2 (Γ ) compactly for s > s1 ;

j=1

m , j=1 m ,

(10.0.24) 1

H −m+s+μj + 2 (Γ ) compactly for s > s1 , 1

H −m+s+μj + 2 (Γ ) compactly for s > s1 .

j=1

(10.0.25) Note that all of these mapping properties remain valid if Ω is replaced by Ω c ∩ K with any compact K  IRn . 4) Pseudodifferential operators on Γ involving traces of Poisson operators, namely:

10.1 Newton Potential Operators for Elliptic Partial Differential Equations

−2m+2μj +1

γV ∈ Lc

(Γ ) :

m ,

1

H −m+s+μj + 2 (Γ ) →

m ,

j=1

j=1

m ,

m ,

625

1

H m+s−μj − 2 (Γ ) ; (10.0.26)

SγV ∈ L0c (Γ ) :

1

H −m+s+μj + 2 (Γ ) →

j=1

1

H m+s+μj + 2 (Γ ) ;

j=1

(10.0.27) RγW ∈ L0c (Γ ) :

m ,

1

H m+s−μj − 2 (Γ ) →

m ,

j=1

j=1

m ,

m ,

1

H m+s−μj − 2 (Γ ) ; (10.0.28)

2m−2μj −1

RSγW ∈ Lc

(Γ ) :

1

H m+s−μj − 2 (Γ ) →

j=1

1

H −m+s+μj + 2 (Γ ) ,

j=1

(10.0.29) for s ∈ IR (see Theorems 9.5.5 and 9.5.7). All the operators in equations (10.0.17) and (10.0.18) are intimately related to the properties of the generalized Newton potentials having symbols of rational type. Their actions on distributions supported by the boundary surface as in Theorems 9.4.3 and 9.5.1 generate all of the boundary integral operators needed for the solution of elliptic boundary problems as in (10.0.17) or (10.0.18). R. Duduchava considered these systems in [104] in more general Sobolev and Besov spaces. The rest of the chapter is devoted to the concrete examples of boundary integral operators in applications. We begin with the Newton potential defined by fundamental solutions, Levi functions or parametrices. These integral operators with Schwartz kernels are pseudodifferential operators in IR3 . Next, we consider the traces of Newton potentials in relation to suitable function spaces for the weak solution of boundary value problems. The boundary potentials can be considered as Newton potentials applied to distributions supported on the boundary manifold. Hence, the jump relations may be computed classically as well as by using pseudodifferential calculus. Moreover, for the boundary integral operators generated by boundary value problems, we find their symbols and invariance properties (Kieser’s theorem). Finally, we present the computation of the solution’s derivatives for the corresponding boundary integral equations.

10.1 Newton Potential Operators for Elliptic Partial Differential Equations and Systems The latter approach can be presented schematically. For illustration let us consider the elliptic partial differential operator P of order 2m given by

626

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

P u(x) = (−1)m



 ⊂ IRn aα (x)Da u(x) = f (x) in Ω ∪ Ω

(10.1.1)

|α|≤2m

 is the tubular neighbourhood of Γ = ∂Ω as defined in Section 9.2. where Ω The ellipticity condition reads here  . aα (x)ξ α = 0 for all ξ ∈ IRn with |ξ| = 1 and x ∈ Ω ∪ Ω |α|=2m

The operator P is obviously a pseudodifferential operator of order 2m with the polynomial symbol  σP (x, ξ) = (−1)m aα (x)(iξ)α . |α|≤2m

−1  is of rational type. In particular, if the coClearly, the inverse σP (x, ξ)  −1 is the symbol of the parametrix efficients aα are constants, then σP (ξ) which also corresponds to the Newton potential operator defined by a fundamental solution. As for the Levi function of order j ∈ IN0 , the corresponding Newton potential is given by (7.2.47), i.e.  L ∼f (x) :=

Lj (x, y)f (y)dy =  Ω∪Ω

 j  =0  Ω∪Ω

N (x, y)f (y)dy =

j  =0

N f (x) ∼

(10.1.2) −−2m  are pseudodifferential operators due to Lemma where N ∈ L (Ω ∪ Ω) c ∼ 7.2.6. Our goal is to show that all of the Newton potentials mentioned above are pseudodifferential operators with symbols of rational type. More generally, we have the following theorem. Theorem 10.1.1. The pseudodifferential operator given by a generalized Newton potential in terms of either a parametrix or Levi function of an elliptic partial differential operator as well as an elliptic system in the sense of Agmon, Douglis and Nirenberg has a symbol of rational type. Remark 10.1.1: Since a fundamental solution is a special parametrix, Theorem 10.1.1 is particularly valid for the Newton potential in terms of a fundamental solution. Proof: We begin with the proof for the case of a parametrix and first consider the scalar elliptic case. From the proof of Theorem 7.2.2 we notice that q−2m (x, ξ) defined by (7.2.8) is of rational type since a(x, ξ) is polynomial. Since all the homogeneous terms in the symbol expansion of the parametrix given by the recursion formula (7.2.9) are finite compositions of differentiation

10.1 Newton Potential Operators for Elliptic Partial Differential Equations

627

and rational type symbols, Lemma 9.4.2 implies that each term q−2m−j (x, ξ) is also of rational type, by induction. Now, for an elliptic system in the sense of Agmon–Douglis–Nirenberg (7.2.10), the characteristic determinant (7.2.14) is polynomial. Therefore, all the elements of the matrix–valued symbol q−2m (x, ξ) given by (7.2.8) are of rational type. Hence, the arguments for the rational type of each term in the matrices defined by the recursion formula (7.2.9) remain the same as in the scalar case. ii) For the Levi functions in the case of scalar elliptic equations we begin  with the kernel L0 with N given by (7.2.35) where Ω is replaced by Ω ∪ Ω ∼0 whose amplitude is  −1 a(x, y; ξ) = a02m (y, ξ) . The latter, clearly is a rational function of ξ depending also on y. Hence the symbol of N is given by the asymptotic expansion (7.1.28) where for each ∼0 homogeneous term only a finite number of differentiations appears. Then Lemma 9.4.2 implies that N ∼ 0 has a symbol of rational type. In order to show −(+2m+1)  given by (7.2.52) in Lemma 7.2.6 has a that N (Ω ∪ Ω) ∼ +1 ∈ Lc symbol of rational type, we use the argument of induction assuming that N ∼ T given by (7.2.31) has the polynomial has a symbol of rational type. Since ∼ amplitude      aα (y) − aα (x) (iξ)α − aα (x)(iξ)α , (−1)m |α|=2m

|α| −2m +

γ0 Af 

H 2m− 2 +σ (Γ )

1

≤

cf H σ (Ω) for σ > σ0 where σ0 =

1 2

max{− 12 , −2m

,

(10.1.3) (10.1.4)

+

1 2} .

In the case of an elliptic system in the sense of Agmon–Douglis–Nirenberg, the appropriate mapping properties for each of the elements of the matrix Newton potential operator can be obtained by again using Theorems 9.6.1 and 9.6.2 in addition to Theorem 7.1.12. 10.1.1 Generalized Newton Potentials for the Helmholtz Equation To illustrate the different generalized Newton potentials in terms of the fundamental solution, parametrices and Levi functions we use the inhomogeneous Helmholtz equation (see Section 2.1),  ⊂ IR3 P u := −(Δ + k2 )u = f in Ω ∪ Ω

(10.1.5)

where the fundamental solution which satisfies the radiation condition (2.1.2) is given explicitly in (2.1.4) as Ek (x, y) =

eik|z| 4π|z|

with z = x − y .

10.1 Newton Potential Operators for Elliptic Partial Differential Equations

So, the classical Newton potential is of the form  N f (x) = Ek (x, y)f (y)dy

629

(10.1.6)

IR3

 ⊂ IR3 . where f has compact support in Ω ∪ Ω Symbol and kernel expansions Since the Helmholtz operator P is a scalar strongly second order operator with constant coefficients with the symbol σP = |ξ|2 − k2 for ξ ∈ IR3 , the Newton potential N in (10.1.6) is a classical pseudodifferential operator N ∈ 3 L−2 c (IR ). In this case, we also have the complete symbol ∞

σN =

∞

 k2p  1 = = a0−2−2p (ξ) for |ξ| > k , 2 2 2+2p |ξ| − k |ξ| p=0 p=0

(10.1.7)

which defines the classical symbol of order −2 as in (7.1.19) and, obviously, is of rational type. As in Section 7.2, we use the cut–off function χ in (7.2.31) 2 to define a parametrix H ∈ L−2 c (IR ) by Hf := Fξ−1 →x

  χ

  1 f (y) . F y → ξ |ξ|2 − k2

(10.1.8)

We remark that in this case, as we shall see, the difference N − H between the Newton potential and the parametrix is a smoothing operator. From the homogeneous expansion σN in (10.1.7), the relation (8.1.49) leads to the homogeneous kernel  k−1+2p (z) = (2π)−2 p.f. eiξ·z a0−2−2p (ξ)dξ . (10.1.9) IR3

By using the explicit Fourier transform in Gelfand and Shilov [143, p. 363] we obtain 1

k−1+2p (z) = k2p (2π)−3 2−2p− 2 (2π)3/2

Γ ( 12 − p) 2p−1 |z| Γ (p + 1)

and, with the properties of the gamma–function Γ , Γ ( 12 − p) Γ (p + 1)

=

k−1+2p (z)

=

√ 1 2p (−1)p π 2 , (2p)! 1 1 |z|2p−1 . k2p (−1)p 4π (2p)!

This defines the asymptotic pseudohomogeneous kernel expansion

(10.1.10)

630

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations ∞ 

k−1+2p (z) =

p=0

∞ 1  (−k2 )p 2p−1 |z| 4π p=0 (2p)!

(10.1.11)

where z = x − y. If we use only a finite number of terms in the expansion (10.1.11) we obtain the Levi functions L2q (x, y) =

q 1  (−k2 )p |x − y|2p−1 4π p=0 (2p)!

(10.1.12)

corresponding to (7.2.46) with j = 2q. The associated volume potentials in terms of Levi functions of order 2q are given by q  (−k2 )p 1  L f (x) = |x − y|2p−1 f (y)dy . ∼2q 4π p=0 (2p)!

(10.1.13)

IR3

We observe that the asymptotic expansion in (10.1.11) even converges for all z ∈ IR3 \ {0} and defines a Schwartz kernel k(z) :=

∞  1 (−k2 )p 2p−1 |z| 4π (2p)! p=0

with k ∈ Ψ hk−1 (IR3 ). Then Theorem 8.1.1 implies that the integral operator  L f (x) := k(x − y)f (y)dy ∼ IR3

is a classical pseudodifferential operator of order −2. On the other hand, the fundamental solution Ek (x, y) has a series expansion in the form Ek (x, y)

=

∞ ∞ 1  (−k2 )p 2p−1 ik  (−k2 )p |z| |z|2p + 4π p=0 (2p)! 4π p=0 (2p + 1)!

=

ik sin(k|z|) 1 1 cos(k|z|) + . 4π |z| 4π k|z|

(10.1.14)

This shows that the difference Ek (x, y) − k(x − y) =

ik sinc(k|x − y|) 4π

sin(k|x − y|) . k|x − y| Clearly, the real part k(x−y) = Re Ek (x, y) also is a fundamental solution which, however, does not satisfy the Sommerfeld radiation condition (2.1.2). Therefore, by using the cut–off function ψ(z) as in Theorem 7.1.16 with

defines a C ∞ –kernel function, namely sinc(k|x − y|) =

10.1 Newton Potential Operators for Elliptic Partial Differential Equations

631

ψ(z) = 1 for |z| ≤ 12 and ψ(z) = 0 for |z| ≥ 1, the Newton potential can be decomposed in the form   ik k(x − y)f (y)dy + sinc(k|x − y|)f (y)dy N f (x) = 4π 3 3 IR IR  = k(x − y)ψ(k|x − y|)f (y)dy IR3

 +

 ik sinc(k|x − y|)}f (y)dy {k(x − y) 1 − ψ(|x − y|) + 4π

IR3

=

N0 f (x) + Rf (x)

where the integral operator N0 is properly supported and R is a smoothing operator corresponding to Theorem 7.1.9. By applying Theorem 7.1.7 to N0 ∈ OP S −2 (IR3 × IR3 ) we find the symbol a(x, ξ) of N0 in the form a(x, ξ)

= =

e−ix·ξ (N0 eiξ• )(x)  ∞  1 −ix·ξ (−k2 )p e |x − y|2p−1 ψ(|x − y|)eiξ·y dy . 4π (2p)! p=0 IR3

Since N0 , N and H all coincide modulo smoothing operators, the asymptotic expansions of their symbols belong to the same equivalence class, i.e., the complete symbol class is characterized by the homogeneous symbol expansion ∞  p=0

a0−2−2p (ξ) =

∞ 

k2p 3 3 ∼ a(x, ξ) ∈ S −2 c (IR × IR ) 2+2p |ξ| p=0

(10.1.15)

as in (10.1.7). L and L all have the mapping properties (10.1.3) Clearly, N0 , N , H , ∼ ∼ 2q and (10.1.4). 10.1.2 The Newton Potential for the Lam´ e System We now consider the simplest nontrivial elliptic system of equations in IR3 in applications, namely the Lam´e system (2.2.1),  ⊂ IR3 . (10.1.16) P u = −Δ∗ u = −μΔu − (λ + μ)graddivu = f in Ω ∪ Ω Its quadratic symbol matrix can be calculated from (5.4.12) as   σP (ξ) = −G  (iξ)CG(iξ) = μ|ξ|2 δjk + (μ + λ)ξj ξk 3×3 . Hence, the characteristic determinant is given by

(10.1.17)

632

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

det σP (ξ) = μ2 (λ + 2μ)|ξ|6 ,

(10.1.18)

P is strongly elliptic and the inverse to σP (ξ) defines the symbol of the Newton potential, i.e. −1  = σP (ξ)

 1  2 (λ + μ) |ξ| ξ δ − ξ . jk j k μ|ξ|4 (λ + 2μ) 3×3

(10.1.19)

The Fourier inverse of (σP )−1 defines the fundamental matrix E(x, y) for P , see (7.2.18), i.e.   −1 i(x−y)·ξ −3 E(x, y) = (2π) p.f. σP (ξ) e dξ =

1 μ(2π)3



IR3



p.f. IR3

+

=

1 i(x−y)·ξ e dξδjk |ξ|2

∂2 λ+μ p.f. λ + 2μ ∂xj ∂xk

 IR3

(10.1.20)

1 i(x−y)·ξ  e dξ |ξ|4 3×3

λ + μ (xj − yj )(xk − yk )  1 (3μ + λ)  δjk + 8πμ (λ + 2μ) |x − y| 3μ + λ |x − y|3 3×3

as in (2.2.2). Since the symbol in (10.1.19) is homogeneous, the fundamental solution defines the Schwartz kernel of the Newton potential. Although one may still define a parametrix as in (10.1.8) by multiplying the homogeneous symbol in (10.1.19) by the cut–off function χ(ξ) one would not gain anything. As in the scalar case, by using the cut–off function Ψ from Theorem 7.1.16, the Newton potential can be decomposed in the form 

N f = N0 f + Rf =   E(x, y)ψ(|x − y|)f (y)dy + E(x, y) 1 − ψ(|x − y|) f (y)dy

IR3

IR3

where N0 ∈ OP S −2 (IR3 × IR3 ) is properly supported and R is smoothing. 10.1.3 The Newton Potential for the Stokes System As we have seen in Section 2.3.3, the fundamental solution of the Stokes system can be obtained from that of the Lam´e system by taking the limit λ → +∞. Hence, we recover (2.3.10) from (10.1.20) and also the corresponding symbols from (10.1.19) and (2.3.10) as σESt (ξ) =

 1  2 1 ξ δ − ξ |ξ| jk j k μ|ξ|4 2 3×3

(10.1.21)

10.2 Surface Potentials for Second Order Equations

633

and σQ (ξ) =

2i ξj . |ξ|2

(10.1.22)

The corresponding Newton potentials then read NSt f = NSt0 f + NSt1 f   = ψ(|x − y|)ESt (x, y)f (y)dy + (1 − ψ(|x − y|) ESt (x, y)f (y)dy IR3

IR3

and Qf = Q0 f + Q1 f with  ψ(|x − y|)Q(x, y) · f (y)dy .

Q0 f = IR3

Clearly, from (10.1.21) and (10.1.22) we conclude that NSt0 ∈ OP S −2 (IR3 × IR ) and Q0 ∈ OP S −1 (IR3 × IR3 ). 3

10.2 Surface Potentials for Second Order Equations For the special case of second order systems, m = 1, we find two kinds of boundary conditions (3.9.10) and (3.9.12). The representation formula (3.7.10) with the fundamental solution E is of the form   u(x) = E(x, y)f (y)dy − E(x, y)P1 γ0 (y)dsy Ω

 −

Γ

E(x, y)P2 (y)γ1 u(y)dsy

(10.2.1)

Γ

−

   ∂  E(x, y) P2 (y)γ0 u(y)dsy . ∂ny

Γ

 ∂  ∂  =− −2H with H = − 12 ∇·n; the mean curvature of Γ (see ∂ny ∂ny (3.5.5)). P1 and P2 are given by (3.4.62). Note that, in general, P1 contains ∂  , we find first order tangential differentiation. By using the definition of ∂n y Here

634

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

 u(x)

=

E(x, y)f (y)dy Ω



−

E(x, y){P1 γ0 u + 2HP2 γ0 u + P2 γ1 u}dsy

(10.2.2)

Γ

+

   ∂ E(x, y) P2 γ0 u(y)dsy , ∂ny γ

i.e., a volume potential, a simple layer surface potential and a double layer potential. Hence, the simplest case of V and W in equations (10.0.20) and (10.0.21) now corresponds to P u(x) = −

n  j,k=1

n ∂  ∂u   ∂u aj (x) + c(x)u ajk (x) + ∂xj ∂xk ∂xj j=1

with the Dirichlet conditions Rγu = u|Γ and the conormal derivative as the complementary boundary operator Sγ =

n 

nj (x)ajk (x)

j,k=1

∂ |Γ . ∂xk

Then the generalized Newton potentials of Section 10.1 have symbols of rational type and generate corresponding simple and double layer potentials supported by Γ . Simple layer potentials In this case, the simple layer potential has the form   Γ λ(x) = Vλ(x) = N (λ ⊗ δΓ ) = Q N (x, x − y  )λ(y  )dsΓ (y  )

(10.2.3)

y∈Γ

where N is the Schwartz kernel of the generalized Newton potential. As a  Γ in (10.2.3) has all the continuous linear mapping, the surface potential Q mapping properties given in Theorem 9.5.8 for an operator of order −2 in (9.5.19). Moreover, this surface potential V has limits for x → Γ from x ∈ Ω  ∩ Ω c because of Theorem 9.5.1: as well as x ∈ Ω   lim N (x , x − y  )λ(y  )dsΓ (y  ) for x ∈ Γ QΓ λ(x) =  x→x ∈Γ

y  ∈Γ

=:

QΓ λ(x ) = V λ(x ) .

(10.2.4)

This boundary integral operator is a pseudodifferential operator V ∈ (Γ ) with m = 1 which is the simple layer boundary integral operator L−2m+1 c introduced in Chapter 2.

10.2 Surface Potentials for Second Order Equations

635

Since the homogeneous principal symbol of the generalized Newton potential is given here by the matrix valued function a0−2 (x, ξ) =

n  

ajk (x)ξj ξk

−1

,

(10.2.5)

j,k=1

Formula (9.4.7) provides us the principal symbol  1 0   {z 2 + bz + a}−1 dza−1 q−1 (x , ξ ) = nn 2π

(10.2.6)

c

where a = a−1 nn

n−1  j,k=1

ajk ξj ξk , b = a−1 nn

n−1 

(ajn + anj )ξj

j=1

and the curve c ∈ C is given in Theorem 9.4.3 and oriented clockwise. Double layer potentials Since in (10.2.3) we have already considered the simple layer potential, it remains to analyze the double layer potential    ∂ E(x, y) P2 (y)u(y)dsy Wu(x) = ∂ny Γ     ∂ = E(x, y) P2 (y) u(y  ) ⊗ δΓ dy ∂ny  Ω

 \ Γ in terms of Newton potentials. Here the latter is defined by for x ∈ Ω   ∂ E(x, y) P2 (y) which generates a pseudodifferential the Schwartz kernel ∂n y  operator A ∈ L−1 c (Ω). Hence, only its principal part will contribute to the tangential differential operator T in Theorem 9.5.5 which here is of order 0. To compute q0 we apply Theorem 9.4.3 in the tubular coordinates  ∈ r for which only the principal symbol of A is needed. Now we use the U representation of the second order differential operator P in the form (3.4.56), i.e. ∂u ∂2u + P2 2 P u = P0 u + P 1 ∂n ∂n where the tangential differential operators P0 and P1 of orders 2, respectively, and 1 and the coefficient P2 are given (3.4.57)–(3.4.59). Hence, the principal symbol of A has the form σAΦr ,0 (x, ξ) = −iξn {p0 (x, ξ  ) + p1 (x, ξ  )ξn + ξn2 }−1 



(10.2.7)

where p0 (x, ξ ) is the principal part of −P0 (x, iξ )/P2 (x) and p1 (x, ξ  ) the principal part of iP1 (x, iξ  )/P2 (x). Furthermore, x = Ψ () with  = ( , k ) ∈ r . Hence, q0 is given by (9.4.7), i.e. U

636

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

i q0 ( , ξ ) = − (2π) 





{p0 (x, ξ  ) + p1 (x, ξ  )z + z 2 }−1 zdz

(10.2.8)

c

where x = Tr ( ) and c is the contour defined by (9.4.9) circumventing the pole in the lower half–plane clockwise. 10.2.1 Strongly Elliptic Differential Equations The simple layer potential for the scalar equation If (10.1.2) is a scalar, real elliptic equation of second order, then it is even a strongly elliptic equation where we have 4a − b2 ≥ γ0 |ξ  |2 with γ0 > 0 .

(10.2.9)

Then the residue theorem implies n−1   n−1 2 − 12   0 = ann (x ) 4ann (x ) ajk (x )ξj ξk − (ajn + anj ξj ) . q−1 j,k=1

j=1

(10.2.10) More generally, if Γ is a given surface, one may prefer to use tubular coordinates (3.3.7). However, since here we only used the homogeneous principal symbol, the conclusion remains the same and the principal symbol in the tubular coordinates can be obtained by the transformation formula (7.1.49). We shall return to the tubular coordinates in the next section. As a consequence of (10.2.9) we conclude the following lemma. Lemma 10.2.1. For a scalar strongly elliptic second order partial differential operator (10.1.1), the simple layer boundary integral operator V ∈ L−1 c (Γ ) is strongly elliptic (see (7.2.62)) in the sense that there exists a function Θ(x ) = 0 on Γ and a positive constant γ0 > 0 such that Re Θ(x )σV0 (x , ξ  ) ≥ γ0 for all ξ  ∈ IRn−1 with |ξ  | = 1 . The simple layer potential for strongly elliptic second order systems In order to guarantee strong ellipticity for the Newton potential, i.e. for the principal symbol a0−2 of a second order system in (10.2.5), we establish the following lemma. Lemma 10.2.2. Let the adjoint differential operator P ∗ P ∗ v(x) = −

n n  ∂  ∗ ∂v   ∂ (a∗j v) + c∗ v ajk (x) − ∂x ∂x ∂x j k j j=1 j=1

(10.2.11)

be strongly elliptic. Then the Newton potential with principal symbol given in (10.2.5) is strongly elliptic.

10.2 Surface Potentials for Second Order Equations

637

Proof: With A := ((ajk ))np×np , which is invertible since A∗ is invertible, for any given ζ ∈ C we choose η = A ζ and find with Θ−1 (x) := A−1 Θ where Θ belongs to A∗ in (7.2.62), Re η  Θ−1 A−1 η = ζ  A(A−1 Θ)A∗ ζ = Re ζ  ΘA∗ ζ ≥ γ0 |ζ|2 ≥ γ0 |η|2 for all η ∈ Cn since |η|2 ≤ c|ζ|2 . Hence, A−1 is strongly elliptic and γ0 =

γ0 c

> 0.



We are now in the position to show strong ellipticity of the simple layer pseudodifferential operator QΓ with the principal symbol given by (10.2.5). Theorem 10.2.3. For a strongly elliptic adjoint differential operator (10.2.11), the simple layer potential operator QΓ corresponding to P with the principal symbol given by (10.2.6) is strongly elliptic on Γ . Proof: In view of the transformation formula (7.1.49) for principal symbols under change of coordinates it suffices to prove the theorem for Γ identified 0 given by (10.2.6), with the tangent hyperplane xn = 0, i.e. for q−1 0 q−1 (x , ξ  )

+

i 2π

−π

1 = 2π

R

 a0−2 (x , 0) ; (ξ  , ξn ) dξn

−R

 a0−2 (x , 0) ; (ξ  , Re eiϑ ) Re eiϑ dϑ

ϑ=0

 with R ≥ R0 sufficiently large so that all the poles of a0−2 (x , 0) ; (ξ  , z) for |ξ  | = 1 are contained in |z| ≤ 12 R0 , Im z ≤ 0. Since a0−2 in (10.2.5) is homogeneous of order −2, the second integral on the right–hand side is of order R−1 . Hence, we obtain 0 q−1 (x , ξ  )

1 = 2π

∞

 a0−2 (x , 0) ; (ξ  , ξn ) dξn .

−∞

Now we first consider the case that Θ = ((δm ))p×p in the definition (7.2.62) of strong ellipticity for P ∗ . Then for η ∈ Cp we have 0 (x , ξ  )η η  q−1

1 = 2π =

with the substitution

1 2π

∞ −∞ ∞

−∞

 η  a0−2 (x , 0) ; (ξ  , ξn ) η dξn  ζ  σP0 ∗ (x , 0) , (ξ  , ξn ) ζ dξn

638

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

 ζ  (ξ  , ξn ) = η  a0−2 (x , 0) ; (ξ  , ξn ) .

(10.2.12)

Strong ellipticity for P ∗ with Θ = ((δm ))p×p implies that 0 (x , ξ  )η Re η  q−1

1 = 2π

∞

  Re ζ  σP0 ∗ (x , 0) , (ξ  , ξn ) ζ dξn

−∞

1 ≥ γ0 2π 1 ≥ γ0 2π

∞

|ζ(ξ  , ξn )|2 (1 + |ξn |2 )dξn

−∞

1

|ζ(ξ  , ξn )|2 dξn for |ξ  | = 1 .

−1

On the other hand, since  η = σP0 ∗ (x , 0) ; (ξ  , ξn ) ζ , we have the estimate |η|2 ≤ |ζ|2

max

|ξ  |=1,ξ

n ∈[−R0 ,R0 ]

 |σP0 ∗ (x , 0) ; (ξ  , ξn ) | = c|ζ|2 .

0 : This implies the strong ellipticity of q−1 0 Re η  q−1 (x , ξ  ) η ≥ γ1 |η|2 for all |ξ  | = 1 and η ∈ Cp

(10.2.13)

with γ1 > 0. For general Θ(x) consider the modified differential operator P∗ = ΘP ∗ instead of P ∗ . Then we find the relation 0 q−1 (x , ξ  )

1 = 2π

=Θ

+∞  3  

−1

dξn

j,k=1

−∞

∗−1

ajk Θ∗ ξj ξk

1 (x , 0) 2π

+∞  3  



−∞

ajk ξj ξk

−1

dξn

j,k=1

0 = Θ∗−1 (x , 0)q−1 (x, ξ  ) .

By using the strong ellipticity (10.2.13) we obtain 0 0 (x , ξ  ) η = Re η  q−1 (x , ξ  ) η ≥ γ12 |η|2 Re η  Θ∗−1 (x , 0)q−1

with γ1 > 0 for all η ∈ Cp and |ξ  | = 1.



10.2 Surface Potentials for Second Order Equations

639

The double layer potential for the scalar equation The poles of the integrand in (10.2.8) are given by 9 1 i z1|2 = − p1 ± 4p0 − p21 2 2 due to the strong ellipticity of the scalar, real differential operator P . Hence, by the residue theorem we find q0 ( , ξ  )

=

i p1 (x, ξ  ) 1 − + 8 2 2 4p0 (x, ξ  ) − p21 (x, ξ  )

q0c ( , ξ  )

=

i p1 (x, ξ  ) 1 + 8 . 2 2 4p0 (x, ξ  ) − p21 (x, ξ  )

and

Thus, the tangential differential operator is given here by T u(x ) =

1 u(x ) . 2

Moreover, for the double layer potential we obtain the jump relation  lim Wu Tr ( ) + n n( ) 0>n →0    ∂ 1 (10.2.14) E(x , y) P2 (y)u(y)dsy = − u(x ) + p.f. 2 ∂ny Γ \{x }

where, in general, the finite part integral operator defines a Cauchy–Mikhlin singular integral operator which is a pseudodifferential operator of order 0 on the boundary Γ . However, since p1 (ξ  ) is the principal part of iP1 (x, iξ  )/P2 (x) as in (10.2.7) and (3.4.58), p1 (ξ  ) is real for a real elliptic scalar differential operator P of second order (3.4.53). Hence,we have Re Θq0 ( , ξ  ) = 1 for Θ = −2 . Consequently, we have the following lemma. Lemma 10.2.4. For a scalar, real elliptic second order differential operator (10.1.1), the corresponding operator − 12 I + K defined by (10.2.14) with the double layer potential K ∈ L0c (Γ ) is a strongly elliptic pseudodifferential operator of order zero. In the special case as for the Laplacian (also the Helmholtz operator), see Remark 3.4.1, we have p1 (x, ξ  ) = 2H(x) and therefore the finite part integral operator K in (10.2.14) even becomes weakly singular.

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

640

10.2.2 Surface Potentials for the Helmholtz Equation As explicit examples for the surface potentials we return to the Helmholtz equation in IR3 . With the Newton potential given by (10.1.5) we define the simple layer surface potential     Γ v(x) = Q Ek (x, y) v(y  ) ⊗ δΓ dy = Ek (x, y)v(y  )dsy . IR3

Γ

For x → Γ we obtain the simple layer boundary integral operator   Γ v(x) = Ek (x, y  )v(y  )dsy Vk (x) = lim Q x→Γ

=

1 4π

 Γ

Γ ik|x−y  |

e v(y  )dsy for x ∈ Γ |x − y  |

(10.2.15)

which is a pseudodifferential operator Vk ∈ L−1 c (Γ ). If Γ is given by xn = 0 then the complete symbol of Vk has a homogeneous asymptotic expansion Σq−1−j (x , ξ  ) and, from Formula (9.4.7), we find q−2p

=

0

(10.2.16)

q−1−2p (x , ξ  )

=

(2π)−1

=

(2π)

−1

=

1  −1−2p 2p |ξ | k for |ξ  | ≥ 1 and p ∈ IN0 2

and



 a0−2−2p (x 0) , (ξ  , z) dz

c

 k2p c

dz |ξ  |2p+2 + z 2

(10.2.17)

by the use of the residue theorem since the only pole in the lower half plane of C is at −i|ξ  |p+1 . For a general surface Γ given by x = T ( ) one first has to employ canonical coordinates and represent the Newton potentials in terms of these coordinates, and then compute the symbol of the boundary potential operator Vk from VkΦr ,0 the transformed operator’s symbol in Theorem 9.4.3. To illustrate the idea let →U  and Φ(x) =  : Ω x = Ψ () = T ( ) + n n( )

(10.2.18)

denote the diffeomorphism (3.3.7) of the canonical coordinates. Then the principal symbol of the Newton potential with respect to the canonical coordinates is transformed according to (7.1.49), i.e.

10.2 Surface Potentials for Second Order Equations

641

  ∂Φ   0 0 Ψ () , σN (, ζ) = σ ζ , Φ,−2 ,−2 N k k ∂x  . The Jacobian where ζ denotes the Fourier variables corresponding to  ∈ U matrix for the canonical coordinates can be computed explicitly by using (3.4.28) and (3.4.29) in the form  ∂Φ  ∂x

  ∂Ψ ∂Ψ ∂Ψ ∂Ψ = g 11 + g 12 , g 12 + g 22 , n . ∂1 ∂2 ∂1 ∂2

(10.2.19)

In what follows, we shall adhere to the Einstein summation convention where for repeated Greek indices λ, ν, α, β, γ. . . the summation index runs from 1 to 2, whereas for Roman indices j, k, , q. . . it runs from 1 to 3. The relation (10.2.19) implies |ξ|2

= =

=

 ∂Φ  2 ∂Φ  ∂Φ  ∂Φ ∂Φj   ζ = ζ ζ = ζ ζj (10.2.20)  ∂x ∂x ∂x ∂xk ∂xk ∂xk m ∂xk qj g g ζj ζ ∂m ∂q ⎛ 11 ⎞ g g 12 0 ζ  ⎝g 12 g 22 0⎠ ζ = {ζ12 g 11 + 2ζ1 ζ2 g 12 + ζ22 g 22 + ζ32 } . 0 0 1

This implies for the principal symbol  ∂Φ  −2   σNk Φ,−2 (, ζ) =  ζ  = {ζ12 g 11 + 2ζ1 ζ2 g 12 + ζ22 g 22 + ζ32 }−1 ∂x since gmq =

∂xk ∂xk ∂m ∂q

j and gmq g qj = δm ,

see (3.4.2) and (3.4.4). For the principal symbol (10.2.20) of Nk we apply Theorem 9.4.3 again for n = 0 and obtain 1

0 ( , ζ  ) = 12 {ζα γ αβ (g  )ζβ }− 2 q−1

(10.2.21)

with ζ  = (ζ1 , ζ2 ) ∈ IR2 \ {0} and γ αβ the contravariant fundamental tensor of Γ , (3.4.14), (3.4.15). Obviously, in view of (10.2.21), the simple layer potential Vk is a strongly elliptic pseudodifferential operator Vk ∈ L−1 c (Γ ) since 1

0 q−1 ( , ζ  ) = 12 {ζα γ αβ ζβ }− 2 ≥ γ0 |ζ  |−1 for all ζ  ∈ IR2

(10.2.22)

with a positive constant γ0 depending on the local fundamental tensor γ αβ . Of course, the strong ellipticity also follows from Lemma 10.2.1.

642

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

The double layer potential The double layer potential Wk of the Helmholtz equation in (2.1.5) will now be considered as a surface potential generated by an appropriate Newton potential applied to the boundary distribution v(x ) ⊗ δΓ . To facilitate the computations, it is more convenient to begin with the adjoint operator   eik|x−y|  1  Wk u(x) := ψ(|x − y|)u(y)dy n(x) · ∇x 4π |x − y| Ω (10.2.23)   eik|x−y|  1 1 − ψ(|x − y|) u(y)dy n(x) · ∇x + 4π |x − y| Ω

where ψ is the same cut–off function as in Theorem 7.1.16. Since the first integral on the right–hand side defines a properly supported pseudodifferential operator in Ω, its symbol can be computed as   σWk (x, ξ) = −n(x) · e−ikξ·(x−y) ∇y Ek (x, y) ψ(y)dy IR3

based on Theorem 7.1.11 after employing ∇x Ek (x, y) = −∇y Ek (x, y) for the fundamental solution Ek = by parts yields  σWk (x, ξ) = i(n(x) · ξ)

1 eik|x−y| (see (2.1.4)). Then integration 4π |x − y|

e−iξ·(x−y) Ek ψdy = n(x) ·

IR3



e−iξ·(x−y) (∇y ψ)Ek dy .

IR3

The second integral on the right–hand side decays, due to the Paley–Wiener Theorem 3.1.3, since ∇y ψ(|x − y|) has a compact support not containing x = 1 1 y. From the decomposition of Ek in (10.1.14) it is clear that only 4π z cos(k|z|) contributes to the symbol σW  and, again, we obtain the complete symbol expansion of rational type  ∞  1 k2p 1 cos(k|z|)ψ(|z|)dz = e−iξ·z + R∞ (ξ) 4π |z| |ξ|2+2p p=0 IR3

where R∞ ∈ S −∞ (IR3 ). Hence, σWk (x, ξ) = i(n(x) · ξ) For p = 0, we find the principal symbol

∞ 

k2p . |ξ|2+2p p=0

(10.2.24)

10.2 Surface Potentials for Second Order Equations

σWk ,−1 (x, ξ) = i(n(x) · ξ)

1 . |ξ|2

643

(10.2.25)

Therefore, by using Formula (7.1.34) in Theorem 7.1.13, we find the symbol of Wk as σ Wk ∼

 k2p   1  ∂ α  ∂ α  −i − i(n(x) · ξ) α! ∂ξ ∂x |ξ|2p+2 p=0

|α|≥0

and its principal symbol 0 σW (x, ξ) = −i(n · ξ) k ,−1

1 . |ξ|2

(10.2.26)

In order to obtain the boundary integral operators on Γ , we again apply Theorem 9.4.3 to Wk and Wk , respectively. Then we have for Γ identified with xn = 0,  zdz 1 1 0   qWk ,0 (x , ξ ) = − (10.2.27) =− 2  2 2π z + |ξ | 2 c

since the only pole in the lower half plane of C is z = −i|ξ  |. In the same way we obtain 1 0   . qW  (x , ξ ) = k ,0 2 These are the principal symbols of QΓ and QΓ defined by the limits of Wk and of Wk for xn → 0− , respectively. We notice that    ∂ 1 Qv(x ) = lim − Wk v(x) = − v(x ) + p.f. Ek (x, y) v(y)dsy 2 ∂ny xn →0 Γ

and Qc v(x )

=

1 lim Wk v(x) = v(x ) + p.f. + 2 xn →0

(10.2.28)    ∂ Ek (x, y) v(y)dsy . ∂ny Γ

Correspondingly, for the transposed operators, one obtains    ∂ 1 lim − Wk v(x) = v(x ) + p.f. Ek (x, y) v(y)dsy Q v(x ) = 2 ∂nx xn →0 Γ

and Qc v(x )

=

1 lim Wk v(x) = − v(x ) + p.f. + 2 xn →0

(10.2.29)    ∂ Ek (x, y) v(y)dsy , ∂nx Γ

resembling the well–known jump relations for the classical layer potential Wk and its adjoint Wk . In particular, we notice

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

644

1 1 [Qc − Q] = T = Tc = I 2 2 0 0 as in (9.3.37). From our symbol computations for qW and qW it follows  k ,0 k ,0 that the acoustic double layer boundary integral operator    ∂ Ek (x, y) v(y)dsy Kk v(y) = p.f. ∂ny Γ

and its adjoint Kk v(y)

=

p.f.

   ∂ Ek (x, y) v(y)dsy ∂nx Γ

both must be pseudodifferential operators on Γ of order at most −1 which ∂ Ek (x, y) and ∂n∂ x Ek (x, y) are in implies that their Schwartz kernels ∂n y Ψ hk−1 (Γ ), i.e., they are weakly singular. For a curved surface Γ , we need to employ the canonical, tubular coordinates for Wk and Wk and then compute the corresponding symbols for x → 0. From our general results in Section 10.2.2 it is clear that the principal symbol of QΓ remains the same. Here, TΓ = − 12 I also on Γ . Now we consider the principal symbols of the operators Kk and Kk ∈ −1 Lc (Γ ) for whose computation we shall need a0Wk ,0 (x, ξ) + a0Wk ,−2 (x, ξ) in the tubular coordinates (10.2.18). These can be obtained by Formula (7.1.34) for the symbol of the transposed operator, i.e., from    1 −4 + O(|ξ| ) σWk (x, ξ) = i n(x) · ξ |ξ|2 which yields σWk (x, ξ)

=

 1  ∂ α  ∂ α σWk (x, −ξ) −i α! ∂ξ ∂x

|α|≥0

= =

1 ∂  ∂  n(x) · ξ  + −i −i + O(|ξ|−3 ) 2 |ξ| ∂ξ ∂x |ξ|2  ∂n ξ ξ  ∇ · n(x) 1 k k  + O(|ξ|−3 ) . +2 −i(n(x) · ξ) 2 − 2 |ξ| |ξ| ∂x |ξ|4

−i(n(x) · ξ)

In the tubular coordinates we use (3.4.19) and, moreover, (3.4.6), (3.4.17) and (3.4.28) to obtain ∂nk ∂yk νλ ∂x ∂x mj ∂nk ∂x mλ ∂nk = g = g = −Lνλ g ∂x ∂m ∂j ∂m ∂λ ∂ν ∂μ and

1 2H(x) + 2 |ξ| |ξ|2 ∂yk ∂x 1 − 2Lνλ g μλ ξk ξ + O(|ξ|−3 ) . ∂ν ∂μ |ξ|4

σWk (x, ξ) = −i(n(x) · ξ)

(10.2.30)

10.2 Surface Potentials for Second Order Equations

645

The transformation of the symbol in terms of the tubular coordinates  and  ζ = ∂Ψ ∂ ξ now reads by the use of Lemma 7.1.17:    1  ∂ α ∂ α iζ·r  σW (x, ξ) ξ= ∂Φ ζ × −i e z=x=Ψ () ∂x α! ∂ξ ∂z

aWk,Φ (, ζ) =

|α|≥0

with r = Φ(z) − Φ(x) − ∂Φ ∂x (x)(z − x). Because of (7.1.52) and since we only ∂x need terms up to the order −2, this yields for (10.2.30) with ζλ = ξ ∂ for λ λ = 1, 2 ; ζ3 = n(x) · ξ and (10.2.20), finally   1 1  μλ ν +O(|ζ|−3 ) (10.2.31) aWk ,Φ (, ζ) = −iζ3 +2H(x) −2 g L ζ ζ μ ν| λ Γ |ξ|2 |ξ|4 where |ξ|2 = ζ32 + d2 (ζ) and d2 (ζ) = ζλ g λν ζν . With this relation (10.2.31), Theorem 9.4.3 can be applied for n → 0− and provides us with the symbols  izdz 1 1 0   =− , q0 ( , ζ ) = − 2π z 2 + d20 2 0 q−1 ( , ζ  )



c

 H(x)(z 2 + d2 ) −

=

1 2π

=

H(x) ζν Lνμ ζμ − d0 (ζ) 2d30 (ζ)

0

2 c

with

2 ν,λ,ν=1 (z 2 + d20 )2

g μν Lνλ ζμ ζν |Γ 

dz

(10.2.32) 1

d0 ( , ζ  ) := {ζλ γ λν ζν } 2 .

(10.2.33)

0 ( , ζ  ) q−1

is now the principal symbol of the acoustic double The symbol layer boundary integral operator Kk ∈ L−1 c (Γ ). 10.2.3 Surface Potentials for the Lam´ e System We begin with the Simple layer potential For the Lam´e system (10.1.16) we define the simple layer potential   Vλ(x) = N (λ × δΓ )(x) = QΓ λ(x) = E(x, y  )λ(y  )dsΓ (y  ) y  ∈Γ

with E given by (2.2.2). For x → Γ , the simple layer boundary integral operator   Γ λ(x) = E(x , y  )λ(y  )dsΓ (y  ) for x ∈ Γ V λ(x ) = lim Q Ω x→Γ

Γ

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

646

defines a pseudodifferential operator V ∈ L−1 c (Γ ).  −1 For Γ identified with x3 = 0, we get with Formula (9.4.7) for σP (ξ) given by (10.1.19) the symbol    −1  δjk 1 1 dz σp (ξ , z) dz = q−1 (x , ξ  ) = 2 2π 2π μ{z + |ξ  |2 } c c ⎛ 2 ⎞  , ξ1 ξ2 , ξ1 z ξ1 λ+μ 1 1 ⎝ξ1 ξ2 , ξ22 , ξ2 z ⎠ dz . − 2π μ(λ + 2μ) {z 2 + |ξ  |2 }2 ξ 1 z , ξ2 z , z 2 c By using the residue  1 dz 2π d20 + z 2 c zdz 1 2π z 2 + d20  c 1 dz 2π (d20 + z 2 )2 c 1 z 3 dz 2π (z 2 + d0 )2 c

formula, we have =

1 −1 d , 2 0

1 2π

=

i − , 2

=

1 −3 d , 4 0

1 2π

=

i − , 2

1 2π

 c c



z 2 dz + d20

=

1 − d0 , 2

z 2 dz 2 (d0 + z 2 )2

=

1 −1 d , 4 0

zdz + d0 ) 2

=

0.

c

z2

(z 2

(10.2.34)

This yields the complete homogeneous symbol matrix of V on Γ , ⎛ 2 ⎞ ξ1 , ξ1 ξ2 , 0 1 λ + μ ⎝ξ1 ξ2 , ξ22 , 0 ⎠ q−1 (ξ  ) = δjk − 2μ|ξ  | 4μ(λ + 2μ)|ξ  |3 0 , 0 , |ξ  |2 ⎛ 2 ⎞ −κξ1 ξ2 , 0 |ξ | + κξ22 , (λ + 3μ) ⎝ −κξ2 ξ1 , |ξ  |2 + κξ12 , 0 ⎠ = 4|ξ  |3 μ(λ + 2μ) 0 , 0 , |ξ  |2 (10.2.35) λ+μ where κ = λ+3μ . Obviously, since 0 < κ < 1, this matrix is positive definite, therefore, q−1 is a strongly elliptic symbol satisfying: Re ζ  q−1 (ξ  )ζ ≥ γ0 |ξ  |−1 |ζ|2 for all 0 = ξ  ∈ IR2 and ζ ∈ C3 (10.2.36) 1 with γ0 = 2(λ+2μ) > 0. We remark that the strong ellipticity (10.2.36) also follows from Lemma 10.2.2. For a general surface Γ we have to use the canonical coordinates (10.2.18)  and for ξ = ∂Φ ∂x ζ the relation (10.2.19). Then the principal symbol (10.1.19) reads

10.2 Surface Potentials for Second Order Equations 0 σN (, ζ) Φ ,−2

=

647

 μ{ζ32 + g νλ ζν ζλ }−2 {ζ32 + ζν g νλ ζλ }δjk (10.2.37)    ∂xj ∂xk  λ+μ nj ζ3 + g νλ ζλ · nk ζ3 + g νλ ζλ . − λ + 2μ ∂ν ∂ν 3×3

For Γ , we now compute the principal symbol of the simple layer boundary integral operator V on Γ according to Theorem 9.4.3:   1 1 1 λ+μ 0 q−1 δjk − ( , ζ  ) = × 2 2 2 2π μ{z + d0 } μ(λ + 2μ) {z + d20 }2 c

 ∂xk ∂xj × nk nj z 2 + nj zγ λν ζλ + nk zγ λν ζλ ∂ν ∂ν  ∂xj  αβ ∂xk  + γ λν ζλ γ ζα dz . ∂ν ∂β 3×3

With (10.2.34) we obtain the explicit symbol matrix of V : 0 q−1 ( , ζ  ) =

 1 1 2 λ+μ δjk − d nj nk 2μd0 4μ(λ + 2μ) d30 0  ∂x  ∂xk   j + γ ιν ζι γ αβ ζβ . ∂ν ∂α 3×3

(10.2.38)

The double layer potential for the Lam´ e system From the Betti–Somigliana representation formula (2.2.4) we now consider the double layer potential given by (2.2.8), i.e.    W ϕ(x) = Ty (x, y)E(x, y) ϕ(y)dsy for x ∈ Γ , Γ

where Ty denotes the boundary traction operator given by (2.2.5). Similar to the case of the Helmholtz operator in Section 10.2.2, the composition with the fundamental solution matrix given by (2.2.2), 

Ty E(x, y)



= NW (x, y)

 ⊂ IR3 of Γ a pseudohomogeneous defines in the tubular neighbourhood Ω Schwartz kernel generating a pseudodifferential operator AW of order −1 with symbol of rational type since both, Ty and the elastic volume potential operators are of rational type. To facilitate the computation, again we begin with the transposed operator A W which has the kernel Tx E(x, y). With the symbol of Tx ,  σTx (x, ξ) = i((λnj (x)ξk + μnk (x)ξj + μ n(x) · ξ δjk ))3×3 (10.2.39) and according to (7.1.37), the complete symbol of A W is given by

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

648

σTx (x, ξ) ◦ σNP −1 (ξ) = σA (x, ξ) W  = i λnj (x)ξk + μnk (x)ξj + μ(ξ · n(x))δjk 3×3  1  2 (λ + μ) |ξ| δk − ξk ξ × 4 μ|ξ| (λ + 2μ) 3×3  λ + μ  i λ|ξ|2 nj (x)ξ + 2μ n(x) · ξ ξj ξ 3×3 =− μ|ξ|4 λ + 2μ  i  λnj (x)ξ + μξj n (x) + μ n(x) · ξ δj 3×3 . + μ|ξ|2

(10.2.40)

For AW , the transposed operator, we then get from (7.1.34) the principal symbol 0 0  (x, ξ) = σA . (10.2.41) σA (x, −ξ) W W

Since W ϕ(x)

=

QΓ ϕ(x)

=

AW (ϕ ⊗ δΓ )

then lim W ϕ(x)

Ω x→Γ

is a pseudodifferential operator QΓ ∈ L0c (Γ ) due to Theorem 9.5.5. To compute the principal symbol of QΓ we apply Theorem 9.4.3. In particular, if xn = 0 corresponds to the surface Γ we obtain  1 0   0 q0 (x , ξ ) = σA (x , 0 ; ξ  , z)dz . W 2π c

By using (10.2.40) with (10.2.41), we first compute the entries of q0 for j = 3 and  = 3 by using n1 (x) = n2 (x) = 0 , n3 (x) = 1:   2(λ + μ) zdz z 2 dz i i q0j0 (x , ξ  ) = − ξ j ξ δ + j 2  2 2 2π z + |ξ | 2π (z + |ξ  |2 )2 λ + 2μ c

c

1 = − δj for j,  = 1, 2 2

(10.2.42)

in view of (10.2.34). If j =  = 3, then with (10.2.34)  zdz  λ (λ + μ) (λ + 2μ)  i q0330 (x , ξ  ) = − 2π z 2 + |ξ  |2 μ (λ + 2μ) μ c  z 3 dz (λ + μ) i 2 + 2  2 2 2π (z + |ξ | ) (λ + 2μ) c

1 1 1 = {λ(λ + μ) − (λ + 2μ)2 + 2(λ + μ)μ} 2 μ λ + 2μ 1 =− . 2

10.2 Surface Potentials for Second Order Equations

649

For  = 3 and j = 1, 2 we find, again with (10.2.34),   z 2 dz dz 2i (λ + μ) i j30   ξj + ξj q0 (x , ξ ) = − 2π (λ + 2μ) (z 2 + |ξ  |2 )2 2π z 2 + |ξ  |2 c

c

μ ξj i . = 2 λ + 2μ |ξ  | Similarly, for j = 3 and  = 1, 2 we obtain μ ξ i q030 (x , ξ  ) = − . 2 λ + 2μ |ξ  | To summarize, we have the principal ⎛ 0 1⎜ 1 0   q0 (x , ξ ) = − I + ⎝ 0 2 2 −iε |ξξ1 |

symbol matrix for a flat surface, ⎞ , 0 , iε |ξξ1 | ⎟ (10.2.43) , 0 , iε |ξξ2 | ⎠ , , −iε |ξξ2 | , 0

μ . where ε = λ+2μ In view of Theorem 9.4.6, the tangential differential operator T in this case is of order zero, i.e. 1 T = I, 2 whereas the skew–symmetric symbolic matrix belongs to the classical singular integral operator of Cauchy–Mikhlin type, namely     Kϕ(x ) = p.v. Ty E(x , y) ϕ(y)dsy , Γ

which is the double layer surface potential operator of linear elasticity (see also (2.2.19)). From this representation we conclude that q0 is a strongly elliptic system according to our definition (7.2.4) since inequality (7.2.16) can easily be derived for Θ(x) = −((δjk ))3×3 and with the help of 0 < ε < 1. In the same manner as for the simple layer potential (1.2.28), we can consider the double layer potential for a curved surface Γ in terms of the canonical coordinates and obtain  ∂Φ   0  −1 Φ (, ζ) = −σA (, ζ) = σ () , ζ σA A W,Φ W,Φ W ∂x  λ + μ     i 1 2 2 λν ∂xj g λ(ζ = + d )n ζ + n ζ  λ j 3 3 2 μ (ζ3 + d2 )2 λ + 2μ ∂ν    ∂x ∂xj (10.2.44) ζλ + n ζ3 g λν ζ ν + nj ζ 3 + 2μζ3 g λν ∂ν ∂ν    i 1 λν ∂xj − λn ζ + n ζ g  ν j 3 μ (ζ32 + d2 ) ∂ν    ∂x + μnj g λν ζλ + n ζ3 + μζ3 δj , ∂ν 3×3

650

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations 1

where d = (ζλ g λν ζν ) 2 . Replacing ζ3 by z and performing the contour integration of formula (9.4.7) we finally again obtain the symbol matrix of q0 ( , ζ  ) on the surface n = 0,  1 ∂x ζλ ∂xj ζλ   i  − δj − ε nj γ λν − n γ λν q00 ( , ζ  ) = 2 2 ∂ν d0 ∂ν d0 3×3 1

with d0 = {γ λν ζλ ζν } 2 . Obviously, q0 remains strongly elliptic if Γ is any smooth surface. 10.2.4 Surface Potentials for the Stokes System The principal symbols of the surface potentials of the Stokes system can be read off explicitly for the simple layer potential by taking λ → +∞ in (10.2.38) and for the double layer potential in (2.3.16) by taling ε → 0 in (10.2.43). For the pressure potential in (2.3.10), however, we insert ξj = nj ζ3 + γ νι

∂xj ζι ∂ν

into (10.1.22) and find the principal symbol  n z + γ νι ∂xj ζ j ∂xj 1 ∂ν ι −1 dz = inj + d−1 γ νι ζι , j = 1, 2, 3 (10.2.45) 2i(2π) z 2 + d20 2 0 ∂ν c

where we employed (10.2.27) and (10.2.34) and where d20 is given in (10.2.33). So, the boundary integral operator of the pressure potential belongs to L0 (Γ ).

10.3 Invariance of Boundary Pseudodifferential Operators In the previous chapters we considered boundary integral operators as pseudodifferential operators on Γ generated by generalized Newton potentials and their compositions with differential operators which are pseudodifferen by applying these to distributions of the tial operators on the domain Ω ∪ Ω special type v ⊗ δΓ . The boundary integral operators then correspond to the traces of these compositions A in the form  QΓ v(x) = T v(x) + p.f. kA (x, x − y)v(y)dsΓ (y) for x ∈ Γ . (10.3.1) Γ

The generalized Newton potential operators have symbols of rational type as established in Theorem 10.1.1, and their composition A with differential operators then have symbols of rational type as well, see Lemma 9.4.2. Hence, Theorem 9.5.7 implies the following invariance property formulated as Kieser’s theorem.

10.3 Invariance of Boundary Pseudodifferential Operators

651

Theorem 10.3.1. The boundary pseudodifferential operators of the form (10.3.1) generated by generalized Newton potentials and their compositions with differential operators are invariant under change of coordinates on Γ , i.e., the tangential differential operators T v are transformed in the usual way by applying the standard chain rule whereas the finite part integral operator transforms according to the classical rule of substitution. We remark that these invariance properties are due to the special properties of operators with symbols of rational type. They satisfy the canonical extension conditions, Definition 9.5.3, the special parity conditions (9.5.14) and the jump relations as in Theorem 9.5.5. In general, as we noticed in Theorem 8.2.2, the transformation of finite part integral operators produces extra terms under the change of coordinates if the special parity conditions are not fulfilled. Theorem 10.3.1 is of particular importance from the computational point of view. In particular, the double layer potentials in applications belong to this class and the corresponding boundary integral operators enjoy the invariance properties. These include the corresponding singular integral operators of Cauchy–Mikhlin type. Clearly, this class of operators also satisfies the Tricomi conditions (9.3.18). 10.3.1 The Hypersingular Boundary Integral Operators for the Helmholtz Equation As a further illustration of Theorem 9.5.5 we now consider the hypersingular surface potential defined by   ∂  ∂ A(v ⊗ δΓ )(x) := Ek (x, y) v(y)dsΓ (y) for x ∈ Γ ∂nx ∂ny Γ (10.3.2)  = kA (x, x − y) v(y  ) ⊗ δΓ (y) dy IR3

where the Schwartz kernel kA can be computed explicitly, kA (x, x−y) =

 1 eikr  n(x)·(x−y)n(y)·(y−x) (ikr−1)−1−(ikr−1)2 5 2π r  + r2 n(x) · n(y)(ikr − 1)

where r = |x − y|. In order to find the corresponding symbol we use (10.2.26)  has the for σWk (x, ξ); then with (7.1.37) the complete symbol of A ∈ L0c (Ω) expansion

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

652

σA (x, ξ) ∼ (−i)2

 1  ∂ β   n(x) · ξ × β! ∂ξ

|β|≥0

×



∞  k2p  ∂ β  1  ∂ α  ∂ α   −i −i n(x) · ξ . ∂x α! ∂ξ ∂x |ξ|2+2p p=0

(10.3.3)

|α|≥0

 has a symbol of rational type, Theorems 9.6.1 and 9.6.2 Since A ∈ L0c (Ω) with m = 0 provide the mapping properties of the hypersingular surface potential (10.3.2). In order to verify Theorem 9.5.5 for the operator A, particularly the jump relations and the derivation of the symbol of QΓ , we need the first terms of the expansion (10.3.3) up to the order O(|ξ|−1 ): σA (x, ξ) = −

2   (n(x) · ξ) ∇ · n  ∂n (n(x) · ξ) + i n(x) · ξ 2 ξ · ξ j |ξ|2 |ξ|2 ∂xj |ξ|4   1 + n(x) · ∇x n(x) · ξ 2 + O(|ξ|−2 ) . |ξ|

∂n = 0 and we find If we identify Γ with xn = 0 then ∂x j  z2 1 1   q(x , ξ ) = − dz + O(|ξ  |−1 ) = |ξ  | + O(|ξ  |−1 ) . 2π z 2 + |ξ|2 2 c

In this special case we conclude that for the tangential operator we have T = 0 in Theorem 9.5.5 and, moreover, lim A(v ⊗ δΓ )(x)

xn →0

= QΓ v(x ) = p.f.



kA (x , x − y  )v(y  )dsΓ (y  ) for x ∈ Γ .

Γ

L1c (Γ )

is the hypersingular boundary integral operator of Here QΓ = D ∈ the Helmholtz equation with the principal symbol q10 (x , ξ  ) =

1  |ξ | and with q00 (x , ξ  ) = 0 . 2

For a general surface Γ , again we transform A into the canonical coordinates (10.2.18) by the use of the transformation formula (7.1.50) together with Formula (7.1.52), which yields (n · ξ)2 (10.3.4) |ξ|2   ∂n  (n · ξ)  1  2H +i − (n · ξ) 2 − 2 ξj ·ξ + (n · ∇ ) n(x) · ξ x |ξ| ∂xj |ξ|4 |ξ|2    2 2  2 ∂ (n(x) · ξ) ∂ Φ(x) 1 − −i · ζ + O(|ξ|−2 ) + 2 ∂ξj ∂ξk |ξ|2 ∂xj ∂xk

σA,Φ (, ζ) = −

10.3 Invariance of Boundary Pseudodifferential Operators

where ξj = g λν

∂xj ζ λ + nj ζ 3 ∂ν

653

(10.3.5)



according to ξ = ∂Φ ∂x ζ and (10.2.19). By applying Theorem 9.4.3, we compute the first term of the symbol of QΓ by employing (9.4.7). Then we obtain the following result. Lemma 10.3.2. The hypersingular boundary integral operator Dk in (2.1.17), namely   ∂  ∂  Dk v(x ) = − p.f. Ek (x , y  ) v(y  )dsΓ (y  ) for x ∈ Γ ∂nx ∂ny Γ

has a symbol of the form ζλ ζα ζγ ∂ √ 1 i ζλ i d0 − √ ( γ γ νλ ) − g λν g αβ Gγνβ + O(d−1 q(x , ζ  ) = 0 ) 2 4 γ ∂ν d0 4 d30 =

where d0

1

{ζλ γ λν ζν } 2 .

(10.3.6)

Proof: Only the first term on the right–hand side of (10.3.4) will contribute to q10 , i.e., by employing (10.2.34) we get  z2 1 1 dz = d0 . (10.3.7) q10 ( , ζ  ) = − 2 2π z + d20 2 c

For the zeroth order terms we compute the contour integrals separately and will need the residues   z 2 dz z 3 dz 1 1 −3 1 d = and =0 (10.3.8) 0 2π (z 2 + d20 )3 16 2π (z 2 + d20 )3 c

c

in addition to those in (10.2.34). Hence,  z 1 −i2H(x) dz = −H(x) . 2π z 2 + d20

(10.3.9)

c

Moreover, from (3.4.6), (3.4.17) and (3.4.28) we obtain ∂nk ∂xj νλ ∂xk =− L ∂xj ∂ν ∂λ and

= =



∂nk z ξk 2 dz ∂xj (z + d20 )2 c  ∂xj νλ ∂xk z 1 2i ξj L ξk dz 2 2π ∂ν ∂λ (z + d20 )2 c  1 z νλ 2iζν L ζλ dz = 0 . 2π (z 2 + d20 )2

1 −2i 2π

(10.3.10)

ξj

c

(10.3.11)

654

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

Furthermore,

 ∂  (n · ∇x ) n(x) · ξ = n( ) · ξ = 0 . ∂3

(10.3.12)

Up to now we have computed all of the contributions to q00 from (7.1.50) for |α| ≤ 1. It remains to compute the contributions from terms in (7.1.50) for |α| = 2. There we first need the explicit expression ∂ ∂  (n(x) · ξ)2  nj (x)nk (x) nj (x)(n(x) · ξ) − = −2 +2 · 2ξk 2 2 ∂ξj ∂ξk |ξ| |ξ| |ξ|4 +2δjk

(n(x) · ξ)2 ξj nk (x)(n(x) · ξ) ξj (n(x) · ξ)2 + 4 − 4 2ξk |ξ|4 |ξ|4 |ξ|6

(10.3.13) for the last term in (10.3.4). Again we collect the terms of the second sum in (10.3.4) and obtain with the first term of (10.3.13) and the chain rule  ∂ 2  1 nj nk ∂ 2 Φ ·ζ =i Φ ·ζ 2 =0 2 |ξ| ∂xj ∂xk ∂n |ξ| ⎛ ⎞ 0 ∂Φ ∂Φ = since = ⎝0⎠ due to (3.4.30). ∂n ∂3 1 For the next product we find, again with the chain rule, i

 i ∂2Φ  2nj ζ3 ζ3 ∂  ∂Φ  − · ζ = −2i ·ζ 2ξ k |ξ|4 2 ∂xj ∂xk |ξ|4 ∂xk ∂n ⎛ ⎞ 0 ζ3  ∂ ⎝ ⎠  0 = −2i 4 · ζ = 0. |ξ| ∂xk 1

(10.3.14)

(10.3.15)

The third product reads −i

∂2Φ ζ2 ζ32 δjk · ζ = −i 32 Δx Φ · ζ (10.3.16) 4 |ξ| ∂xj ∂xk |ξ| ∂Φ ∂ 2 Φ  ζ2  ·ζ + = −i 34 P0 Φ − 2H |ξ| ∂n ∂n2  1 ∂ √  ∂Φ  ζ2 = −i 2 3 2 2 √ · ζ − 2Hζ3 + 0 γγ κλ (z + d0 ) γ ∂κ ∂λ   2 ∂ √ κλ 1 ζ ζ3 = −i 2 3 2 2 √ ζλ + 2iH 2 3 2 2 . γγ (z + d0 ) γ ∂κ (z + d0 )

For (10.3.16), the integration over c then gives the contribution corresponding to this third product:

10.3 Invariance of Boundary Pseudodifferential Operators

655

⎛ ⎞ ζ1 z ∂ Φ 1 ⎝ ζ2 ⎠ dz −i δ · jk 2π (z 2 + d20 )2 ∂xj ∂xk z c     ∂ √ κλ 1 z2 z3 1 1 γγ dz + 2iH dz ζλ = −i √ γ ∂κ 2π (z 2 + d20 )2 2π (z 2 + d20 )2 

2

2

c

=

1  ∂ √ κλ  ζλ −i √ γγ +H. 4 γ ∂κ d0

c

For the fourth product from (10.3.13) we obtain ∂  ∂Φ  ξj 4ξj nk (x)ζ3  i ∂ 2 Φ  − · ζ = −2i =0 ζ 3 |ξ|4 2 ∂xj ∂xk |ξ|4 ∂xj ∂n

(10.3.17)

(10.3.18)

as in (10.3.15). Finally, for the last product from (10.3.13), we have 4i

∂2Φ ζ32 ξ ξ ·ζ j k |ξ|6 ∂xj ∂xk   ∂2Φ ∂xj ∂xk ζ2  q ζλ + nj ζ3 g αβ ζ α + nk ζ 3 ζq = 4i 36 g λν |ξ| ∂ν ∂β ∂xj ∂xk  ∂xj ∂xk ∂ 2 Φq ζ2 = 4i 2 3 2 3 g λν g αβ ζλ ζα ζq (ζ3 + d0 ) ∂ν ∂β ∂xj ∂xk  ∂ 2 Φq ∂ 2 Φq ∂xj +2g λν nk ζλ ζq ζ3 + nj · nk ζ32 ζq ∂ν ∂xj ∂xk ∂xj ∂xk =: Σ1 + Σ2 + Σ3 . (10.3.19)

Now we examine each term in (10.3.19) separately. We begin with Σ1 . There we have ∂xj ∂xk ∂ 2 Φq g λν g αβ ∂ν ∂β ∂xj ∂xk  ∂  ∂Φ  ∂x q k = g λν g αβ ∂ν ∂xk ∂β ∂  ∂Φq ∂xk  ∂Φq ∂ 2 xk − g λν g αβ = g λν g αβ ∂ν ∂xk ∂β ∂xk ∂ν ∂β  ∂  ∂ ∂Φ q q r ∂xk − g λν g αβ = g λν g αβ G ∂ν ∂β ∂xk νβ ∂r from (3.4.8) with the Christoffel symbols Grνβ . ∂q Hence, with = δβq and differentiation, ∂β g λν g αβ

∂xj ∂xk ∂ 2 Φq ∂Φq = −g λν g αβ Grνβ = −g λν g αβ Grνβ δrq ∂ν ∂β ∂xk ∂xk ∂r = −g λν g αβ Gqνβ .

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

656

By applying the contour integration, the first term in (10.3.19) becomes with the residues in (10.3.8):  1 Σ1 ( ; ζ  , z)dz = 2π c   1   z3 z2 1 γ 3 dzG + dzG ζ −4ig λν g αβ ζλ ζα νβ νβ γ 2π (z 2 + d20 )3 2π (z 2 + d20 )3 c

c

ζλ ζα ζγ i − g λν g αβ Gγνβ . 4 d30

=

(10.3.20)

Next, we analyze Σ2 in (10.3.19) by using (3.4.25) and (3.4.26). We begin with ∂xj ∂  ∂Φq  2g λν ζ λ ζq ζ3 ∂ν ∂n ∂xj  ∂  ∂q  ∂Φq ∂ 2 xj  − 2g λν ζ λ ζq ζ3 = 2g λν ∂3 ∂ν ∂xj ∂ν ∂3 ∂xj ∂Φq = 0 − 2g λν Grν3 ζλ ζq ζ3 ∂r ∂xj ∂q = −2g λν Grν3 ζλ ζq ζ3 = −2g λν Grν3 δrq ζλ ζq ζ3 ∂r

Hence, 1 2π

 c

=

−2g λν Gβν3 ζλ ζβ ζ3 = (2g λν Lβν − 3 2g λν Gν3 Lβ )ζλ ζβ ζ3

=

2γ λν Lβν ζλ ζβ ζ3 for 3 = 0 .

Σ2 ( ; ζ  , z)dz =

8i 2π

 c

z 3 dz γ λν Lβν ζλ ζβ = 0 . + d20 )3

(z 2

For the last term Σ3 of (10.3.19) we get with nj n k

∂ 2 Φq ∂  ∂Φq  nk ζq ζ32 ζq ζ32 = ∂xj ∂xk ∂3 ∂xk ∂  ∂q  2 ζq ζ3 = ∂3 ∂3

∂ ∂3 nk

(10.3.21)

= 0:

∂  ∂Φq  2 n k ζ q ζ3 ∂3 ∂xk ∂ = (δ q )ζq ζ32 = 0 , ∂3 3

=

(10.3.22)

which yields Σ3 = 0. By collecting all terms, in particular, (10.3.7)–(10.3.9), (10.3.11)–(10.3.22), the proposed relation (10.3.6) follows.  Now we show that in this case the tangential differential operator T = 0. Lemma 10.3.3. For the hypersingular boundary potential operator as given in (10.3.2), the tangential operator T in the jump relation (9.4.25) vanishes: T = Tc = 0 .

10.3 Invariance of Boundary Pseudodifferential Operators

657

Proof: Here we can apply Theorem 9.4.6, i.e. (9.4.26); but we can also compute the jumps of the symbols by (9.4.27) which means nothing but replacing all the contour integrations over c in the proof of Theorem 9.4.3 by corresponding counter clockwise oriented contour integrals over the complete circle |z| = R > 2d0 . It is easy to see that these integrals corresponding to (10.3.7), (10.3.11), (10.3.20) and (10.3.21) all vanish. In addition, we see that (10.3.9) gives : z 1 −i2H(x) dz = 2H(x) 2 2π z + d20 |z|=R

whereas for the corresponding integrals in (10.3.17) we find : : z3 z2 1 1 dz = −2H(x) and dz = 0 . −iH(x) 2 2 2 2 2π (z + d0 ) 2π (z + d20 )2 |z|=R

|z|=R

So, for the hypersingular operator of the Helmholtz equation we find [q10 + q00 ]( , ξ  ) = 2T = 0 ,

(10.3.23) 

as expected. 10.3.2 The Hypersingular Operator for the Lam´ e System

Similar to the case of the Helmholtz equation, we now consider the surface potentials of the hypersingular operator D as in (2.2.31) and define    A(v ⊗ δΓ )(x) := − Tx Ty E(x, y) v(y)dsΓ (y)  =

Γ

 kA (x, x − y) v(y  ) ⊗ δΓ (y) dy for x ∈ Γ .

(10.3.24)

IR3

Here the Schwartz kernel is given by μ 1  1 3z · n(y){2μnj (x)zk 4π λ + 2μ |z|5 5 + λnk (x)zj + λz · n(x)δjk − 2 z · n(x)zj zk } |z| + 3λ{n(x) · n(y)zj zk + z · n(x)nj (y)zk }   + 2μ 3z · n(x)zj nk (y) + |z|2 nj (y)nk (x) + |z|2 n(x) · n(y)δjk − 2(μ − λ)nj (x)nk (y)|z|2 3×3 (10.3.25) where z = x − y (see Brebbia et al [44, p.191]). kA (x, x − y) =

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

658

To compute the symbol, we use the complete symbol of AW by applying (7.1.34) to the symbol of the transposed to the elastic double layer potential (10.2.40) and the symbol of the traction operator Tx to find the symbol of the hypersingular operator by employing (7.1.37) in Theorem 7.1.14: σ−Tx ◦AW (x, ξ) =   1  ∂ β    − i λnj (x)ξk + μξj nk (x) + μ ξ · n(x) δjk 3×3 × β! ∂ξ |β|≥0

 ∂ α ∂ β  1  ∂ α  × −i × −i ∂x α! ∂ξ ∂x |α|≥0



 i λ + μ  λ|ξ|2 n (x)ξm + 2μ n(x) · ξ)ξ ξm 3×3 μ|ξ|4 λ + 2μ   i  λn − (x)ξ + μξ n (x) + μ n(x) · ξ δ .  m  m m 3×3 μ|ξ|2

×

(10.3.26)

With the complete symbol available, it is clear, that the jump conditions can be investigated in the same manner as for the Helmholtz equation. Nevertheless, this is very tedious and cumbersome and we therefore shall return to the jump relations by using a different approach later on, based on a different representation formula for the double layer and the hypersingular boundary potentials (see Han [170] and Kupradze et al [253]). For simplicity, we now confine ourselves to the computation of the principal symbol of −Tx ◦ AW and of D on Γ by choosing α = β = 0 in (10.3.26). A straightforward computation yields 1  2 4 λ+μ λ |ξ| nj (x)nk (x) μ(λ + 2μ) |ξ|4    2 +2λμ|ξ|2 n(x) · ξ ξj nk (x) + nj (x)ξk + 4μ2 n(x) · ξ ξj ξk 3×3   1  2 2 − λ |ξ| nj (x)nk (x) + μ(2λ + μ) n(x) · ξ ξj nk (x) + nj (x)ξk 2 μ|ξ|  2 (10.3.27) +μ2 ξj ξk + μ2 n(x) · ξ δjk 3×3 .

0 (x, ξ) = σ−T x ◦AW

By setting n1 (x) = n2 (x) = 0, n3 (x) = 1 and ξ3 = z, we obtain 0 σαβ

=

0 σα3

=

0 σ33

=

 4μ(λ + μ) z2 z2 1  ξ α ξβ , δ + − μ αβ z 2 + d20 λ + 2μ (z 2 + d20 )2 z 2 + d20 z3 λ+μ  z  0 ξα for α, β = 1, 2 ; σ3α = 4μ − λ + 2μ (z 2 + d20 )2 z 2 + d20 z4 λ2 4μ(λ + μ)  z2  − . (10.3.28) − 2 λ + 2μ (z 2 + d20 )2 z 2 + d20 λ + 2μ −μ

Now we employ Formula (9.4.7) and use (10.2.34), (10.3.8) and

10.3 Invariance of Boundary Pseudodifferential Operators

1 2π

 c

659

z4 3 dz = − d0 2 2 2 (z + d0 ) 4

to obtain the principal symbol of the hypersingular boundary integral operator D: ⎛ 2 ⎞ |ξ | + ε1 ξ12 , ε1 ξ1 ξ2 , 0 μ 0 ⎝ ε 1 ξ2 ξ1 ⎠ (10.3.29) , |ξ  |2 + ε1 ξ22 , 0 qD,1 = 2|ξ  | 0 , 0 , (1 + ε1 )|ξ  |2 λ and, hence, |ε1 | < 1. where ε1 = λ+2μ 0 Obviously, for any ξ  ∈ IR2 with |ξ  | = 1, the matrix qD,1 is symmetric and positive definite. Consequently, inequality (9.1.19) is satisfied with Θ = ((δjk ))3×3 and γ0 = (1 − |ε1 |) > 0. Therefore the hypersingular boundary integral operator matrix D of linear elasticity is strongly elliptic and satisfies a G˚ arding inequality (9.1.21) on Γ where t = s = 12 . For the jump relation, i.e. for obtaining the operator T in (9.4.26) we could follow the calculations as for the Helmholtz equation which will be cumbersome, as mentioned before. Alternatively, we use the representation (2.2.34) of the hypersingular operator, i.e.  ∂2  1  μ p.f. v(y)dsy Dv(x)= −Tx W v(x) = − 4π ∂nx ∂ny |x − y| Γ      1 μ I M ∂y , n(y) v(y)dsy − p.f. M x, n(x) 4μ2 E(x, y) − 2π |x − y| Γ     1  μ M ∂x , n(x) p.f. IM ∂y , n(y) + v(y)dsy . (10.3.30) 4π |x − y| Γ

We notice that the first term on the right–hand side of (10.3.30) corresponds to the hypersingular boundary potential (10.3.2) of the Laplacian, i.e. of the Helmholtz equation for k = 0. For the latter, T = Tc = 0 due to Lemma 10.3.3, and from Lemma 10.3.2 we obtain its symbol in the form μq(x , ξ  )((δjk ))3×3 with q(x , ξ  ) given by (10.3.6). For the next two terms we use the property that the elements of the operator matrix   M ∂x , n(x) = mjk (∂x , n(x)) 3×3 where

  ∂ ∂ − nj (y) = −mkj ∂y , n(y) (10.3.31) mjk ∂y , n(y) = nk (y) ∂yj ∂yk

are tangential differential operators and apply Corollary 9.5.6 and Theorem 9.5.7. To facilitate the presentation we first write the second term in the form

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

660



− μ ∂x , n(x)







4μE(x, y  ) −

y  ∈Γ

  1 1 I M ∂y , n(y  ) v(y  )dsy 2π |x − y  | 1 ◦ M(v ⊗ δΓ )(x) = −μM ◦ N

where 1 f (x) = N





4μE(x, y) −

 Ω∪Ω

(10.3.32)

 1 1 I f (y)dy 2π |x − y|

1 ∈ L−2 (IR2 ) with weakly singular kernel whose is a Newton potential N c corresponding boundary potential with f (y) = w(y  ) ⊗ δΓ (y) has a continuous extension to Γ without jump (see (10.2.4)). Consequently, Corollary 9.5.6 and Theorem 9.5.7 imply, that 1 ◦ M(v ⊗ δΓ )(x) − μM ◦ N x→x ∈Γ      1 1  = −M ∂x , n(x ) 4μE(x , y  ) − M ∂y , n(y  ) v(y  )dsy 2π |x − y  | lim 



Γ

where for the corresponding tangential operator we have T = 0. For the last term we write this operator matrix in terms of its components which are compositions of tangential operators with the Newton potential V of the Laplacian, i.e.  3 μ  4π

=1  Ω∪Ω

 mk ∂x , n(x)

  1 mj ∂y , n(y) vk (y  ) ⊗ δΓ dy |x − y| =μ

3  

∂x , n(x) ◦ V ◦ mj (vk ⊗ δΓ )

(10.3.33)

=1

where we have employed the Corollary 9.5.6 for interchanging the tangential 1 we have differentiation and integration. In the same manner as for N lim

x→x ∈Γ

  μmk ∂x , n(x) ◦ V ◦ mj (vk ⊗ δΓ ) (x)    1 μ = mk ∂x , n(x ) mj ∂y , n(y  ) vk (y  )dsy ;   4π |x − y | Γ

the corresponding tangential operator T again vanishes. To conclude this section we summarize one of the main results for the hypersingular boundary potentials in elasticity in the following lemma.

10.4 Derivatives of Boundary Potentials

661

Lemma 10.3.4. The hypersingular boundary potential Dv satisfies    Ty E(x , y) v(y)dsy (10.3.34) Dv(x ) := −Tx p.f.  =

− p.f.

=

μ p.f. − 4π

Γ

Γ

  Tx Ty E(x , y) v(y)dsy   Γ

∂2 1  v(y)dsy  ∂nx ∂ny |x − y|

   − p.f. M ∂x , n(x ) 4μ2 E(x , y  ) − Γ

+ p.f.

μ 4π

 

 M ∂x , n(x )

Γ

  μ I M ∂y , n(y  ) v(y  )dsy 2π|x − y  |

   1   M ∂ , n(y ) v(y )dsy y |x − y  |

for x ∈ Γ . Moreover, D has a strongly elliptic principal symbol given by (10.3.29). We remark that the last two finite part integrals are Cauchy principal value integrals due to the corresponding Tricomi conditions (9.3.8) with m = −1 and j = |α| = 0 from (10.3.32) and (10.3.33), respectively, since all the operators involved here have symbols of rational type (see Lemma 9.4.1). 10.3.3 The Hypersingular Operator for the Stokes System In view of Section 2.3.3, in particular relation (2.3.48) between the hypersingular operator of the Lam´e and that of the Stokes system it is clear that the principal symbol of the Stokes system, in the tubular Hadamard coordinates is given by (10.3.29) with ε1 = 1 = limλ→+∞ λ/(λ + 2μ), i.e., ⎛ 2 ⎞ ξ 1 ξ2 , 0 |ξ | + ξ12 , μ 0 ⎝ ξ 1 ξ2 , |ξ  |2 + ξ22 , 0 ⎠ (10.3.35) = qD st ,1 2|ξ  | 0 , 0 , 2|ξ  |2 which is a strongly elliptic symbol matrix. Correspondingly, the hypersingular operator of the Stokes system can be obtained from (10.3.34) by sending λ → +∞, i.e., sending c → 0 in (2.3.47) and (2.3.48). Therefore, one obtains the last relation in (10.3.34) where now E(x, y) is to be chosen as in (2.3.10) according to the Stokes system.

10.4 Derivatives of Boundary Potentials In practice one often needs to compute not only the Cauchy data of the solution of boundary value problems and their potentials but also their gradients or higher order derivatives in Ω (or Ω c ) as well as near and up to the

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

662

boundary Γ . As is well known, these derivatives can be considered as linear  the vicinity of Γ . In combinations of normal and tangential derivatives in Ω, the framework of pseudodifferential operators, for derivatives of potentials, the following lemma provides us with the jump relations for the derivatives of boundary potentials based on Theorem 9.4.6.  Lemma 10.4.1. Let A ∈ Lm c (Ω) have a symbol of rational type. Further let mjk (x, ∂x ) = nk (x )

∂ ∂ − nj (x ) ∂xj ∂xk

(10.4.1)

denote the extension of the G¨ unter derivative form Γ into its neighbourhood. Then  lim mjk (x, ∂x ) ◦ A(u ⊗ δΓ ) x + n n(x ) 0>n →0   (10.4.2) mjk (x , ∂x )kA (x , x − y  ) u(y  )dsy − mjk ◦ T u(x ) = p.f. y  ∈Γ

where x ∈ Γ and T is the tangential differential operator given in Theorem 9.4.6. For the normal derivative we find  ∂ ◦ A(u ⊗ δΓ ) x + n n(x ) (10.4.3) ∂n     ∂ kA (x, x − y  ) x=x u(y  )dsy − Tn u(x ) for x ∈ Γ . = p.f. ∂nx lim

0>n →0

y  ∈Γ

Here Tn is the tangential differential operator corresponding to the new pseu∂ ◦ A with the Schwartz kernel ∂n∂ x kA (x, x − y). In dodifferential operator ∂n ∂ = ∂∂n , the operator Tn reads terms of the symbol of AΦr and ∂n Tn =

1 2

1 + 2

 0≤j≤m+2



 ∂ α     ( a , 0), ξ Dα m−j ξ=(0,1)  ∂ξ  |α|=m+2−j  ∂ α ∂    (−i)|α|+1 a (, ξ) Dα . m−j ξ=(0,1) ∂ξ  ∂n  

0≤j≤m+1 |α|=m+1−j

(−i)|α|

=( ,0)

(10.4.4) Clearly, the corresponding limits for x ∈ Ω c also satisfy the equations (10.4.2) and (10.4.3) with + instead of − in front of the tangential operators. Proof: The first limit, (10.4.2) is a special case of Corollary 9.5.6. For the second limit in (10.4.3), we apply Theorem 9.4.6 to the composed operator ∂ ∂n AΦr in the tubular coordinates, which means in the parametric half space n < 0. With the symbol of ∂∂n ⊗ AΦr obtained from (7.1.37),

10.4 Derivatives of Boundary Potentials

σ

∂ ∂ n

◦A

= iξn



am−j (, ξ)

j≥0

663

 ∂   ∂  (ξn ) − i am−j (, ξ) . ∂ξn ∂n j≥0

A straightforward application of (9.4.26) to this symbol gives the desired formula (10.4.4).  If mjk (x, ∂x )u(x) and

∂u ∂n (x)

are available, then the relation

 ∂v ∂v (x) = nk (x)mjk (x, ∂x )v(x) + nj (x) (x) ∂xj ∂n n

k=1

(10.4.5)

∂v = Dj v + nj (x) (x) ∂n (see (3.4.32) and (3.4.33)) allows one to compute arbitrary derivatives   ∂ A(u ⊗ δΓ ) x + n (x ) . 0>n →0 ∂xj lim

The general procedure for computing derivatives of the solution of the boundary value problems in terms of boundary potentials and the Cauchy data can be summarized as follows: First one computes the Cauchy data by using any kind of boundary integral equations as desired. Next, from the ∂u which either is given or can Cauchy data we compute Mu. Together with ∂n ∂u by using (10.4.5). be obtained from the Cauchy data, we then compute ∂x j This approach can be repeated and yields an appropriate bootstrapping procedure for finding higher order derivatives (see Schwab et al [381]). For computing Mu we employ the commutator [M, A](u ⊗ δΓ )(x) = (M ◦ A − A ◦ M)(u ⊗ δΓ )(x)

(10.4.6)

 which has the same order as A ∈ Lm c (Ω) due to Corollary 7.1.15 since M ∈  L1c (Ω). Correspondingly, on the manifold Γ , we have the commutator [MΓ , QΓ ] ∈ Lm+1 (Γ ) c where QΓ ∈ Lm+1 (Γ ) is defined by (9.5.4), (9.5.15). c  Theorem 10.4.2. (Schwab et al [381]) Let A ∈ Lm c (Ω) with symbol of rational type. Then the commutators in (10.4.6) in terms of the Schwartz kernel kA (x, x − y) can be expressed explicitly as follows:

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

664

[mj , A](u ⊗ δΓ )(x)    = mj (x, ∂x )kA (x, x − y) − kA (x, x − y)mj (y, ∂y ) u(y)dsy y∈Γ

  mj (x, ∂x )kA (x, z)

= Γ

+



n (x) − n (y)

 ∂  ∂  − nj (x) − nj (y) kA (x, z) z=x−y u(y)dsy ∂zj ∂z (10.4.7)

for x ∈ Γ . Moreover, for Ω  x → Γ and QΓ defined by (9.5.4) and (9.5.15), we have    [mj , QΓ ]u(x) = −[mj , T ]u(x ) + p.f. mj (x , ∂x )kA (x , z) +

Γ



 ∂  ∂  n (x )−n (y ) − nj (x )−nj (y  ) kA (x , z) z=x −y u(y  )dsy ∂zj ∂z (10.4.8) 



for x ∈ Γ . In local coordinates for x ∈ Γ ,  = χr (x ) and for AΦr ,0 , the corresponding formula reads $ ∂ % % $ ∂ , QΓ,Φr u( ) = − , TΦr u( ) (10.4.9) ∂ν ∂ν       ∂ + p.f. kAΦr ,0 ( , 0), z z=( −η ,0) u Tr (η  ) dη  , ν = 1, . . . , n − 1 . ∂ν Ur

Here kAΦr ,0 is given by (9.5.7) in terms of the parametric representation of the boundary manifold Γ . Proof: Since in (10.4.7) x ∈ Γ , differentiation and integration are interchangeable and integration by parts is allowed. To prove (10.4.8) we first apply Lemma 10.4.1 and obtain with Theorem 9.5.4 mj (x , ∂x )QΓ u(x ) = −mj (x , ∂x )T u(x )   + p.f. mj (x , ∂x )kA (x , x − y  ) u(y  )dsy . Γ

Next, for x ∈ Γ we consider the boundary potentials    Γ (mj u)(x) = kA (x, x − y  ) mj (y  , ∂y )u(y  ) dsy Q Γ

and

(10.4.10)

10.4 Derivatives of Boundary Potentials

  u(x) Q Γ





mj (y  , ∂y )kA (x, x − y  ) u(y  )dsy .

−

:=

665

Γ

For the latter we note that its kernel is a Schwartz kernel of a pseudodiffer with symbol of rational type. Moreover, (Ω) ential operator in Lm+1 c  Γ (mj u)(x) − Q   u(x) = 0 for all x ∈ Γ . Q Γ By applying Theorem 9.5.5 and subsequently Theorem 9.5.7, we then obtain  Γ (mj u) − Q   u]Γ = 0 T (mj u)(x ) − T  u(x ) = 12 [Q Γ which implies  Γ (mj u)(x) lim Q

Ω x→Γ



kA (x , x − y  )mj (y  , ∂y )u(y  )dsy

= −T (mj u) + p.f. y  ∈Γ

=

lim

Ω x→Γ 

 Γ u(x) Q 



= −T u(x ) − p.f.



mj (y  , ∂y )kA (x , x − y  ) u(y  )dsy .

y  ∈Γ

Since T  u = T mj u, we find QΓ mj u(x ) = −T mj u(x )   − p.f. mj (y  , ∂y ) kA (x , x − y  )u(y  )dsy for x ∈ Γ .

(10.4.11)

y  ∈Γ,

Hence, the commutator has the representation [mj , QΓ ]u(x ) = −[mj , T ]u(x )    + p.f. mj (x , ∂x ) + mj (y  , ∂y ) kA (x , x − y  ) u(y  )dsy . Γ

By using the definition of mjk (x, ∂x ) in (10.4.1) and z = x−y, the formula (10.4.8) follows with the standard chain rule. For the formula (10.4.9), we note that in the tubular coordinates the transformed pseudodifferential operator AΦr ,0 defines a boundary potential  Γ,Φ in the half space where n(x ) = n(y  ). Hence (10.4.9) follows from Q r ∂ kA (x , z) vanish.  (10.4.8) since the last terms involving ∂z ∂ To illustrate how to compute the derivatives ∂x on Γ , let us consider one  of the general equations (10.0.15) or (10.0.16), respectively, e.g. Equation

666

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

(10.0.15) where ϕ is given and λ is obtained from solving the system (10.0.15) for u in Ω and for λ on Γ . Then we apply mjk (x, ∂x ) to the equation (10.0.15) and obtain T u = mjk Rγ N f + RγV(mjk λ) mjk ϕ − mjk Rγ ∼ ∼

+ [mjk , RγV]λ − mjk RγWϕ

(10.4.12)

which is an integral equation on Γ for mjk λ if we compute mjk ϕ directly from the given data ϕ on Γ . Once we have mjk ϕ and mjk λ available, with the matrix N of the tangential differential operators given in (3.9.5), we obtain      mjk ϕ ϕ ∂u ∂ m−1  . + [mjk , N ] mjk u , mjk , . . . , mjk m−1 u =N λ mjk λ ∂n ∂n (10.4.13) With (10.4.5), we finally recover all the derivatives of the first order on Γ ,  ∂u ∂u |Γ = nk (x)mjk (x, ∂x )u(x) + nj (x) (x) . ∂xj ∂n n

k=1

For calculating the second or higher order derivatives, we employ a bootstrapping procedure by repeating the above procedure with ϕ and λ replaced by Dt ϕ and Dt λ (see (3.4.40)) for t = 1, . . . , m − 1 consecutively and obtain the derivatives n  n , ∂ α nk (x)mk (x, ∂x ) + n (x) u|Γ (10.4.14) Dα u|Γ = ∂n =1

k=1

for |α| ≤ 2m − 1 where 2m is the order of P . If one needs the 2m–th order derivatives on Γ , then for |α| = 2m, in  ∂ 2m (10.4.14), ∂n u cannot be computed from the boundary integral operators alone. Since u is a solution of the partial differential equation (10.1.1), P u = f , we may use this equation in the tubular coordinates along Γ in the form (3.4.49) (or (A.0.20)). By inserting (10.4.14) into one of these relations, we finally obtain with (3.4.51) 2m−1   ∂ 2m   ∂ k u  −1 P f− uΓ = (P ) ∼2m ∼k ∂nk Γ ∂n

(10.4.15)

k=0

are tangential operators of orders at most 2m − k. The derivawhere P ∼k k u ∂ for k = 0, . . . , m − 1 have been obtained by the bootstrapping tives P ∼k ∂nk procedure described above. Of course, instead of using equation (10.0.15), one may use (10.0.16) instead with the same procedure. In the following we return to the examples of the Helmholtz equation and the Lam´e system.

10.4 Derivatives of Boundary Potentials

667

10.4.1 Derivatives of the Solution to the Helmholtz Equation We begin with the representation of the solution of the Helmholtz equation,     ∂ ∂u Ek (x, y) u(y)dsy for x ∈ Ω . u(x) = Ek (x, y) (y)dsy − ∂n ∂ny Γ

Γ

(10.4.16) ∂u ∂u Here the Cauchy data are ϕ = u|Γ and σ = ∂n |Γ . If σ = ∂n |Γ is given in H s−1 (Γ ) then we first solve for ϕ the integral equation of the second kind (2.1.11) ( or of the first kind with the hypersingular operator, respectively, see Table 2.1.1). Equation (2.1.11) reads     ∂ 1 ϕ(x) + Ek (x, y) ϕ(y)dsy = Ek (x, y)σ(y)dsy ; (10.4.17) 2 ∂ny Γ

Γ

and ϕ ∈ H s (Γ ) due to the mapping properties of the boundary integral operators and to the shift theorem 9.7.3. Then we may apply the operator mj (x, ∂x ) on u(x) in (10.4.16),   mj (x, ∂x )Ek (x, y) σ(y)dsy ϕj (x) := mj (x, ∂x )u(x) =  −

Γ



mj (x, ∂x )

∂ Ek (x, y) u(y)dsy for x ∈ Ω . ∂ny

(10.4.18)

Γ

Taking the limit x → Γ , we obtain from (10.4.18) a resulting boundary integral equation of the second kind for the desired tangential derivative ϕj (x) = mj (x, ∂x )u(x):    ∂ Ek (x, y) ϕj σ(y)dsy ∂ny Γ   mj (x, ∂x )Ek (x, y) σ(y)dsy − ([mj , Kk ]ϕ)(x) = p.f.

1 ϕj (x) + 2

(10.4.19)

Γ

where ϕ is now the solution of the boundary integral equation (10.4.17) and σ is given. Here [mj , Kk ] is the commutator given by (10.4.20) ([mj , Kk ]ϕ)(x) = (mj ◦ Kk − Kk ◦ mj )ϕ(x)    ∂ ∂ = p.f. mj (x, ∂x ) Ek (x, y) − Ek (x, y)mj (y, ∂y ) ϕ(y)dsy ∂ny ∂ny Γ    ∂ ∂ = p.f. Ek (x, y) + mj (y, ∂y ) Ek (x, y) ϕ(y)dsy mj (x, ∂x ) ∂ny ∂ny Γ

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

668

which is a weakly singular integral operator on ϕ due to Theorem 10.4.2. Now, ϕj (x) can be obtained by solving the integral equation (10.4.19) of the ∂ on Γ can be obtained from (10.4.5). In second kind and all derivatives ∂x j fact, in the following we show directly that the kernel of the commutator in (10.4.20) is of order O(r−1 ) where r = |x − y|. To this end, let us write ∂Ek (x, y) = (y − x) · n(y)f (r) ∂ny where f (r) =

1 ikr (ikr 4πr 3 e

− 1) = O(r−3 ) and r = |x − y|. Then

∂Ek  = (y − x) · n(y) mj (x, ∂x ) + mj (y∂y ) f (r) mj (x, ∂x ) + mj (y, ∂y ) ∂ny  + f (r) mj (x, ∂x ) + mj (y, ∂y ) (y − x) · n(y) . (10.4.21)



For the first term it is easy to show that it is weakly singular for smooth Γ since  (y − x) · n(y) mj (x, ∂x ) + mj (y, ∂y ) f (r)  ∂  ∂  f (r) = (y − x) · n(y) nj (x) − nj (y) − n (x) − n (y) ∂y ∂yj =

O(r2 ) · O(r−3 ) = O(r−1 ) .

(10.4.22)

For the second term we use the chain rule and Taylor expansion about y to obtain   mjk (x, ∂x ) + mjk (y, ∂y ) n(y) · (y − x) (10.4.23)   ∂n ∂n (y) − nj (y) (y) · (y − x) = −nk (x)nj (y) + nj (x)nk (y) + nk (y) ∂yj ∂yk n   ∂nk ∂nj ∂n ∂n   (y − x ) + O(r2 ) . nj − nk + nk − nj = ∂y ∂y ∂yj ∂yk  y

=1

From (3.4.6) with ∂n ∂yk

∂nk ∂n

=

= 0 and (3.4.28) we obtain n−1 

μ,λ=1

=

−

n−1  ∂yk ∂yk μλ ∂n ∂y g =− g μλ Lνλ ∂μ ∂λ ∂μ ∂ν

n−1 

μ,λ,ν=1

∂yk μν ∂y ∂nk L = ∂ ∂ ∂y μ ν μ,ν=1

(10.4.24)

since Lμν = Lνμ , where we have employed (3.4.17). Inserting (10.4.24) into (10.4.23), we find

10.4 Derivatives of Boundary Potentials

669



 mjk (x, ∂x ) + mjk (y, ∂y ) n(y) · (y − x) n n−1    ∂y μν ∂yk ∂y μν ∂yj = − nj (y) L + nk (y) L ∂μ ∂ν ∂μ ∂ν =1 μ,ν=1  ∂yj μν ∂y ∂yk μν ∂y (y − x ) + O(r2 ) −nk (y) L + nj (y) L ∂μ ∂ν ∂μ ∂ν =

O(r2 ) .

(10.4.25)

−3

With f (r) = O(r ) and collecting (10.4.25) and (10.4.22) we finally obtain that (10.4.21), i.e. the kernel of (10.4.20), is of order O(r−1 ). Since ϕ and σ on Γ are known, boundary integral equation (10.4.19) is the same as (10.4.17), only the right–hand side has been modified. Clearly, if we apply the operator mrs again, now to equation (10.4.19), we may apply the same arguments as before to compute higher order tangential derivatives of ϕ. For higher order normal derivatives we use the Helmholtz equation in the form (10.4.15) and later on apply mrs to the given data ∂u |Γ . σ = ∂n 10.4.2 Computation of Stress and Strain on the Boundary for the Lam´ e System In applications one is not only interested in the boundary traction and boundary displacement of the elastic field but also in the complete stress and strain on Γ . We now show how the procedure for computing derivatives of potentials in this section can be applied to this situation. As in the case of the Helmholtz equation let us begin with the representation formula (2.2.4) with f = 0,     Ty E(x, y) ϕ(y)dsy for x ∈ Ω (10.4.26) v(x) = E(x, y)t(y)dsy − Γ

Γ

where t(y) = Ty v(y) now denotes the boundary traction and ϕ = v|Γ the boundary displacement. We assume that both are known from solving (2.2.16) or (2.2.17) for the relevant missing Cauchy data with σ = t. To obtain the complete stress or strain up to the boundary, we need all of the first ∂v derivatives ∂xji on Γ . For this we need to compute ϕjk := mj ϕk on Γ which can be obtained from one of the boundary integral equations, e.g. from (2.2.16), i.e. ϕj (x)

=

( 12 I − K)ϕj



(10.4.27) mj (x, ∂x )E(x, y)t(y)dsy for x ∈ Γ .

−[mj , K]ϕ + p.v. Γ 

Here ϕj = (ϕj , . . . , ϕjn ) is the desired new unknown. Note that this is the same system of singular integral equations for all the ϕj as in (2.2.16)

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

670

for ϕ, with only modified right–hand sides. In particular, we note that the commutator     [mj , K]ϕ(x) = p.v. mj (x, ∂x ) + mj (y, ∂y ) Ty E(x, y) ϕ(y)dsy Γ

(10.4.28) is of the same type singularity as K, hence, it is an operator of the class L0c (Γ ). We shall further discuss the commutator in more detail at the end of this section. After solving (10.4.27) for ϕjk with , j, k = 1, . . . , n, we now may compute the stress tensor σij and the strain tensor εij on Γ as follows: If ϕjk on Γ are available, we introduce the surface divergence operator  ∂ ∂  DivΓ ϕ = np ϕrpr = np np ϕr − nr ∂xr ∂xp (10.4.29)  ∂v  = divv − n · on Γ . ∂n Γ Here and in what follows we use again the Einstein summation convention. Then the boundary traction (2.2.5) can be rewritten in the form ∂v + μϕrr t = (λ + μ)(divv)n + μ ∂n  ∂v  n + μϕrr . = (λ + μ) DivΓ ϕ + n · ∂n

(10.4.30)

Multiplication by n and summation lead to the identity (λ + 2μ)n ·

∂v |Γ = n · t − λDivΓ ϕ ∂n

(10.4.31)

from which with (10.4.30) we obtain μ

∂v λ+μ λ+μ |Γ = t − n · tn − 2μ DivΓ ϕn − μϕrr . ∂n λ + 2μ λ + 2μ

(10.4.32)

Now the stress tensor on Γ can be written in the form   ∂v ∂vj  ∂v  i σij = λdivvδij + μ = λ DivΓ ϕ + n · δij + ∂xj ∂xi ∂n  ∂v ∂vi   ∂vj ∂vj  ∂vi ∂vj  i + + nj + ni +μ − nj − ni ∂xj ∂n ∂xi ∂n ∂n ∂n λ = λ(DivΓ ϕ)δij + (n · t − λDivΓ ϕ)δij λ + 2μ + μ(nq ϕjqi + nq ϕiqj + nj ϕrir + ni ϕrjr )   λ+μ 2μ(λ + μ) n · t − ni DivΓ ϕ + nj ti − ni λ + 2μ λ + 2μ   λ+μ 2μ(λ + μ) n · t − nj DivΓ ϕ ; + ni tj − nj λ + 2μ λ + 2μ

10.4 Derivatives of Boundary Potentials

671

or explicitly in terms of the computed ϕjk as σij | = Γ

 1 (n · t + 2μDivΓ ϕ) λδij − 2(λ + μ)ni nj λ + 2μ + nj ti + ni tj + μ(nr ϕjri + nr ϕirj + nj ϕrir + ni ϕrjr ) .

(10.4.33)

Then from Hooke’s law we also obtain the strain tensor on Γ , i.e. 2μεij |

Γ

= =

∂vi  ∂xi ∂xj Γ μ(nk ϕikj + nk ϕjki + nj ϕrir + ni ϕrjr ) (10.4.34) λ+μ +ni tj + nj ti − 2 (n · t + 2μDivΓ ϕ)ni nj . λ + 2μ μ

 ∂v

j

+

To this end, for three–dimensional problems with n = 3, let us first rewrite the kernel of K according to our formula (2.2.30):   kK (x, x − y) = (Ty E x, y)   1 1  ∂  1 1 I + mst (y, ∂y ) = ∂ny 4π r 4π r 3×3  − 2μ msr (y, ∂y )Ert (x, y) 3×3 where E(x, y) =



Ert (x, y)

3×3

(10.4.35)

. Then the elements of the commutator

[mj , Kst ]ϕt (x) =    ∂ 1 1  ϕs (y)dsy mj (x, ∂x ) + mj (y, ∂y ) ∂ny 4π r Γ   1 1    ϕt (y)dsy mj (x, ∂x ) + mj (y, ∂y ) mst (y, ∂y ) + p.v. 4π r Γ    − 2μp.v. mj (x, ∂x ) + mj (y, ∂y ) msr (y, ∂y )Ert (x, y) ϕt (y)dsy Γ

for s, j,  = 1, 2, 3. Now we examine each term on the right–hand side. The first term corresponds to the commutator (10.4.20) for k = 0 and is a weakly singular kernel defining on Γ a pseudodifferential operator in the class L−1 c (Γ ). The next two terms, however, define kernels of classical singular integral operators which are in the class L0c . For the kernel of the second term we have

672

10. Integral Equations on Γ ⊂ IR3 Recast as Pseudodifferential Equations

1  1 mj (x, ∂x ) + mj (y, ∂y ) mst (y, ∂y ) 4π r  1 1 = mst (y, ∂y ) mj (x, ∂x ) + mj (y, ∂y ) 4π r 1 1  mj (y, ∂y )mst (y, ∂y ) + mst (y, ∂y )mj (y, ∂y ) + 4π r  ∂  ∂ 1 1 mst (y, ∂y ) n (y) − n (x) + nj (y) − nj (x) = 4π ∂yj ∂y r ! " 1 1 mj (y, ∂y ) , mst (y, ∂y ) + 4π r with the commutator [mj , mst ]|Γ . If we employ (10.4.24) on Γ then [mj (y, ∂y ) , mst (y, ∂y )]|Γ ∂nt ∂nk ∂n ∂nj = mk (y, ∂y ) + mtj (y, ∂y ) + mjk (y, ∂y ) + mt (y, ∂y ) ∂yj ∂y ∂yt ∂yk defines a tangential differential operator of the first order on Γ . Collecting terms, we conclude that for the second term we have a singular integral operator in the form    1 1 p.v. mjk (x, ∂x ) + mjk (y, ∂y ) mst (y, ∂y ) ϕt (y)dsy 4π r Γ   1 1  mst (y, ∂y ) p.v. n (y) − n (x) (yj − xj ) = 4π r Γ   + nj (y) − nj (x) (y − x ϕt (y)dsy    1 1 [mj (y, ∂y ) , mst (y, ∂y )] ϕt (y)dsy . p.v. + 4π r Γ

The last term of the commutator can be treated in the same manner resulting in the singular integral operator    mj (x, ∂x ) + mj (y, ∂y ) msr (y, ∂y )Ert (x, y) ϕt (y)dsy −2μ p.v. Γ

= −2μ p.v.

  mst (y, ∂y ) Γ

 ∂Ert (x, y) n (y) − n (x) ∂yj

 ∂Ert (x, y)  ϕt (y)dsy + nj (y) − nj (x) ∂y    − 2μ p.v. [mj (y, ∂y ) , mst (y, ∂y )]Ert (x, y) ϕt (y)dsy . Γ

10.5 Remarks

673

10.5 Remarks For higher order tangential derivatives of ϕ we may repeatedly apply mj (y, ∂y ) to (10.4.27) again. Of course, the same bootstrapping procedure can be used for computing tangential derivatives of the boundary tractions t via one of the boundary integral operators (2.2.16) or (2.2.17) between the Cauchy data. For computing higher order normal derivatives, we need to employ the corresponding formula (10.4.15) for the Lam´e system in local coordinates together with their normal derivatives as desired. In concluding this section, we see how we can compute the derivatives of the boundary potentials on the boundary by solving the same boundary integral equations for the tangential derivatives as for the Cauchy data by appropriate modifications of the right–hand sides but without additional effort. This means that available computer packages can be adopted with only slight modifications. Moreover, from the mathematical point of view, the boundary integral equations for the tangential derivatives in the variational formulation satisfy the same G˚ arding inequalities as the bilinear forms of the original boundary integral equations.

11. Boundary Integral Equations on Curves in IR2

In Chapter 10 we presented the essence of boundary integral equations recast as pseudodifferential operators on boundary manifolds Γ ⊂ IRn for n = 3. In this chapter we present the two–dimensional theory of classical pseudodifferential and boundary integral operators based on Fourier analysis. In general, the representations of boundary potentials are based on the local charts and local coordinates (3.3.3)–(3.3.5). For n = 2, however, every closed Jordan curve Γj as a part of Γ admits a global parametric representation (4.2.45). Moreover, any function defined on a closed curve can be identified with a 1– periodic function on IR. These global representations allow one to use Fourier series expansions in the theory of pseudodifferential operators. The latter lead to explicit expressions in terms of corresponding Fourier coefficients. Moreover, the Sobolev spaces on Γ can be characterized in terms of the function’s Fourier coefficients as in the Lemmata 4.2.4, 4.2.5 and Corollary 4.2.6. If the boundary Γ consists of p distinct, simply closed, smooth curves Γj , j = 1, . . . , p, then for each curve Γj we may consider the periodic extension of a function defined on Γj and identify the original function on Γ with a p–vector valued periodic function. For simplicity, we first consider Ω a bounded, simply connected domain having one simply closed smooth boundary curve Γ which admits a global regular parametric 1–periodic representation (3.3.7) or (4.2.45), i.e. Γ : x = T (t) for t ∈ IR and T (t + 1) = T (t) (11.0.1)  dT  satisfying  dt  ≥ c0 > 0 for all t ∈ IR. Hence, we need just one global chart for the representation of Γ , and the local representation in Section 9.1 can now be identified with the global representation and t stands for  . The  of Γ now also has one global chart defined by tubular neighbourhood Ω x = Ψ (t, n ) = T (t) + n n(t)

(11.0.2)

 Clearly, all of the formulations in for t ∈ IR , n ∈ [−ε, ε] and x ∈ Ω. Chapter 9 in terms of local charts (Or , Ur , χr ) can now be reduced to global  , Ur = IR × (−ε, ε) and χr the formulations based on (11.0.2), i.e. Or = Ω inverse to T . © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_11

676

11. Boundary Integral Equations on Curves in IR2

11.1 Representation of the basic operators for the 2D–Laplacian in terms of Fourier series Before we give the general formulations concerning the pseudodifferential operators on Γ in terms of Fourier series expansions let us illustrate the main ideas by analyzing the four basic boundary integral operators V, K, K  and D of the two–dimensional Laplacian as introduced in Section 1.2. Further examples can be found in [197] and will also be treated below. In particular, we want to compute their symbols in terms of Fourier coefficients as well as the action of these operators on Fourier series. For the simple layer potential V on the closed curve Γ , parameterized globally according to (11.0.1), we write  1 1 σ(y)dsy log (11.1.1) V σ(x) = 2π |x − y| Γ

=

−

1 4π

1

 1 log 4 sin2 π(t − τ ) σ (τ )dτ + 4π

0

 k∞ (t, τ ) σ (τ )dτ Γ



where x = T (t) , y = T (τ ) , σ (τ ) = σ T (τ ) k∞ (t, τ ) := 2 log

dsy dτ

, and

 |2 sin π(t − τ )|  |T (t) − T (τ )|

(11.1.2)

by using the identity |ei2πt − ei2πτ | = 2| sin π(t − τ )| in the representation (11.1.1). Moreover, we let τ L be the arc length parameterization on Γ where L is the length of Γ . For Γ ∈ C ∞ , the kernel k∞ ∈ C ∞ (Γ × Γ ) is biperiodic and defines a smoothing operator in L−∞ (Γ ). If the periodic function σ  is represented by its Fourier series expansion σ (τ ) =



1 σ * e

with σ * = L

2πiτ

=Z

e−2πiτ σ(τ )dτ

(11.1.3)

0

(see (4.2.46) and (4.2.47)) then we obtain the Fourier series representation of the simple layer potential operator V , 

V σ( T (t)



=

1 − 4π

1

  log 4 sin2 π(t − τ ) σ * e2πiτ dτ ∈Z

0

 1  + k∞ (t, τ )e2πiτ dτ σ * 4π =

 =0

∈Z Γ

1 σ * e2πit + (K∞ σ)(t) . 4π||

(11.1.4)

11.1 Fourier Series Representation of the Basic Operators

677

By using the Fourier series form of the Sobolev space norms from Lemma 4.2.4 we see from (11.1.4) that for σ  ∈ H s (Γ ), 2   1  V σ  − K∞ σ2H s+1 (Γ ) = σ *  ||2s+2  4π|| =0  1 2   L 2 ≤ |* σ |2 ( + δ0 )2s = σ2H s (Γ ) . 4π 4π ∈Z

This implies the mapping property V σH s+1 (Γ ) ≤ c1 σH s (Γ ) + c2 (s, s0 )σH s0 (Γ )

(11.1.5)

for any s0 ≤ s; and the order of V is −1 as this explicit computation shows. Consequently, in view of Parseval’s equality, (11.1.4) also implies L2

1 σ2 − 1 H 2 (Γ ) 4π

=

1  1 1 |* σ |2 + |* σ0 | 2 4π || 4π =0 ∈Z

=

V, σ, σL2 (Γ ) + Cσ, σL2 (Γ )

(11.1.6)

with the compact operator C defined by L Cσ = −K∞ σ + 4π

1 σdτ . 0

Clearly, (11.1.6) is G˚ arding’s inequality for V and V is strongly elliptic. The symbol of V can be computed from Theorem 7.1.11 since V is in −1 L−1 c (Γ ) because it stems from the trace of the two–dimensional operator Δ 2 having symbol of rational type in IR and, moreover, is properly supported since Γ is compact. Formula (7.1.27) then yields, for  1 1 ψ(|x − y|)σ(y)dsy + RΓ σ log (11.1.7) V σ(x) = 2π |x − y| Γ

with an appropriate cut–off function ψ and RΓ having a C ∞ kernel, the symbol in the global parameterization can be computed in the form aψ (t, ξ) = e

−itξ

1  0

1 =− 2π

 IR

1 − 2π



IR

−

 1 log |2 sin π(t − τ )| ψ(|t − τ |)eiξτ dτ 2π



ψ(|t − τ |)e−iξ(t−τ ) log |t − τ | dτ

 2 sin π(t − τ )    e−iξ(t−τ ) log  ψ(|t − τ |)dτ . t−τ

678

11. Boundary Integral Equations on Curves in IR2

Since the last term on the right–hand side is a C ∞ -function with compact support, its Fourier transform is C ∞ and decays faster than any order of |ξ|−N for N ∈ IN according to the Paley–Wiener–Schwartz Theorem 3.1.3. Hence, for the symbol we only need to consider the first term, which is given by  1 eiξz log |z|ψ(|z|)dz . (11.1.8) aψ (t, ξ) = − 2π IR

According to Definition 8.1.1, the kernel k0 (x, z) = log |z| in (11.1.8) is a pseudohomogeneous function of degree 0. Therefore, for finding the complete homogeneous symbol we employ formula (8.1.29) and find a−1 (t, ξ) = lim aψ (t, ξ) . →+∞

(11.1.9)

However, it is not clear how to compute this limit without detailed information on the oscillating integral in (11.1.8). We therefore integrate by parts and obtain  1  1 1 p.f. eiξz ψ(|z|) + log |z|ψ  dz aψ (t, ξ) = iξ 2π z IR  1 1 = p.f. eiξz ψ(|z|)dz 2πiξ z IR  1 1 + ψ  + log |z|ψ  dz . 2 2πξ z 1 2 ≤|z|≤a

Now the limit in (11.1.9) can be performed and yields with the Fourier transform of z −1 (see Gelfand and Shilov [143, p.360])    1 a−1 (t, ξ) = lim p.f. eizξ ψ(|z|)dz →+∞ 2πiξ z IR     1 1 iξz 1   + ψ e + log |z|ψ dz 2π ξ 2 z 1 2 ≤|z|≤1

 1 1  |z|  p.f. dz eizξ ψ →∞ 2πiξ z  IR  1 1 1 1 = p.f. iπ sign ξ = . eizξ dz = 2πiξ z 2πiξ 2|ξ|

= lim

IR

Hence, a−1 (t, ξ) =

1 2|ξ|

(11.1.10)

11.1 Fourier Series Representation of the Basic Operators

679

is the complete symbol of V and is homogeneous of order −1. Consequently, we may write V according to Formula (7.1.7), i.e.,  V σ T (t) = LFξ−1 (11.1.11) →t a(t, ξ)Fτ →ξ σ(τ ) + R∞ σ 1 χ(ξ) with a cut–off function χ as in (7.1.5). With the where a(t, σ) = 2|ξ| Fourier series representation of σ , we have  1    V σ T (t) = Fξ−1 χ(ξ)Fτ →ξ σ * ei2πτ + R∞ σ →t 2|ξ| ∈Z  1   −1 χ(ξ)Fτ →ξ ei2πτ + R∞ σ .(11.1.12) σ * Fξ→t = 2|ξ| ∈Z

By using the identity Fτ →ξ ei2πτ =

√ 2πδ(ξ − 2π) for  ∈ Z ,

(11.1.13)

(see Gelfand and Shilov [143, p.359]) we obtain  V σ T (t)

=



σ * Fξ−1 →t

√

2π

∈Z

=



σ * a(t, 2π)e

 χ(ξ) δ(ξ − 2π) + R∞ σ 2|ξ|

i2πt

∈Z

+

 ∈Z

(11.1.14)

1 σ * L

k∞ (t, τ )e2πτ dτ

τ =0

where a(t, ξ) = χ(ξ) 2|ξ| is a symbol of V . In comparison with (11.1.4), we see that the two representations are identical. This suggests that for periodic functions σ and A ∈ Lm c (Γ ) we may have a representation in the form   a(t, 2π)* σ ei2πt + R∞ σ (11.1.15) Aσ T (t) = ∈Z

where a ∈ S m c (Γ × IR) is a representative of the classical symbol of A. We shall return to this point in Section 11.2 For the double layer potential operators K and K  in IR2 , we first represent Γ in terms of its arc length parameterizations with s = Lt. Then x = T (t) , y = T (τ ) and dT (t) (11.1.16) dt d2 T (t − τ )3 d3 T (t − τ )4 d4 T (t) + (t) − (t) + TR (t, τ ) 2 3 dt 3! dt 4! dt4

T (t) − T (τ ) = (t − τ ) −

(t − τ )2 2

680

11. Boundary Integral Equations on Curves in IR2

where the remainder TR ∈ C ∞ (Γ × Γ ) and is of the form TR = O5 = (t − τ )5 f∞ (t, τ ), where f∞ (t, τ ) is a smooth function in C ∞ in both variables. We recall (3.4.20) in IR2 which yields L11 = κ with the curve’s curvature K = κ; and with (3.4.17) and (3.4.18) we obtain Frenet’s formulae dt dn d2 t dκ = Lκn , = −Lκt and = L n − L 2 κ2 t dt dt dt2 dt

(11.1.17)

with t the unit tangent and n the unit normal vectors of Γ . For ease of reading, we introduce the generic notation O = (t − τ ) f∞ (t, τ ) ,  ∈ IN0 where f∞ (t, τ ) is a smooth function in C ∞ of both variables. Then we obtain for (11.1.16) the canonical representation of Γ ,  dκ  x − y = (t − τ )L − 16 (t − τ )3 L3 κ2 (t) + 18 (t − τ )4 L3 κ t(t) dt   d2 κ  dκ 1 − 12 L2 (t − τ )2 κ − 16 (t − τ )3 L2 + 24 (t − τ )4 L2 2 − L4 κ3 n(t) + O5 . dt dt (11.1.18) Then, as a consequence we obtain $ % dκ 1 1 (t − τ )2 L2 κ2 (t) + 12 (t − τ )3 L2 κ (t) + O4 , r2 = |y − x|2 = L2 (t − τ )2 1 − 12 dt (11.1.19) and, together with the Frenet formulae (11.1.17), we have  dκ  n(y)= 1 − 12 L2 κ2 (t)(t − τ )2 + 21 (t − τ )3 L2 κ (t) n(t) (11.1.20) dt  dκ 1 2 + Lκ(t)(t − τ ) − L (t) 2 (t − τ ) dt   d2 κ + 16 (t − τ )3 L 2 (t) − L3 κ3 (t) t(t) + O4 , dt  dκ  2 21 2 (11.1.21) t(y)= 1 − L κ 2 (t − τ ) + 12 (t − τ )3 L2 κ (t) t(t) dt  dκ − Lκ(t − τ ) − L (t) 12 (t − τ )2 dt   d2 κ 3 1 + 6 (t − τ ) L 2 (t) − L3 κ3 (t) n(t) + O4 . dt Moreover, dκ 1 1 n(y) · (y − x) = − κ(t)L2 (t − τ )2 + (t − τ )3 L2 (t) + O4 . 2 3 dt

(11.1.22)

Hence, with (11.1.19) and (11.1.22) the kernel of the double layer potential reads

11.1 Fourier Series Representation of the Basic Operators

1 ∂ log |x − y| 2π ∂ny  L2 1  L2 2 3 dκ (t − τ ) (t − τ ) (t) + O κ(t) − = 4 2πr2 2 3 dt 1 dκ 1 κ(t) − (t − τ ) (t) + k∞,1 (t0 , t) = k∞ (t, τ ) = 4π 6π dt

681

−

(11.1.23)

with a doubly periodic kernel function k∞ ∈ C ∞ (Γ × Γ ). Consequently, Kϕ has the representation 1





k∞ (t, τ )ϕ T (τ ) dτ =

Kϕ(t) = L

1 

1 ϕ * L

∈Z

0

k∞ (t, τ )e2πiτ dτ

(11.1.24)

0

where K is a smoothing operator for Γ ∈ C ∞ . Clearly, the adjoint operator K  has the corresponding representation 1



K σ(t) = L

 k∞ (τ, t)σ T (τ ) dτ

(11.1.25)

0

with the same smooth kernel k∞ . The symbol of both operators, therefore, is equal to zero. For the hypersingular operator D of the Laplacian, as in (1.2.14), we use the Fourier series representation (11.1.4) of V and obtain   1 d  dϕ  V = L−1 π ϕ * ||e2πit + (K1∞ ϕ)(t) (11.1.26) Dϕ T (t) = − 2 L dt dτ ∈Z

where

d dϕ  K∞ (t) . dt dτ In the same manner as for the simple layer potential V , we compute K1∞ ϕ(t) = L−2

(Dϕ − K1∞ ϕ)2H s−1 (Γ ) = L−2 π 2 =L

−2 2

π



|ϕ * |2 ||2s

∈Z 2 (ϕH s (Γ ) − |ϕ *0 |2 )

≤ L−2 π 2 ϕ2H s (Γ ) .

(11.1.27)

This inequality implies the mapping property of D, namely DϕH s−1 (Γ ) ≤ c1 ϕH s (Γ ) + c2 (s, s0 )ϕH s0 (Γ )

(11.1.28)

for any s0 ≤ s. Hence, the order of D is +1. For the G˚ arding inequality we also obtain from (11.1.27) with Parseval’s equality,

682

11. Boundary Integral Equations on Curves in IR2

1 Dϕ, ϕL2 (Γ ) = L

(Dϕ)(t)ϕ(t)dt 0

= Lπ



|ϕ * |2 | + δ0 | + Cϕ, ϕL2 (Γ )

(11.1.29)

∈Z

= Lπϕ2

1

H 2 (Γ )

+ Cϕ, ϕL2 (Γ ) ,

where the smoothing operator is defined by 1 Cϕ = K∞ ϕ − L π 2

 ϕ T (t) dt .

0

This equality (11.1.29) verifies the strong ellipticity for the hypersingular operator D. On the other hand, from (1.2.14) we have with (11.1.12)  d  dϕ  Dϕ T (t) = − T (t) V dsx ds 1 d  dϕ  V (t) =− 2 L dt dτ  χ(ξ) dϕ dϕ  1 d Fξ−1 F + R (t) =− 2 τ →ξ ∞ →t L dt 2|ξ| dτ dτ  χ(ξ)  dϕ  1 d 1 2 (iξ) R (t) . F ϕ(τ ) − = − 2 Fξ−1 τ →  ξ ∞ →t L 2|ξ| L2 dt dτ Hence, the symbol of D is given by σD (t, ξ) =

1 χ(ξ) |ξ| . L 2

Substituting into (11.1.26), we have   ϕ * σD (t, 2π)ei2πt + R1∞ ϕ(t) , Dϕ T (t) =

(11.1.30)

(11.1.31)

∈Z

which again is of the form (11.1.15).

11.2 The Fourier Series Representation of Periodic Operators A ∈ Lm c (Γ ) We return to the general formula (11.1.15) and will show that this is indeed the general representation form for any periodic classical pseudodifferential m operator A ∈ Lm c (Γ ). According to Theorem 8.1.8, any given A ∈ Lc (Γ ) has the representation

11.2 The Fourier Series Representation of Periodic Operators A ∈ Lm c (Γ )

 k(x, x − y)v(y)dsy +

Av(x) = p.f. Γ

 ∂ j aj (x) − i v(x) ∂sx j=0

m 

683

(11.2.1)

where it is understood that the last term only appears when m ∈ IN0 . The Schwartz kernel belongs to the class Ψ hkκ (Γ ) with κ = −m − 1. In terms of the periodic representation (11.0.1), with v(t) := v T (t) , the operator takes the form 1 AΦ v(t) = p.f.

kΦ (t, t − τ ) v (τ )dτ +

m 

 aj (t)

 −id j

j=0

τ =0

dt

v(t) ,

(11.2.2)

where v,  aj and the Schwartz kernel kΦ are 1–periodic. Again, the last term only appears for m ∈ IN0 and, moreover, the kernel kΦ and the coefficients  aj (t) are obtained from the local transformation from sx to t. A symbol aΦ ∈ S m c (Γ ×IR) (see (7.1.39)) of AΦ is given by formula (7.1.27) in Theorem 7.1.11, i.e., aΦ (t, ξ)

= =

e−itξ (AΦ eiξ• )(t) 1 m  e−iξ p.f. kΦ (t, t − τ )eiξτ dτ +  aj (t)ξ j (11.2.3) j=0

τ =0

since A is properly supported because of the compact curve Γ . With this symbol, A can also be written in the form AΦ v(t)



1

=

1 2π

=

Fξ−1  + R∞ v(t) →t aΦ (t, ξ)Fτ →ξ v

ei(t−τ )ξ aΦ (t, ξ) v (τ )dτ dξ + R∞ v(t) IR τ =0

(11.2.4)

for any v ∈ C0∞ ([0, 1]) ⊂ C0∞ (IR) where R∞ is an appropriate smoothing operator. Theorem 11.2.1. Every operator A ∈ Lm c (Γ ) has a Fourier series representation in the form   aΦ (t, 2π)* v ei2πt + R∞ v (11.2.5) Av T (t) = ∈Z

where aΦ ∈ S m c (T × IR) is a classical symbol of AΦ given by (11.2.3), and 1 v* := L

 e−2πiτ v T (τ ) dτ

for  ∈ Z

(11.2.6)

0

are the Fourier coefficients of v(t) (see Agranovich [4],Saranen et al [370]).

11. Boundary Integral Equations on Curves in IR2

684

Remark 11.2.1: The Fourier series representation (11.2.4) of A ∈ Lm c (Γ ) coincides with the definition of periodic pseudodifferential operators defined in the book by Saranen and Vainikko [369, Definition 7.2.1]. Proof: By substituting the Fourier series expansion of v, i.e.  v* e2πiτ v(τ ) = ∈Z

into (11.2.4) and exchanging the order of summation and integration — since v ∈ C0∞ ([0, 1]) and the Fourier series converges in every Sobolev space — we have  2πiτ AΦ v(t) = v* Fξ−1 + R∞ v(t) . →t aΦ (t, ξ)Fτ →ξ e ∈Z

By using the identity (11.1.13), the result follows.



The explicit representation (11.2.5) in terms of Fourier series relies on the symbol aΦ (t, ξ). If the operator A ∈ Lm c (Γ ) is given as in (11.2.2) then the Schwartz kernel kΦ (t, t − τ ) admits the pseudohomogeneous asymptotic expansion L  kj−m−1 (t, t − τ ) + RL (t, τ ) (11.2.7) kΦ (t, t − τ ) = j=0

with kj−m−1 ∈ Ψ hfj−m−1 , and one may employ the one–dimensional Fourier transform in (11.2.4) to obtain the following explicit formulae for corresponding homogeneous expansion terms of the symbol. Theorem 11.2.2. Let kj−m−1 ∈ Ψ hfj−m−1 be given where j ∈ IN0 . In the special case m ∈ IN0 and 0 ≤ j ≤ m, let kj−m−1 also satisfy the Tricomi condition (8.1.56) (where n = 1). Then the homogeneous symbol a0m−j corresponding to kj−m−1 is explicitly given by: If m ∈ Z:  mπ a0m−j (t, ξ) = |ξ|m−j Γ (j − m) ij e−i 2 kj−m−1 (t, − sign ξ) + (−i)j ei

mπ 2

 kj−m−1 (t, sign ξ) .

(11.2.8)

If m ∈ Z and m + 1 ≤ j ∈ IN0 :  a0m−j (t, ξ) = (j − m − 1)!ξ m−j ij−m fj−m−1 (t, −1) + (−1)j−m fj−m−1 (t, 1)  +iπpj−m−1 (t, 1)(sign ξ) (11.2.9) where kj−m−1 (t, z) = fj−m−1 (t, z) + pj−m−1 (t, 1)z j−m−1 log |z| .

(11.2.10)

11.2 The Fourier Series Representation of Periodic Operators A ∈ Lm c (Γ )

685

If m ∈ IN0 and 0 ≤ j ≤ m: a0m−j (t, ξ) = ξ m−j (sign ξ)

im+1−j πkj−m−1 (t, −1) . (m − j)!

(11.2.11)

Proof: Let us begin with the case m ∈ Z. Then (8.1.28) for j > m and (8.1.53) for j < m can be written in the compressed form  aψ,m−j (t, ξ) = p.f. kj−m−1 (t, t − τ )eiξ(τ −t) ψ(τ − t)dτ IR

∞ = kj−m−1 (t, −1) p.f.

z j−m−1 eiξz ψ(z)dz 0 ∞

+ kj−m−1 (t, 1) p.f.

z j−m−1 e−iξz ψ(z)dz ,

0

and (8.1.29) yields a0m−1 (t, ξ)

= lim  →+∞

j−m

∞



kj−m−1 (t, −1) p.f.

z j−m−1 eiξz ψ(z)dz 0

∞ + kj−m−1 (t, 1) p.f.

z j−m−1 e−iξz ψ(z)dz



0

∞ = kj−m−1 (t, −1) lim

→+∞

z j−m−1 eiξz ψ

p.f.

z 

dz

0

∞ + kj−m−1 (t, 1) lim p.f. →∞

z j−m−1 e−iξz ψ

z 

dz .

0

Hence, we need the one–dimensional Fourier transform of z j−m−1 which can explicitly be found in Gelfand and Shilov [143, p. 360], i.e.,  iπ a0m−j (t, ξ) = |ξ|m−j iΓ (j − m) kj−m−1 (t, −1)(sign ξ)i(sign ξ)(j−1) · e−(sign ξ) 2 m  iπ −kj−m−1 (t, 1)(sign ξ)i−(sign ξ)(j−1) · e(sign ξ) 2 m . Writing down this result for ξ > 0 and ξ < 0 separately gives the desired equation (11.2.8). Next, we consider the case m ∈ Z and m + 1 ≤ j ∈ IN0 . Then kj−m−1 ∈ ψhfj−m−1 has the form (11.2.10). With (8.1.28) we find for the homogeneous term

11. Boundary Integral Equations on Curves in IR2

686

∞ ahψ,m−j (t, ξ) = fj−m−1 (t, 1)

z j−m−1 e−iξz ψ(z)dz

0

(11.2.12)

∞ + fj−m−1 (t, −1)

z j−m−1 eiξz ψ(z)dz . 0

Hence, for ξ = 0 we need the limits of ∞ z k eiξz ψ(z)dz with k = j − m − 1 ∈ IN0

Ik (ξ, ) := j−m 0

for  → +∞. Integration by parts for k ≥ 1 yields i Ik (ξ, ) = j−m−1 k ξ

∞ z

k−1 iξz

e

i ψ(z)dz + j−m−1 ξ

1

z k ψ  (z)eiξz dz .

1 2

0

The second integral is the Fourier transform of a C0∞ ( 12 , 1)–function and, due to the Palay–Wiener–Schwartz theorem (Theorem 3.1.3), therefore, decays faster than −(j−m) . Hence, for ξ = 0 fixed, we find the recursion relation Ik =

i kIk−1 (ξ, ) + O(−1 ) , ξ

which gives Ik (ξ, ) =

ik k! I0 (ξ, ) + O(−1 ) , ξk

where ∞ eizξ ψ(z)dz =

I0 (ξ, ) = 

i i + ξ ξ

1

eizξ ψ  (z)dz =

i + O(−1 ) . ξ

1 2

0

Hence, lim Ij−m−1 (ξ, ) =

→+∞

ij−m (j − m − 1)! . ξ j−m

(11.2.13)

Inserting (11.2.13) in (11.2.12) gives a0h,m−j (t, ξ) =

(j − m − 1)!  (−i)j−m fj−m−1 (t, 1) ξ j−m  +ij−m fj−m−1 (t, −1) .

For the logarithmic part in (11.2.10) we introduce

(11.2.14)

11.2 The Fourier Series Representation of Periodic Operators A ∈ Lm c (Γ )

687

 z k log |z|ψ(z)eizξ dz

k+1

Jk (ξ, ) := 

IR

and integrate by parts finding  ik k z k−1 log |z|ψ(z)eizξ dz Jk (ξ, ) =  ξ IR  i k i +  z k−1 ψ(z)eiξz dz + k ξ ξ

z k log |z|ψ  (z)eiξz dz ,

1 2 ≤|z|≤1

IR

Jk (ξ, ) =



ik Jk−1 + O(−1 ) for k ≥ 1 ξ

and ξ = 0 fixed since for the last two integrals again the Paley–Wiener– Schwartz theorem (Theorem 3.1.3) can be applied. Thus, ik k! J0 (ξ, ) + O(−1 ) . ξk

Jk (ξ, ) = For J0 one finds 

log |z|eizξ ψ

J0 (ξ, ) = IR



log |z|eizξ ψ

=



z 

dz −  log  IR

z 

ψ(z)eizξ dz

dz + O(−1 ) .

IR

With  → +∞ and ξ = 0 fixed k = j − m − 1 ≥ 0, and the Fourier transformed of log |z|, we obtain lim Jk (ξ, ) = −

→+∞

π (j − m − 1)! ij−m−1 . |ξ| ξ j−m−1

Together with (11.2.14), this yields (11.2.9). In the remaining case k = m + 1 − j ∈ IN we have from (8.1.53) with the homogeneous kernel k−m−1+j : ∞ a0m−j (t, ξ)

= p.f.

k−m−1+j (t, 1)z −m−1+j e−izξ dz

0

∞ + p.f.

k−m−1+j (t, −1)z −m−1+j eizξ dz

0

where we employ the Tricomi condition (8.1.56) for n = 1:

688

11. Boundary Integral Equations on Curves in IR2

k−m−1+j (t, 1) = (−1)m−j+1 k−m−1+j (t, −1) obtaining a0m−j (t, ξ)

∞



= k−m−1+j (t, −1) p.f.  = k−m−1+j (t, −1) p.f.

z

−k izξ

e

∞ k

dz + (−1) p.f.

0

z −k e−izξ dz



0

z −k eizξ dz .

IR

If we integrate

 Lk := p.f.

z −k eizξ dz

IR

by parts then we obtain for k ≥ 2: Lk =

iξ Lk−1 . k−1

Hence, with the Fourier transformed of z −1 , we find Lk =

(iξ)k−1 im−j L1 = πξ m−j sign ξ , (k − 1)! (m − j)! 

i.e. (11.2.11).

11.3 Ellipticity Conditions for Periodic Operators on Γ The representation (11.2.5) of periodic operators A ∈ Lm c (Γ ) gives rise to simple characterization of ellipticity concepts such as ellipticity, strong ellipticity as well as the odd ellipticity. To this end, we employ the asymptotic expansion of the symbol, aΦ (t, ξ)

=

a0Φ,m (t, ξ) + a0Φ,m−1 (t, ξ) + . . .  a0Φ,m (t, ξ) + aΦ (t, ξ) − a0Φ,m (t, ξ)

=

a0Φ,m (t, ξ)

=

(11.3.1)

+ aΦ (t, ξ)

where  aΦ ∈ S m−1 (Γ × IR). The principal symbol a0Φ,m (t, ξ) is positively hoc mogeneous of degree m and periodic with respect to t. In view of (11.3.1), we may now write    a0Φ,m (t, 2π)* v ei2πt + Rv(t) (11.3.2) Av T (t) = 0=∈Z

 ∈ Lm−1 (Γ ); and by decomposition: where R c

11.3 Ellipticity Conditions for Periodic Operators on Γ

 Av T (t)



=

≥1

+

689

a0Φ,m (t, 1)(2π)m v* ei2πt 

a0Φ,m (t, −1)(−2π)m v* ei2πt

(11.3.3)

≤−1

+



1 v* p.f.

∈Z

 τ )ei2πτ dτ . R(t,

0

11.3.1 Scalar Equations In accordance with (7.2.3), the operator A is elliptic, if the two conditions a0Φ,m (t, 1) = 0

and

a0Φ,m (t, −1) = 0

(11.3.4)

are both satisfied for all t. Since for an elliptic operator there exists a parametrix and Γ is compact, Theorem 5.3.11 can be applied and A is, hence, a Fredholm operator. Lemma 11.3.1. For the elliptic operator A ∈ Lm c (Γ ), the Fredholm index is given explicitly by   a0 (t, 1)  1 Φ,m index(A) = − . (11.3.5) d arg 0 2π aΦ,m (t, −1) Γ

Here Γ = ∂Ω may consist of a finite number of simple, plane, closed and smooth Jordan curves which are oriented in such a way that Ω always lies to the left of Γ . Before we begin with the proof, let us first rewrite the representation (11.3.3) in the form     (11.3.6) |2π|m v* e2πit + v*0 Av T (t) = b+ (t) ∈Z\{0}

+ b− (t)

 

  |2π|m (sign )* v e2πit + v*0 + Rv

∈Z\{0}

where the functions b± (t) are defined by b± (t) :=

1 0 {a (t, 1) ± a0Φ,m (t, −1)} . 2 Φ,m

(11.3.7)

Further let us use the Fourier representation of the Bessel potential operator Λm : H s (Γ ) → H s−m (Γ ) for periodic functions, namely  Λm v(t) := |2π|m v* e2πit + v*0 . (11.3.8) ∈Z\{0}

690

11. Boundary Integral Equations on Curves in IR2

Then

Λ−m ◦ Λm v = Λm ◦ Λ−m v =



v* e2πit = v .

(11.3.9)

∈Z

We also shall use the Fourier representation of the Hilbert transform H : H s (Γ ) → H s (Γ ) for periodic functions, i.e.,   v* e2πit − v* e2πit . (11.3.10) Hv(t) = ≥0

Clearly, H ◦ Hv(t) =

 0 such that

11.3 Ellipticity Conditions for Periodic Operators on Γ

inf min{ Re Θ(t)a0Φ,m (t, 1) , Re Θ(t)a0Φ,m (t, −1)} ≥ β0 . t

691

(11.3.15)

Clearly, for a strongly elliptic operator, the zero Fredholm index follows with a direct computation from the index formula (11.3.5) which is consistent with Remark 9.1.2. We emphasize that the strongly elliptic equation Au = f on Γ m

with f ∈ H − 2 (Γ ) for any fixed  ∈ IR can be treated by the variational method in terms of the corresponding bilinear forms as in Chapter 5. Specifically we may define the bilinear form as a(u, v) = (ΘAu, v)H  (Γ ) = (Θf, v)H  (Γ ) m

(11.3.16)

m

where H1 = H + 2 (Γ ) and H2 = H − 2 (Γ ) and u, v ∈ H1 . This variational formulation provides the foundation of corresponding Galerkin methods. One special feature of periodic pseudodifferential operators on closed curves is the additional concept of the odd ellipticity as introduced in Sloan [390]. Definition 11.3.1. The operator A ∈ Lm c (Γ ) is called oddly elliptic if there exists a smooth function Θ ∈ C ∞ (Γ ) and a constant β0 > 0 such that   inf min{ Re Θ(t)a0Φ,m (t, 1) , − Re Θ(t)a0Φ,m (t, −1) } ≥ β0 > 0 . (11.3.17) t

We remark that the inequality (11.3.17) cannot  begeneralized to higher ξ 0 since then the unit dimensions of Γ for a continuous symbol aΦ,m t, |ξ| sphere in IRn−1 is not disconnected anymore. Both, strong as well as odd ellipticity can also be characterized in terms of the functions b+ (t) and b− (t) defined in (11.3.7). Theorem 11.3.2. i) A ∈ Lm c (Γ ) is strongly elliptic if and only if the condition b+ (t) + λb− (t) = 0 for all t and all λ ∈ [−1, 1]

(11.3.18)

is satisfied. ii) A ∈ Lm c (Γ ) is oddly elliptic if and only if the condition b− (t) + λb+ (t) = 0 for all t and all λ ∈ [−1, 1] is satisfied.

(11.3.19)

692

11. Boundary Integral Equations on Curves in IR2

Remark 11.3.1: Both conditions (11.3.18) or (11.3.19) can easily be verified by inspecting the corresponding graphs in the complex plane, see Figure 11.3.1. These conditions play an important rˆole in the analysis of singular integral equations and their approximation and numerical treatment (see Gohberg et al [151, 152], Pr¨ ossdorf and Silbermann [358, 359], Sloan et al [391] and the references therein). In particular, the function Θ(t) is not needed in (11.3.18) or (11.3.19). Proof: (See Pr¨ossdorf and Schmidt [356, Lemma 4.4].) We first prove the proposition i) for the strongly elliptic case. Suppose that A is strongly elliptic and (11.3.15) is satisfied. Then we show (11.3.18) by contradiction, i.e., for some λ0 ∈ [−1, 1] and t0 we have 1 1 b+ (t0 ) + λ0 b− (t0 ) = a+ (t0 ) + a− (t0 ) + λ0 a+ (t0 ) − a− (t0 ) = 0 2 2 0 0 where a+ (t) := aΦ,m (t, 1) , a− (t) := aΦ,m (t, −1). This implies after multiplication by Θ(t0 ) and taking the real part,   (1 + λ0 ) Re Θ(t0 )a0 (t0 ) + (1 − λ0 ) Re Θ(t0 )a− (t0 ) = 0 . Condition (11.3.15) together with λ0 ∈ [−1, 1] then would imply   0 ≤ (1 + λ0 ) Re Θ(t0 )a+ (t0 ) = −(1 − λ0 ) Re Θ(t0 )a− (t0 ) ≤ 0 , i.e. (1 + λ0 ) = 0 together with (1 − λ0 ) = 0 which is impossible. Conversely, suppose that (11.3.18) is fulfilled. This means that the convex combination satisfies μa+ (t) + (1 − μ)a− (t) = 0 for all t and for all μ = 12 (1 − λ) ∈ [0, 1] .

a+ (t) •

μa+ (t) + (1 − μ)a− (t)

a− (t)

Figure 11.3.1: Strong ellipticity condition.

11.3 Ellipticity Conditions for Periodic Operators on Γ

 Therefore,  arg

693



a+  a−

< π and a+ = 0 , a− = 0. We may now define  i (11.3.20) Θ(t) := exp − 2 (α+ + α− ) where α± := arg a± .

Then Θ(t)a+ (t) = |a+ (t)| exp

i



(α+ − α− ) , 2 i |a− (t)| exp − 2 (α+ − α− )

Θ(t)a− (t) = and   Re Θ(t)a+ (t) = |a+ (t)| cos 12 (α+ − α− ) > 0,   1 Re Θ(t)a− (t)) = |a− (t)| cos − 2 (α+ − α− ) > 0 , which is (11.3.15). For odd ellipticity, the proof proceeds in the same manner as in the strongly elliptic case by interchanging b+ and b− and replacing a− by (−a− ).  A simple example of an oddly elliptic equation is the Cauchy singular integral equation with the Hilbert transform,     d 1 1 log |x − y| u(y)dsy + u(y)dsy = f (x) . HΓ u(x) = − π dsy 2π Γ

Γ

Here the principal symbol is given by a0Φ,0 (t, ξ) = i

ξ |ξ|

for |ξ| ≥ 1 .

Hence, b+ (t) = 0 and b− (t) = i and (11.3.19) is satisfied. However, HΓ is not strongly elliptic. On the other hand, with the Hilbert transform (11.3.10) as a preconditioner, any oddly elliptic equation Au = f on Γ can be treated by using the variational methods as in the strongly elliptic case if one defines an associated bilinear equation by a(u, v) := (ΘHAu, v)H  (Γ ) = (ΘHf, v)H  (Γ ) m

m

(11.3.21)

where again H1 = H + 2 (Γ ) , H2 = H − 2 (Γ ) and u, v ∈ H1 . Now equation (11.3.21) defines a strongly elliptic bilinear form which fulfills a G˚ arding inequality and HA ∈ Lm (Γ ) is strongly elliptic. Clearly, the Fredholm index c of the oddly elliptic operator A is zero since H defined by (11.3.10) is an isomorphism with Fredholm index zero and index(HA) = index(A) due to (5.3.23) in Theorem 5.3.11.

11. Boundary Integral Equations on Curves in IR2

694

11.3.2 Systems of Equations We now extend the concept for one equation to a system of p pseudodifferential equations recalling Section 7.2.1, but now on Γ , p 

Ajk uk = fj for j = 1, . . . , p on Γ

(11.3.22)

k=1 s +t

with operators Ajk ∈ Lcj k (Γ ). Now let us assume that the system (11.3.22) is elliptic in the sense of Agmon–Douglis–Nirenberg, see Definition 7.2.3. Here, fj ∈ H sj (Γ ) for j = 1, . . . , p are given and uk ∈ H tk (Γ ) for k = 1, . . . , p are the unknowns. By ajk0 Φ,sj +tk (t, ξ) we denote the homogeneous principal symbols of Ajk on Γ which are 1–periodic in t. By using the representation (11.3.3) for each of the operators in (11.3.22) we obtain p 

p  ∞   sj +tk Ajk u T (t) = ajk0 u *k e2πit Φ,sj +tk (t, 1)(2π)

k=1

k=1

+

(11.3.23)

=1



p   sj +tk 2πit ajk0 (t, −1)(−2π) u * e Rjk uk + k Φ,sj +tk

≤−1

k=1 s +t −1

where the remaining operators Rjk ∈ Lcj k (Γ ). The system (11.3.22) is elliptic in the sense of Agmon–Douglis–Nirenberg if both conditions B(t, 1) := det ((ajk0 Φ,sj +tk (t, 1)))p×p = 0 and B(t, −1) := det ((ajk0 Φ,sj +tk (t, −1)))p×p = 0 for every t ∈ IR

(11.3.24)

are satisfied (see Definition 7.2.3). In this case, a parametrix Q0 exists acp & H tk (Γ ) → cording to Theorem 7.2.3 and, hence, A = ((Ajk ))p×p : H1 := H2 :=

p &

k=1

H

−sj

(Γ ) is a Fredholm operator. The Fredholm index of ((Ajk ))p×p

j=1

can be computed as follows. Lemma 11.3.3. For the elliptic system (11.3.22) in the sense of Agmon– Douglis–Nirenberg, the Fredholm index is given by   B(t, 1)  1 . (11.3.25) index((Ajk ))p×p = − d arg 2π B(t, −1) Γ

Proof: For the proof we may proceed in the same manner as for the scalar case. In particular, we see that the system may be rewritten in the form

11.3 Ellipticity Conditions for Periodic Operators on Γ p 

 Ajk uk T (t)

=

k=1

p  

b+,jk (t)Λsj +tk uk + b−,jk (t)Λsj HΛtk uk

695



k=1

+

p 

jk uk R

(11.3.26)

k=1 s +t −1

jk ∈ L j k (Γ ) which includes Rjk and smoothing operators in with R c terms of u *k0 . If we introduce new unknowns wk := Λtk uk , with the help of the isomorphic Bessel potential operator (11.3.8), we obtain the system of classical Cauchy singular integral equations p 

≈

b+,jk (t)wk (t) + b−,jk (t)Hwk + Rjk wk = Λ−sj fj , j = 1, . . . , p . (11.3.27)

k=1

Here ≈

Rjk + Λ−sj (b+,jk Λsj − Λsj b+,jk ) + Λ−sj (b−,jk Λsj − Λsj b−,jk ) ∈ L−1 c (Γ ) . The last two terms are the commutators of the pseudodifferential operator s b±,jk ∈ L0c (Γ ) with Λsj ∈ Lcj (Γ ). By applying Corollary 7.1.15 we see that ≈

s −1

these commutators are in the class Lcj (Γ ) and therefore Rjk ∈ L−1 c (Γ ). The Fredholm index of the system (11.3.27) of Cauchy singular equations can be found in the book by Muskhelishvili [320, formula (180.22)] which yields ≈

(11.3.25) since the mappings Λsj +tk are isomorphisms and Rjk are compact operators.  The condition (7.2.62) for strong ellipticity now reads:  There exist a C ∞ –matrix–valued 1–periodic function Θ(t) = Θjk (t) p×p and a constant β0 > 0 such that inf min

t∈IR



Re

Re

p  ,j,k=1 p 

ζ Θj (t)ajk0 Φ,sj +tk (t, +1)ζ k , ζ Θj ajk0 Φ,sj +tk (t, −1)ζ k



,j,k=1

≥ β0 |ζ|2 for all ζ ∈ Cp . s +tk

Again, for a strongly elliptic system with Ajk ∈ Lcj index is zero since any associated bilinear form p a(v, u) := (v, Λσ ΘΛ−σ Au) 

(11.3.28) (Γ ), the Fredholm

H (t −s )/2 (Γ )

(11.3.29)

=1

satisfies a G˚ arding inequality such as (7.2.64) on the boundary curve Γ . As for one scalar equation, we may define odd ellipticity for the system (11.3.23), namely:

11. Boundary Integral Equations on Curves in IR2

696

Definition 11.3.2. The operator A = ((Ajk ))p×p is called oddly elliptic if there exist a smooth matrix–valued 1–periodic function Θ(t) and a constant β0 > 0 such that  inf min

t∈IR

p 

Re

ζ Θj (t)ajk,0 Φ,sj +tk (t, 1)ζ k ,

,j,k=1 p 

− Re

ζ Θj (t)ajk,0 Φ,sj +tk (t, −1)ζ k

(11.3.30) 

≥ β0 |ζ|2 for all ζ ∈ Cp .

,j,k=1

Again, Theorem 11.3.2 remains valid for the system as follows. Theorem 11.3.4. s +t

i. The system of operators ((Ajk ))p×p with Ajk ∈ Lcj k (Γ ) is strongly elliptic if and only if the condition  det b+,jk (t) + λb−,jk (t) p×p = 0 for all t and for all λ ∈ [−1, 1] (11.3.31) is satisfied. ii. The system is oddly elliptic if and only if the condition  det b−,jk (t) + λb+,jk (t) p×p = 0 for all t and for all λ ∈ [−1, 1] (11.3.32) is satisfied. Proof: The basic ideas of the proof for the system are similar to the scalar case in Theorem 11.3.2. However, one needs the following results of matrix algebra and functional calculus. The detailed proof for systems was given by Pr¨ ossdorf and Schmidt in [357]. (See also Pr¨ossdorf and Rathsfeld [355] for coefficients with discontinuities.) i.

Let us assume that (11.3.31) is violated, i.e., for some t0 and λ0 ∈ [−1, 1] we have  det b+,jk (t0 ) + λ0 b−,jk (t0 ) p×p = 0 . Then there exists at least one vector ζ0 ∈ Cp , ζ0 = 0 with p  

b+,jk (t0 ) + λ0 b−,jk (t0 ) ζ 0k = 0 , j = 1, . . . , p .

k=1

Hence,

11.3 Ellipticity Conditions for Periodic Operators on Γ p 

ajk,0 Φ,sj +tk (t0 , 1)ζ 0k =

k=1

697

p  

b+,jk (t0 ) + b−,jk (t0 ) ζ 0k

k=1

= (1 − λ0 )

p 

b−,jk (t0 )ζ 0k

k=1

and p 

ajk,0 Φ,sj +tk (t0 , −1)ζ 0k =

k=1

p  

b+,jk (t0 ) − b−,jk (t0 ) ζ 0k

k=1

= −(1 − λ0 )

p 

b−,jk (t0 )ζ 0k .

k=1

Now, let Θ(t0 ) be any matrix satisfying   Re ζ0 Θ,j (t0 )ajk,0 Φ,sj +tk (t0 , 1)ζ 0k ,j,k

= (1 − λ0 ) Re



 ζ0 Θj (t0 )b−,jk (t0 )ζ 0k > 0

,j,k

then for this matrix will hold   Re ζ0 Θj (t0 )ajk,0 Φ,sj +tk (t0 , −1)ζ 0k ,j,k=1



= −(1 − λ0 ) Re

 ζ0 Θj (t0 )b−,jk (t0 )ζ 0k < 0 .

,j,k

Therefore, for any choice of Θj (t0 ), both conditions on (11.3.28) can never be satisfied at the same time. Hence, (11.3.28) implies (11.3.31). ii. Now, let (11.3.31) be satisfied. Then we need to construct the matrix– valued function Θ(t), providing (11.3.28). To this end, define the matrix– valued function  −1  U (t) := b+,jk (t) b−,jk (t) which is well defined since (11.3.31) with λ = 0 implies the existence of −1  . Then let b+,jk (t) V (t) :=



I + U (t)

12 

I − U (t)

12 −1

where the square roots are defined by means of the Dunford–Taylor integral with eigenvalues having positive real parts Kato [223, Section I.5.6.]. With V (t) define  N+ (t) := V (t) I + U (t) . Then the matrix function

698

11. Boundary Integral Equations on Curves in IR2

∞ H(t) :=

  ∗ exp − N+ (t)s exp − N+ (t)s ds

0

can be defined by Bellman [26] and  Θ(t) = H(t)V (t) b+,jk (t) gives the desired matrix providing the two inequalities in (11.3.28). For the oddly elliptic case, just exchange the roles of b+,jk , and b−,jk and of (11.3.28) and (11.3.31).  Again, with the Hilbert transform (11.3.10) as preconditioner, the oddly elliptic system of equations (11.3.22) may be treated by variational methods; in this case the associated variational equation with a strongly elliptic bilinear form assumes the form p a(u, v) := (Λσ ΘΛ−σ HAu, v) 

H (t −s )/2 (Γ )

=1 p = (Λσ ΘΛ−σ Hf, v) 

H (t −s )/2 (Γ )

,

(11.3.33)

=1

where H1 =

p &

H tk (Γ ) and H2 =

p &

H −sj (Γ ).

j=1

k=1

Now HA : H1 → H2 is a strongly elliptic system; and since H is an isomorphism, we again have for the Fredholm index of the oddly elliptic system operator A: index(HA) = index(A) = 0 in view of Theorem 5.3.11. 11.3.3 Multiply Connected Domains Before we treat the general system of pseudodifferential equation on Γ = ∂Ω of a multiply connected domain Ω ⊂ IR2 let us begin with the following simple example, the Dirichlet problem Δu = 0 in Ω and u|Γ = ϕ on Γ where Γ =

q #

Γj and each of the curves Γj is a closed smooth Jordan curve.

j=1

Moreover, Γ1 contains all of the other curves in the interior domain bounded by Γ1 and the curves are mutually disjoint. They all are oriented in such a way that Ω always lies to the left of Γ . On each of the closed curves we use a 1–periodic global parametric representation y|Γj = Tj (z). By using the direct method, the solution can be expressed by boundary potentials as in

11.3 Ellipticity Conditions for Periodic Operators on Γ

699

(1.1.7) which leads to the boundary integral equation (1.3.3) for the unknown ∂u |Γ , i.e., missing Cauchy datum, the boundary charge σ = ∂n V σ = f (x) := 12 ϕ(x) + Kϕ(x) for x ∈ Γ .

(11.3.34)

In order to write (11.3.34) as a system of periodic integral equations, we introduce the notation  dT (t) σ  (t) := σ T (t) dt

(11.3.35)

and write (11.3.34) as q  =1

q 1  

 1 log |Tr (t) − T (τ )| σ  (τ )dτ 2π =1τ =0  = fr (t) := f Tr (t) for r = 1, . . . , q .

Vr σ  (t) :=

−

(11.3.36)

This system is a particular example of a system of pseudodifferential equations as considered in Section 11.3.2. We note that Vr for r =  has a C ∞ –kernel. Hence, the symbol matrix has diagonal form, i.e., ar0 Φ,−1 (t, ξ) =

1 δr 2|ξ|

(11.3.37)

according to (11.1.10). In terms of the representation (11.3.6) we have the Fourier series representation for the system q 

 (Vr σ ) T (t)

=

=1

q 

b+,r (t)

 

=1

=

r σ |2πj|−1 σ *j e2πijt + σ *0 + R



0=j∈Z

1 −1 Λ σ r + 2

q 

r σ R

(11.3.38)

=1

where b+,r = 12 δr and, in this case, b−,r = 0 ; s = t = − 12 and r ∈ L−∞ (Γ ) for  = 1, . . . , q. The last expression corresponds to (11.3.27). R c Clearly, this is a strongly elliptic system. For a system of the form (11.3.22) on the boundary Γ of the multiply connected domain Ω we now write uk, (t)

:=

Ajk,r uk,

:=

and

uk |Γ , fj,r := fj |Γr

(11.3.39)

Ajk |Γ uk |Γ |Γr .

(11.3.40)



In terms of these notations, the system (11.3.22) for the multiply connected domain now reads

700

11. Boundary Integral Equations on Curves in IR2

p q  

Ajk,r uk, = fj,r on Γr for r = 1, . . . , q and j = 1, . . . , p .

=1 k=1

(11.3.41) For  = r, the integration in the integral operator representation of Ajk,r ∈ s +t Lcj k (Γ ) (see Theorem 8.1.8) acts on Γ whereas the observation point x ∈ Γr , which is separated from Γ and |x − y| ≥ dist (Γr , Γ ) > 0. Here the Schwartz kernel Ajk,r is C ∞ –smooth. Hence, for  = r, the symbol matrix elements are zero, i.e., ajk,r0 Φ,sj +tk (t, ξ) = 0 for r =  and j, k, = 1, . . . , p . As a consequence, the principal symbol of Ajk,r has a block diagonal structure as ⎞ ⎛ jk,110 0 ... 0 aΦ,sj +tk (t, ξ) ⎟ ⎜ ⎟ ⎜ 0 0 ajk,220 Φ,sj +tk (t, ξ) ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ . (11.3.42) ⎟ ⎜ ⎟ ⎜ .. .. ⎟ ⎜ . . ⎟ ⎜ ⎠ ⎝ jk,qq0 0 0 . . . aΦ,sj +tk (t, ξ) and j, k = 1, . . . , p. The coupling operators Ajk,r for r =  between different boundary components Γ = Γr are all smoothing operators. Hence, with respect to mapping and Fredholm properties, the complete system can be separated into the single systems with corresponding principal symbols Ajk,rr Φ,sj +tk (t, ξ). In particular, det =

q ,

det





ajk,r Φ,sj +tk (t, ξ)

ajk,rr Φ,sj +tk (t, ξ)

p×p

pq×pq

for all t and ξ = ±1 .

r=1

Therefore, the index formulae (11.3.5) and (11.3.25) also remain valid for the multiply connected case when the integrals are to be taken over all of the components of Γ . In a similar manner, the concepts of strong and odd ellipticity of the system can now be used for each of the boundary components Γr separately.

11.4 Fourier Series Representation of some Particular Operators

701

11.4 Fourier Series Representation of some Particular Operators To conclude this chapter we now apply Theorem 11.2.2 to a few more basic operators in addition to those of the Laplacian. The general procedure will be to find the symbol’s expansion by making use of the boundary integral operators considered and their pseudohomogeneous kernel expansions. 11.4.1 The Helmholtz Equation We begin with the simple layer potential operator Vk in (2.1.6). We need the pseudohomogeneous kernel expansion of Ek (x, y) in (2.1.4). From (2.1.18) we see that Ek (x, y)

=

1 (log k + γ0 ) (11.4.1) 2π ∞ ∞    1  − log((kr) am (kr)2m + bm (kr)2m . 2π m=1 m=1

E(x, y) −

By using the 1–periodic parameterization x = T (t) and y = T (τ ) we may rewrite Ek (x, y) in the form of a pseudohomogeneous expansion   1 log |t − τ | 1 + a1 k2 L2 (t − τ )2 Ek T (t), T (τ ) = − 2π  1 − L4 (t − τ )4 a1 k2 κ2 (t) − a2 k 4 12  dκ 2 1 4 L κ(t) (t)(t − τ )5 + O6 k4 + a1 k 12 dt + k∞ (k, t, τ ) =

5 

(11.4.2)

kj+1−1 (t, t − τ ) + R5 (t, τ )

j=0

according to (11.2.7) where k∞ is a C ∞ –function of t and τ and kj (t, t − τ ) = pj (t, t − τ ) log |t − τ | with homogeneous polynomials pj of degree j ≥ 0 . It is clear that the individual terms of the expansion are no longer 1–periodic in t and τ . Then Theorem 11.2.2 provides us with the first six terms of the homogeneous symbol corresponding to the parametric representation x = T (t) defining the mapping Φ, suppressed in the following notation:

702

11. Boundary Integral Equations on Curves in IR2

a0−1 (t, ξ) = −

1 −1 1 ξ iiπ1(sign ξ) = , 2π 2|ξ|

a0−2 (t, ξ) = 0 , a0−3 (t, ξ) = −

1 1 1 a1 k2 L2 2!ξ −3 i3 iπ sign ξ = k2 L2 3 , 2π 4 |ξ|

a0−4 (t, ξ) = 0 ,

L4  2 1 2 a1 k κ (t) − a2 k4 4!ξ −5 i5 iπ sign ξ 2π 12 L4  2 2 3 1 = k κ (t) + k4 , 4 4 |ξ|5 1 1 dκ a1 k2 L4 κ(t) (t)5!ξ −6 i6 iπ sign ξ a0−6 (t, ξ) = 2π 12 dt 1 dκ 5 2 4 = −i k L κ(t) (t) 6 sign ξ , 4 dt |ξ| a−5 (t, ξ) =

(11.4.3)

(see Agranovich [6, p. 35 (2.45)]). Consequently, the simple layer potential operator Vk ∈ L−1 c (Γ ) admits a periodic Fourier series representation in the form  Vk σ T (t) =



4 

a0−1−j (t, 2π)* σ ei2πt + R5 σ

(11.4.4)

∈Z\{0} j=0

where σ * is the Fourier coefficient given by 1 σ * = L

 e−2πiτ σ T (t) dτ

0

and  (R5 σ)(x) =

R5 (x, y)σ(y)dsy = Γ

 ∈Z

1 σ *



T (t), T (τ ) e2πiτ dτ .

0

with R5 ∈ C (Γ × Γ ) defining a pseudodifferential operator belonging to L−7 c (Γ ). For the double layer potential operator Kk we use the representation with (2.1.20), i.e. ∞  ∂      ∂ Ek (x, y) = E(x, y) 1+ am 1+2m log(kr) +2mbm (kr)2m . ∂ny ∂ny m=1 5

Then with (11.1.23) we get   ∂ 1 dκ 1 1 κ(t) − (t − τ ) (t) + k∞,1 (t, τ ) × Ek T (t), T (τ ) = ∂ny 2π 2 3 dt  2 2  4 a1 k r log r + (kr) log rR∞,1 (k2 ; r2 ) + R∞,2 (k; r2 )

11.4 Fourier Series Representation of some Particular Operators

703

where the kernels R∞,1 and R∞,2 are C ∞ –functions of x and y. Then r2 will be replaced by (11.1.19) and we obtain the first four terms of the asymptotic kernel expansion:   ∂ dκ  1 2 a1 L2 k2 κ(t)(t − τ )2 − (t − τ )3 (t) log |t − τ | Ek T (t), T (τ ) = ∂ny 2π 3 dt + (t − τ )4 log |t − τ |k∞,4 (k, t, τ ) + k∞,2 (k, t, τ ) . Here, the order of the double layer potential Kk ∈ L0c (Γ ) should be m = 0, but for the symbol expansion we find the terms a00 = a0−1 = a0−2 = 0 , 1 L2 a0−3 = − k2 κ(t) 3 , 4 |ξ| L2 i dκ a0−4 = k2 (t) 4 sign ξ . 2 dt |ξ|

(11.4.5)

Consequently, the double layer potential operator Kk admits the 1–periodic Fourier series representation  Kk ϕ T (t)    1 1 1 i dκ + k2 (t) sign  ϕ * ei2πt + R5 ϕ . − k2 κ(t) = L2 3 4 4 |2π| 2 dt |2π| ∈Z\{0}

Here, the smooth remaining operator R5 ∈ L−5 c (Γ ) is given by  R5 ϕ(x) =

R5 (x, y)ϕ(y)dsy = Γ

 ∈Z

1 ϕ *



T (t), T (τ ) e2πτ dτ ,

0

and has a doubly periodical kernel R5 ∈ C 3 (Γ × Γ ). For the operator Kk which is the transposed to Kk , one may compute the first four terms of its symbol expansion by employing the formula (7.1.34) directly. However, for illustration, we first compute the pseudohomogeneous kernel expansion of Kk in the same manner as for Kk . A simple computation shows that ∂Ek ∂E = + Rk (x, y) ∂nx ∂nx ∞     ∂E  = 1+ am 1 + 2m log(kr) + 2mbm (kr)2m . ∂nx m=1 With dκ 1 1 n(x) · (x − y) = − (t − τ )2 κ(t)L2 + (t − τ )3 L2 (t) + O4 2 6 dt

(11.4.6)

11. Boundary Integral Equations on Curves in IR2

704

we obtain

∂E 1 dκ  1  κ(t) − (t − τ ) (t) + O2 . (x, y) = ∂nx 4π 3 dt

Hence,  ∂ dκ  L2 1 a1 k2 log |t − τ | (t − τ )2 κ(t) − (t − τ )3 (t) Ek (x, y) = ∂nx 2π 3 dt  (k, t, τ ) + O0 , + (t − τ )4 log |t − τ |k∞,4  where k∞,4 and O0 are C ∞ –functions of t and τ . With Theorem 11.2.2 we now get the symbols

a00 = a0−1 = a0−2 = 0 and L2 1 L2 i dκ a0−3 = − k2 κ(t) 3 , a0−4 = k2 (t) 4 sign ξ , 4 |ξ| 4 dt |ξ| which also follow with (7.1.34) from the symbols of Kk in (11.4.5). Again, for Kk we have the 1–periodic Fourier series representation   1  1 (Kk σ) T (t) = L2 − k2 κ(t) 4 |2π|3 ∈Z\{0}  1 i dκ + k2 (t) sign  σ *ei2πt + R5 σ , 4 dt |2π|4 1      R5 σ = R5 (x, y)σ(y)dsy = σ * R5 T (τ ), T (t) e2πiτ dτ . ∈Z

Γ

0

For the hypersingular operator  ∂ ∂ Ek (x, y)ϕ(y)dsy with x ∈ Γ Dk ϕ = − ∂nx ∂ny Γ

we use the Maue formula [284] Dk ϕ = −

d d 1 d dϕ Vk − k 2 Vk ϕ Vk ϕ − k 2 Vk ϕ = − 2 dsx dsy L dt dτ

(11.4.7)

which can be obtained by a slight modification of the proof of Lemma 1.2.2 for the Laplacian. With the representation (11.4.4) of Vk we then find four terms of the 1–periodic Fourier series representation of Dk , i.e. Dk ϕ

=

−

4   1 d 0 a (t, 2π) (11.4.8) 2 dt −1−j L ∈Z\{0} j=0    1 * ei2πt + 2 (2πi)2 + k2 a0−1−j (t, 2π) ϕ L 1    1 d    2πiτ 2 R T (t), T (τ ) (2π) + k T (t), T (τ ) e ϕ * R dτ 5 5 L2 dt

−

∈Z



0

11.4 Fourier Series Representation of some Particular Operators

705

where a0−1−j (t, ξ) are the symbol terms of Vk given in (11.4.3). 11.4.2 The Lam´ e System For the two–dimensional space the fundamental solution of the Lam´e system is given in (2.2.2), E(x, y) =

 λ + 3μ  λ+μ 1 − log rδik + (x − y )(x − y ) (11.4.9) i i k k 4πμ(λ + 2μ) λ + 3μ r2

Hence, the simple layer potential can be rewritten as  λ + 3μ  1 (V σ)(x) = − log rσ(y)dsy 2μ(λ + 2μ) 2π Γ   λ+μ 1 L(x, y)σ(y)dsy + λ + 3μ 2π

(11.4.10)

Γ

where the tensor L has the form L(x, y) =

1 (x − y)(x − y) . r2

With the parametric representation we have    1 L T (t), T (τ ) = 2 T (τ ) − T (t) T (τ ) − T (t) r and obtain with (11.1.18) and (11.1.23)  (11.4.11) L T (t), T (τ ) = [1 + (τ − t)2 c(t, τ )]−1 ×    × a(t, τ )t(t) + b(t, τ )n(t) a(t, τ )t(t) + b(t, τ )n(t) + L∞ (t, τ ) where

(τ − t)2 2 2 L κ (t) , 6 (11.4.12) (τ − t) (τ − t)2 dκ b(t, τ ) = Lκ(t) + L (t) 2 6 dt ∞ and c(t, τ ) from (11.1.19) all are  C –functions of t and τ . Consequently, L T (t), T (τ ) is a C ∞ –function of t and τ . The symbol of the elastic simple layer potential operator is the same as that of the Laplacian up to a multiplicative constant. More precisely, we have the 1–periodic Fourier series representation a(t, τ ) = 1 −

 (Vσ) T (t) =

 λ + 3μ  1 *  e2πit + R∞ σ T (t) σ 2μ(λ + 2μ) 4π|| =0

(11.4.13)

11. Boundary Integral Equations on Curves in IR2

706

where  R∞ σ(x) =

R∞ (x, y)σ(y)dsy =

 ∈Z

Γ

1 * σ

 R∞ T (t), T (τ ) e2πiτ dτ

0

 with R∞ ∈ C ∞ (Γ × Γ ), and R∞ T (t), T (τ ) is 1–periodic in t and τ . For the double layer operator of elasticity (2.2.19) we use the expression (2.2.21) with (2.2.23) and obtain the corresponding kernel as 

Ty E(x, y)



=

 ∂   2(λ + μ) 1 μ  I+ L(x, y) log r − λ + 2μ μ ∂ny 2π     1 0 1 d − − log r . (11.4.14) −1 0 dsy 2π

As we have seen in (11.1.23) and in (11.4.10),      μ 1 0 1 d  Ty E(x, y) = − log r + T∞ (x, y) − λ + 2μ −1 0 dsy 2π where T∞ is in C ∞ (Γ ×Γ ). So it remains to compute the symbol of the operator with the kernel given by the first term. For the latter we use integration by parts and (11.1.4) to obtain      1 dσ μ 0 1  − log |x − y| (y)dsy (Kσ) T (t) = − λ + 2μ −1 0 2π dsy Γ  + T∞ (x, y)σ(y)dsy (11.4.15) Γ

μ = λ + 2μ



0 −1

  1  i sign  *  e2πit + K ∞ (x, y)σ(y)dsy σ 0 2L =0

Γ

where K ∞ ∈ C ∞ (Γ × Γ ) and 1





K ∞ T (t)T (τ ) σ(τ )dτ = 0



1

 * . K ∞ T (t), T (τ ) e2πiτ dτ σ

∈Z 0

The complete symbol of the double layer elastic operator K is therefore given by   μ 0 1 i σK (t, ξ) = sign ξ . (11.4.16) λ + 2μ −1 0 2L For the transposed operator K  in (2.2.20) we apply (7.1.34) to find the symbol of K  , i.e.

11.4 Fourier Series Representation of some Particular Operators

σK  (t, ξ) = σK (t, −ξ) = −σK (t, ξ) .

707

(11.4.17)

Hence, the 1–periodic Fourier expansion representation of K  assumes the form    μ 0 −1  i sign   *  e2πit (K ϕ) T (t) = ϕ λ + 2μ 1 0 3L =0

+

1 



(11.4.18)



* . K ∞ T (τ ), T (t) e2πiτ dτ ϕ

∈Z 0

For the elastic hypersingular operator we use the relations (2.2.32) with (2.3.36), i.e.   dϕ d 2μ(λ + μ)  1 1 (Dϕ)(x) = − log |x − y| + L(x, y) (y)dsy , − dsx (λ + 2μ) 2π 2π dsy Γ   1 2μ(λ + μ) d  i 2πiz sign * ϕ e + D ∞ (x, y)ϕ(y)dsy (Dϕ) T (t) = − L (λ + 2μ) dt 2L =0

Γ

π 2μ(λ + μ)  = 2 ||* ϕ ei2πt L (λ + 2μ)

(11.4.19)

=0



1

+

 * . D ∞ T (t), T (τ ) ei2πτ dτ ϕ

∈Z 0

The complete symbol of D is therefore given by σD (t, ξ) =

1 μ(λ + μ) |ξ|I . L2 (λ + 2μ)

11.4.3 The Stokes System The fundamental solution and the kernel of the pressure operator of the Stokes system in two dimensions are given by (2.3.10), i.e., 1  (x − y)(x − y)  − (log r)I + , 4πμ r2 1 1 Q(x, y) = − (∇y ln r) = (x − y) π 2πr2

E(x, y) =

(11.4.20) (11.4.21)

where (11.4.20) can also obtained from (11.4.9) as the limit for λ → +∞. Hence, for the Stokes simple layer potential we obtain from (11.4.10)    1 1  1 Vst σ = − (log r2 )σ(y)dsy + L(x, y)σ(y)dsy 2μ 4π 2π Γ

Γ

708

11. Boundary Integral Equations on Curves in IR2

with r2 given by (11.1.19) and L by (11.4.11) and (11.4.12). The corresponding Fourier series representation,   1  1 *  e2πit + R∞st σ T (t) Vst σ T (t) = σ 2μ 4π||

(11.4.22)

=0

follows from (11.1.13) with λ → +∞. For the kernel of the pressure operator Q (11.4.21) in local coordinates we find with (11.1.18) and (11.1.19)  1 1 1 − (t − τ )2 L2 κ2 (t) L(t − τ ) 12  dκ 1 + (t − τ )3 L2 κ(t) (t) + O4 t (t) 24 dt 1  2 1 1 2d κ 3 2 2 dκ  − κ(t) + (t − τ ) (t − τ ) + O (t) − L κ 4 n(t) 2 24 dt2 18 dt 1 t(t) + k1∞ (t, τ )t(t) + k2∞ (t, τ )n(t) . = L(t − τ )

 Q T (t), T (τ ) =

Hence, with Theorem 11.2.2 we find the corresponding complete symbol as aQ (t, ξ) = −

iπ (sign ξ)t(t) , L

(11.4.23)

and the Fourier representation  i  sign()* σ  e2πit Qσ T (t) = −  L =0

1 +



 k1∞ (t, τ )t(t) + k2∞ (t, τ )n(t) · σ T (τ ) dτ .

0

The double layer potential operator of the Stokes system can be obtained from (11.4.15) by letting λ → +∞. Hence, it is a smoothing operator as well as its adjoint. Since the hypersingular operator D st of the Stokes system equals the one of elasticity (11.4.19) with λ → +∞ we obtain    2μπ  i2πt * (D st ϕ) T (t) = 2 ||ϕ * e + D ∞st T (t), (T (τ ) e2πit dτ ϕ L 1

=Z 0

=0

and for its complete symbol aDst (t, ξ) =

1 μ|ξ|I . L2

11.4 Fourier Series Representation of some Particular Operators

709

11.4.4 The Biharmonic Equation The fundamental solution of the biharmonic operator is given by (2.4.7), i.e. E(x, y) =

1 2 8π r

log r with r2 = |x − y|2 .

(11.4.24)

We recall that the boundary integral operators in the Calder´on projector CΩ in (2.4.23) consist of 16 different operators on the boundary curve Γ . In the same manner as in the previous examples we will first present the kernels kj (x, y) of these operators and later on the corresponding symbols based on the pseudohomogeneous kernel expansions, as well as 1–periodic Fourier series representations. We begin with the first row of the matrix (2.4.23). 1 2 = 8π r log r , (11.4.25) k14 (x, y)=E(x, y) ∂ 1 n(y) · (y − x)(2 log r + 1) , (11.4.26) E(x, y) = k13 (x, y)= ∂ny 8π k12 (x, y)= − My E(x, y) (11.4.27) 1 + 3ν 1  (1 + ν) log r + =− 4π 2  2 1  , +(1 − ν) n(y) · (y − x) r2 (11.4.28) k11 (x, y)=Ny E(x, y) 1  1 =− 2n(y) · (y − x) 2 4π r d  1  +(1 − ν) n(z) · (y − x) t(z) · (y − x) 2 z=y . dsy r

For the second row, we have the corresponding kernels as: k24 (x, y)= k23 (x, y)= =

k22 (x, y)= =

∂ 1 n(x) · (x − y)(2 log r + 1) , (11.4.29) E(x, y) = ∂nx 8π ∂ ∂ E(x, y) (11.4.30) ∂nx ∂ny 1  − n(x) · n(y)(2 log r + 1) 8π 1 +2n(y) · (y − x) n(x) · (x − y) 2 , r ∂ − My E(x, y) (11.4.31) ∂nx 1 1  (1 + ν)n(x) · (x − y) 2 − 4π r  1 −2(1 − ν) n(x) · n(y) n(y) · (y − x) 2 r  2 1  + n(y) · (y − x) n(x) · (x − y) 4 , r

710

11. Boundary Integral Equations on Curves in IR2

k21 (x, y)= =

∂ Ny E(x, y) (11.4.32) ∂nx  1 1 1 n(x) · n(y) 2 + 2n(y) · (y − x) n(x) · (x − y) 4 − 2π r r 1 1−ν d  n(x) · t(z) n(z) · (y − x) 2 − 4π dsy r 1 + n(x) · n(z) t(z) · (y − x) 2 r 1  + 2t(z) · (y − x) n(z) · (y − x) n(x) · (x − y) 4 z=y . r −

The kernels in the third row are given by: k34 (x, y) = Mx E(x, y) (11.4.33)  2 1  1 + 3ν 1  , (1 + ν) log r + + (1 − ν) n(x) · (x − y) = 4π 2 r2 ∂ Mx E(x, y) (11.4.34) k33 (x, y) = ∂ny 1 1  (1 + ν)n(y) · (y − x) 2 = 4π r  1 −2(1 − ν) n(x) · n(y) n(x) · (x − y) 2 r  2 1  , + n(x) · (x − y) n(y) · (y − x) 4 r k32 (x, y) = − Mx My E(x, y) (11.4.35)   2 1 (1 − ν)  1 + 3ν + 2(1 − ν) n(x) · n(y) = − 4π r2    1 +8(1 − ν) n(x) · n(y) n(y) · (y − x) n(x) · (x − y) 4 r 2 1  2  +8(1 − ν) n(x) · (x − y) n(y) · (y − x) r6  2  2  1  , −2(1 + ν) n(y) · (y − x) + n(x) · (x − y) r4

11.4 Fourier Series Representation of some Particular Operators

711

k31 (x, y)=− Mx Ny E(x, y) (11.4.36) 2 (1 − ν)  − n(y) · (y − x) + 2n(x) · n(y) n(x) · (x − y) 4 = 2π r 1 d  − (1 + ν)t(z) · (y − x) n(z) · (y − x) 4 + dsy r 1 + (1 − ν)n(x) · t(z) n(x) · n(z) 2 r 1 + 2(1 − ν)t(z) · (y − x) n(x) · (x − y) n(x) · n(z) 4 r  2 1 + 8 n(x) · (x − y) n(y) · (y − x) 6 r 1 d  2(1 − ν)t(z) · n(x) n(x) · (x − y) n(z) · (y − x) 4 + dsy r  2 1  + 4(1 − ν)t(z) · (y − x) n(x) · (x − y) n(z) · (y − x) 6 . z=y r Similarly, for the last row we obtain the kernels k44 (x, y) = Nx E(x, y) (11.4.37) 1 1  2(n(x) · (x − y) 2 =− 4π r 1  d  n(z) · (x − y) t(z) · (x − y) 2 z=x , + (1 − ν) dsx r ∂ E(x, y) (11.4.38) k43 (x, y) = Nx ∂ny 1 d  1  1  2n(x) · n(y) 2 + (1 − ν) = n(z) · n(y) t(z) · (x − y) 2 z=x 4π r dsx r 1 1  4n(x) · (x − y) n(y) · (y − x) 4 + 4π r 1 d $ n(z) · (x − y) n(y) · t(z) 2 + (1 − ν) dsx r 1 % + 2n(z) · (x − y) n(y) · (y − x) t(z) · (x − y) 4 z=x , r

712

11. Boundary Integral Equations on Curves in IR2

k42 (x, y) = −Nx My E(x, y) (11.4.39) 2 (1 − ν)  − n(x) · (x − y) + 2n(x) · n(y) n(y) · (y − x) 4 = 2π r 1 d  − (1 + ν)t(z) · (x − y) n(z) · (x − y) 4 + dsx r 1 + (1 − ν)n(y) · t(z) n(z) · n(y) 2 r 1 + 2(1 − ν)t(z) · (x − y) n(y) · (y − x) n(z) · n(y) 4 r  2 1 + 8 n(y) · (y − x) n(x) · (x − y) 6 r  1 d 2(1 − ν)t(z) · n(y) n(y) · (y − x) n(z) · (x − y) 4 + dsx r  2 1  + 4(1 − ν)t(z) · (x − y) n(y) · (y − x) n(z) · (x − y) 6 , z=x r k41 (x, y) = −Nx Ny E(x, y) (11.4.40)  2 d d 1 (1 − ν) n(ζ) · n(z) t(ζ) · t(z) 2 =− 4π dsx dsy r 1  + 2n(ζ) · n(z) t(ζ) · (y − x) t(z) · (x − y) 4 z=x ζ=y r 1  (1 − ν)  d  n(z) · n(x) t(z) · (x − y) 4 − z=x π dsx r     1 (1 − ν) d n(ζ) · n(x) t(ζ) · (y − x) 4 ζ=y − π dsy r (1 − ν)  d  1 − n(z) · (x − y) t(z) · n(y) 4 π dsx r 1  + 4n(z) · (x − y) t(z) · (x − y) n(y) · (y − x) 6 z=x r 1 (1 − ν)  d  n(ζ) · (y − x) t(ζ) · n(x) 4 − π dsy r 1  + 4n(ζ) · (y − x) t(ζ) · (y − x) n(x) · (x − y) 6 ζ=y r  2 d d (1 − ν) 1 − n(ζ) · t(z) n(z) · t(ζ) 2 4π dsx dsy r  + 2 n(ζ) · (y − x) t(ζ) · n(z) t(z) · (x − y) 1 + n(z) · (x − y) t(z) · n(ζ) t(ζ) · (y − x) 4 r 1 + 2t(z) · t(ζ) n(z) · (x − y) n(ζ) · (y − x) 4 r 1  +8n(ζ) · (y − x) t(ζ) · (y − x) n(z) · (x − y) t(z) · (x − y) 6 z=x . ζ=y r

11.4 Fourier Series Representation of some Particular Operators

713

In order to calculate the pseudohomogeneous expansions of these kernels we collect some simple relations by using the 1–periodic parametric representation of the boundary Γ based on arclength. In addition to (11.1.19)–(11.1.22), we shall also need the following expressions, all of them modulo O4 :  dκ  n(y) · n(x)= 1 − L2 κ2 12 (t − τ )2 + 12 (t − τ )3 L2 κ (t) ,(11.4.41) dt   2 21 2 3 2 dκ 1 t(y) · t(x)= 1 − L κ 2 (t − τ ) + 2 (t − τ ) L κ (t) ,(11.4.42) dt  dκ 1 t(x) · n(y)=L(t − τ ) κ(t) − 2 (t − τ ) (t) dt d2 κ  − 16 (t − τ )2 (L2 κ3 (t) − 2 (t) , (11.4.43) dt  dκ n(x) · t(y)= − L(t − τ ) κ(t) − 12 (t − τ ) (t) dt  2 3 d2 κ  2 1 − 6 (t − τ ) L κ (t) − 2 (t) , (11.4.44) dt dκ (11.4.45) n(y) · (y − x)= − 12 κ(t)L2 (t − τ )2 + 13 (t − τ )3 L2 (t) , dt dκ (11.4.46) n(x) · (x − y)= − 12 κ(t)L2 (t − τ )2 + 16 (t − τ )3 L2 (t) , dt  t(y) · (y − x)= −L(t − τ ) 1 − 16 (t − τ )2 L2 κ2 (t) , (11.4.47)  2 2 2 1 t(x) · (x − y)= L(t − τ ) 1 − 6 (t − τ ) L κ (t) . (11.4.48) Now we are in the position to present the pseudohomogeneous expansions on projector CΩ in (2.4.23); of the operator’s kernels kj (t, t−τ ) in the Calder´ and with Theorem 11.2.2, the corresponding first terms of their symbol expansions. From formulae (11.4.25)–(11.4.28) we obtain   1 1 4 log |t − τ | L2 (t − τ )2 − 12 L (t − τ )4 κ2 (t) k14 T (t), T (τ ) = 8π  dκ 1 3 L (t − τ )5 κ(t) (t) + O6 + O0 , + 12 dt σ14 (t, ξ) = L2 14 |ξ|−3 + L4 14 κ2 (t)|ξ|−5 dκ − iL4 54 κ(t) (t)|ξ|−6 sign ξ + O(|ξ|−7 ) , (11.4.49) dt  −1 log |t − τ |× k13 T (t), T (τ ) = 4π   dκ × 12 κ(t)L2 (t − τ )2 − 13 (t − τ )3 L2 (t) + O4 + O0 , dt σ13 (t, ξ) = − 14 L2 κ(t)|ξ|−3 dκ + i 12 L2 (t)|ξ|−4 sign ξ + O(|ξ|−5 ) , (11.4.50) dt

714

11. Boundary Integral Equations on Curves in IR2

 1 k12 T (t), T (τ ) = − (1 + ν) log |t − τ | + O0 , 4π σ12 (t, ξ) = (1 + ν) 14 |ξ|−1 ,  k11 T (t), T (τ ) σ11 (t, ξ) = 0 .

(11.4.51) O0 , (11.4.52)

Now the next row of CΩ is obtained from the representations (11.4.29)– (11.4.32):   1 log |t − τ | 12 κ(t)L2 (t − τ )2 k24 T (t), T (τ ) = − 4π  dκ − 16 (t − τ )3 L2 (t) + O4 + O0 , dt (11.4.53) σ24 (t, ξ) = − 14 L2 κ(t)|ξ|−3 dκ + i 14 L2 (t)|ξ|−4 sign ξ + O(|ξ|−5 ) , dt   1 k23 T (t), (τ ) = − 4π log |t − τ | 1 − 12 L2 κ2 (t)(t − τ )2  dκ + 12 (t − τ )3 L2 κ (t) + O4 , dt −1 −3 1 1 2 2 σ23 (t, ξ) = 4 |ξ| + 4 L κ (t)|ξ| dκ − i 34 L2 κ (t)|ξ|−4 sign ξ + O(|ξ|−5 ) , (11.4.54) dt  k22 T (t), T (τ ) = O0 , σ22 (t, ξ) = 0 , (11.4.55)    1 1 1 (1 − ν) 1 d + O0 + k21 T (t), T (τ ) = − 2 2 2π L (t − τ ) 4π L dτ L(t − τ ) 1 1+ν =− + O0 , 4π L2 (t − τ )2 1 (11.4.56) σ21 (t, ξ) = (1 + ν) 14 2 |ξ| . L Next, we consider the third row of CΩ , (11.4.33)–(11.4.36).  1+ν k34 T (t), T (τ ) = log |t − τ | + O0 , 4π σ34 (t, ξ) = −(1 + ν) 14 |ξ|−1 , (11.4.57)  k33 T (t), T (τ ) = O0 , σ33 (t, ξ) = 0 , (11.4.58)  1 1 + O0 , k32 T (t), T (τ ) = − (1 − ν)(3 + ν) 2 4π L (t − τ )2 (1 − ν)(3 + ν) |ξ| , (11.4.59) σ32 (t, ξ) = 4L2

11.4 Fourier Series Representation of some Particular Operators

715

 κ(t) (1 − ν)  k31 T (t), T (τ ) = − 2 2π L (t − τ )2  d 1 1 1+ν κ(t) + O0 + 2 L 2 dτ (t − τ ) 1 (1 − ν)(3 + ν) κ(t) 2 + O0 , =− 4π L (t − τ )2 (1 − ν)(3 + ν) σ31 (t, ξ) = κ(t)|ξ| . (11.4.60) 4L2 Finally, we consider the last row (11.4.37)–(11.4.40).  k44 T (t), T (τ ) = O0 , σ44 (t, ξ) = 0 , (11.4.61) 1 1 (1 − ν) d 1 + O0 + k43 T (t), T (τ ) = 2π L2 (t − τ )2 4πL2 dt (t − τ ) 1 1 (1 + ν) 2 = + O0 , 4π L (t − τ )2 (1 + ν) |ξ| , (11.4.62) σ43 (t, ξ) − 4L2   κ(t) 1 1−ν dκ k42 T (t), T (τ ) = − (t) − 2π L2 (t − τ )2 dt L2 (t − τ ) (1 + ν) 1 d  κ(t)  + O0 − 2 L2 dt t − τ   κ(t) 1 1 dκ (t) + O0 , = − (1 − ν)(3 + ν) 2 − 4π L (t − τ )2 dt L2 (t − τ ) (1 − ν)(3 + ν) σ42 (t, ξ) = κ(t)|ξ| 4L2 (1 − ν)(3 + ν) dκ (t) sign ξ , (11.4.63) −i 4L2 dt  (1 − ν)2 d d 1 k41 T (t), T (τ ) = 4 4πL dt dτ (t − τ )2  1 (1 − ν) 1 d  1 2 1 κ − − (t) π L dt L3 (t − τ )3 6 L(t − τ )  1 1 2 1 (1 − ν) 1 d  κ + O0 − (t) + π L dτ L3 (t − τ )3 6 L(t − τ ) 1 6 1 1 (1 − ν)(3 + ν) 4 (1 − ν)κ2 (t) 2 = − 4π L (t − τ )4 3π L (t − τ )2 dκ 1 1 (1 − ν)κ (t) 2 + O0 , + 3π dt L (t − τ ) 

11. Boundary Integral Equations on Curves in IR2

716

1 1 |ξ|3 + (1 − ν) 3 κ2 (t)|ξ| 4L4 3L dκ 1 (11.4.64) − i(1 − ν) 2 κ(t) (t) sign ξ . 3L dt

σ41 (t, ξ) = (1 − ν)(3 + ν)

Since all of the symbol expansions now are available, the action of all these operators on 1–periodic Fourier series have the representation in the form of (11.2.5) where aΦ (t, ξ) in (11.2.3) is to be replaced by σjk (t, ξ), as in the other examples. The first three terms of the asymptotic symbol expansion of the Calder´on projector in (2.4.23) read: σCΩ (t, ξ) ⎛

0

1−ν −1 4 |ξ| 1 2L (1−ν)(3+ν) |ξ| 4L2

(1−ν)(3+ν) |ξ|3 4L4

0

1 2L 1+ν 4L2 |ξ|

⎜ ⎜ ⎜ =⎜ ⎜ ⎝ ⎛

⎛

1 −1 4 |ξ| 1 2L − 1+ν 4L2 |ξ|

⎞

⎟ ⎟ ⎟ ⎟ −1 ⎟ − 1+ν |ξ| 4 ⎠ 0

1 2L 2

0

− L4 κ(t)|ξ|−3

0

0

0

0

(1−ν)(3+ν) κ(t)|ξ| 4L2

0

0

⎜ ⎜ 0 ⎜ + ⎜ (1−ν)(3+ν) ⎜ κ(t)|ξ| ⎝ 4L2 0

L2 −3 4 |ξ|

0

0

⎟ 2 − L4 κ(t)|ξ|−3 ⎟ ⎟ ⎟ ⎟ 0 ⎠ 0 ⎞

iL2 dκ L4 2 −4 −5 sign ξ 2 dt |ξ| 4 κ (t)|ξ| ⎟ 2 2 L iL dκ 2 −3 −4 sign ξ ⎟ ⎟ 4 κ (t)|ξ| 4 dt |ξ|

0

0

0

0

0

0

0

0

1−ν 2 3L2 κ |ξ|

dκ −i (1−ν)(3+ν) 4L2 dt sign ξ

0

0

⎜ ⎜ ⎜ +⎜ ⎜ ⎝

⎞

⎟ ⎟ ⎠

+ ... To conclude this chapter, we now return to the boundary integral equations associated with the boundary value problems of the biharmonic equation. From the symbols σjk (t, ξ) we see that, in contrast to the previous examples, these systems of equations have different orders. We begin with the interior problems as considered in Section 2.4.2. For the interior Dirichlet problem, Theorem 2.4.1, the corresponding system of boundary integral equations (2.4.26), (2.4.27) of the first kind, which are modified to always guarantee unique solvability (independent of so–called Γ –contours), now reads

11.4 Fourier Series Representation of some Particular Operators

V23 σ1 + V24 σ2 + n1 ω1 + n2 ω2 = f2 , V13 σ1 + V14 σ2 + x1 ω1 + x2 ω2 + ω3 = f1 1

717

(11.4.65)

where the solution (σ1 , σ2 ) = (M u, N u) ∈ H − 2 (Γ ) × H − 2 (Γ ) has to satisfy the additional compatibility conditions    ∂pj σ1 + σ2 pj ds = 0 with p1 = 1 , p2 = x1 , p3 = x2 . (11.4.66) ∂n 3

Γ

As we have seen, the solution (ω, σ1 , σ2 ) of the the system (11.4.65), (11.4.66) is unique. In terms of pseudodifferential operators, this is a 2 × 2 system of Agmon–Douglis–Nirenberg type with the orders s1 = t1 = − 12 and s2 = t2 = − 32 ; hence the principal symbol matrix (7.2.12) is given by   1 |ξ|−1 0 (11.4.67) 0 L2 |ξ|−3 4 which obviously is strongly elliptic with Θ(t) = I in (7.2.62). Consequently, 3 due to Theorem 5.3.10, for any given right–hand side (f1 , f2 ) ∈ H 2 +s (Γ ) × 1 H 2 +s (Γ ), the system (11.4.65), (11.4.66) has a unique solution (ω, σ1 , σ2 ) ∈ 1 3 IR3 × H − 2 +s (Γ ) × H − 2 +s (Γ ) with any s ∈ IR. For the exterior Dirichlet problem we have the same system of equations, however, the right–hand sides in (11.4.65) and (11.4.66) are different, see (2.4.28). Similarly, for the interior Neumann problem, the system of boundary integral equations of the first kind is now D41 ϕ1 + D42 ϕ2 = g2 D31 ϕ1 + D32 ϕ2 = g1 1

(11.4.68) 3

where the right–hand sides (g1 , g2 ) ∈ H − 2 +s (Γ ) × H − 2 +s (Γ ) are given in Table 2.4.5. These equations always have a three–dimensional kernel given by R=

3 2

 v,

 ∂v   |Γ v = a1 x1 + a2 x2 + b with (a1 , a2 , b) ∈ IR3 . (11.4.69) ∂n

The system (11.4.68) is of Agmon–Douglis–Nirenberg type with s1 = t1 = and s2 = t2 = 12 . The principal symbol is given by   1 L14 |ξ|3 0 (11.4.70) (1 − ν)(3 + ν) 1 0 4 L2 |ξ|

which obviously is strongly elliptic with Θ = I since ν ∈ [0, 1). Therefore, the Fredholm alternative holds for (11.4.68) and the solution is unique only

11. Boundary Integral Equations on Curves in IR2

718 3

1

in (H 2 +s (Γ ) × H 2 +s (Γ ))/R, and (g1 , g2 ) have to satisfy orthogonality con1 ditions which are satisfied if the given boundary data M u = ψ1 ∈ H − 2 +s (Γ ) 3 and N u = ψ2 ∈ H − 2 +s (Γ ) with s ∈ IR satisfy the compatibility conditions (2.4.16). As we discussed in Section 2.4.2, the right–hand sides (g1 , g2 ) in (11.4.68) satisfy the orthogonality conditions and the system (11.4.68) then has a solution which is not unique due to the kernel R of this operator system.

11.5 Remarks Boundary integral equation methods for two–dimensional boundary value problems and various examples are considered by Yu in [442] based on the work by Feng Kang [117]. Boundary integral equations on periodic functions and corresponding Fourier analysis can be found in the books by Gohberg and Fel’dman [151] and Gohberg and Krupnik [152]. In McLean et al [291] and Lamp et al [259], periodic pseudodifferential operators are treated with Fourier approximations. For a slightly larger class of periodic pseudodifferential equations than those in 11.1–11.4, Saranen and Vainikko present in their book [369] a very extensive analysis employing Fourier series representations.

12. Remarks on Pseudodifferential Operators Related to the Time Harmonic Maxwell Equations

12.1 Introduction The class of pseudodifferential operators is essentially the smallest algebra of operators which contains all differential operators, all fundamental solutions of elliptic differential operators and all integral operators with pseudohomogeneous kernel expansions. The linear pseudodifferential operators can be characterized by generalized Fourier multipliers, known as symbols. The development of the theory of pseudodifferential operators has made it possible to provide a unified treatment of differential and integral operators. These materials are subjects of Chapter 7 to Chapter 11. However, for a quick glance of the materials, we refer to our paper [206] with emphasis on boundary integral equations recast as pseudodifferential equations. In this chapter, our aim is to follow the approach in Chapter 9 to analyze the Maxwell system from a pseudodifferential equation’s point of view. We begin with the Maxwell equations for time–harmonic electromagnetic fields given by (6.4.1)–(6.4.4), i.e., curl E(x) = iωμH(x) curl H(x) = (σ − iωε)E(x) + J0 div (εE) = ,

(12.1.1)

div (μH) = 0 where J0 and  are given, satisfying the compatibility condition (6.4.5), ? (12.1.2) (x) = div J0 (x) (iω − σ/ε). For simplicity we assume, μ, ε, σ and ω are given as real positive constants. We reduce the system (12.1.1) to a second order partial differential equation for E in the form PE := curl curl E − k 2 E = f := iωμ J0 in R3 where k 2 = (iμω)(σ − iωε). Since curl curl E = −ΔE + grad div E, © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6_12

(12.1.3)

720

12. Remarks on Pseudodifferential Operators for Maxwell Equations

(12.1.3) can now be written as PE = −(ΔE + k 2 E) + grad div E = f in R3 .

(12.1.4)

In particular, we find the symbol of P, its inverse and the fundamental solution, and define the surface potentials. As will be seen, the standard Sobolev spaces here do not lead to the correct solution space for the Maxwell system.

12.2 Symbols of P and the corresponding Newton potentials A simple computation shows that the symbol of P is given by ⎞ ⎛ 2 |ξξ | − k 2 − ξ12 , −ξ2 ξ1 , −ξ3 ξ1 ⎠ −ξ1 ξ2 , |ξξ |2 − k 2 − ξ22 , −ξ3 ξ2 σP (ξξ ) = ⎝ −ξ1 ξ3 , −ξ2 ξ3 , |ξξ |2 − k 2 − ξ32   = (|ξξ |2 − k 2 )δj − ((ξj · ξ )) .

(12.2.1)

j,=1,2,3

For the determinant of σP (ξξ ) one obtains  2 Det σP (ξξ ) = −k 2 |ξξ |2 − k 2 ,

(12.2.2)

and for the inverse of σP (ξξ ) have −1  −1 2  = Det σP (ξξ ) (|ξξ | − k2 )× σP (ξξ ) ⎛ 2 ⎞ (|ξξ | − k 2 ) − (ξ22 + ξ32 ), ξ1 ξ 2 ξ1 ξ3 ⎝ ⎠ ξ1 ξ2 (|ξξ |2 − k 2 ) − (ξ12 + ξ32 ), ξ3 ξ 2 ξ1 ξ3 ξ2 ξ3 (|ξξ |2 − k 2 ) − (ξ12 + ξ22 ) ⎛ 2 ⎞ ξ 2 ξ1 ξ3 ξ1 ξ1 − k 2 , −1 ⎝ ξ1 ξ2 ξ22 − k 2 ξ3 ξ2 ⎠ = 2 2 k (|ξξ | − k 2 ) ξ1 ξ3 , ξ2 ξ 3 ξ32 − k 2 −1    1 1 = δj − 2 ((ξj ξ )) ; j,  ∈ {1, 2, 3}. σP (ξξ ) k |ξξ |2 − k 2

(12.2.3)

This is also the symbol of the Newton potential associated with P. We notice that this symbol of the Newton potential consists of two parts, in contrast to the elasticity and the Stokes systems as well as the scalar Helmholtz equation:  −1 σN (ξξ ) = σP (ξξ ) = N1 (ξξ ) + N2 (ξξ ) with N1 (ξξ ) =

1 1 1 δj , N2 (ξξ ) = − 2 ((ξj ξ )) 2 . 2 2 ξ ξ |ξ | − k k |ξ | − k 2

(12.2.4)

12.3 Representation formulae

Correspondingly, two parametrices are given by   1 ξ Fy→ξ f (y) , H1 f := Fξ−1 →x χ(|ξ |) ξ 2 2 |ξ | − k 1 −1  1 Fy→ξ f (y)) H2 f := − 2 Fξ →x χ(|ξξ |)((ξj ξ )) 2 k |ξξ | − k 2

721

(12.2.5)

with the cut–off function χ(η) is given in (7.1.5), where H1 ∈ OP S −2 (R3 × R3 ) and H2 ∈ OP S 0 (R3 × R3 ) are properly supported. The asymptotic expansions for N1 and N2 are:  1  −6 4  k2 + + O |ξξ | k δj σN1 (ξξ ) = |ξξ |2 |ξξ |2 ∞  k 2p−2 σN2 (ξξ ) = −((ξj ξ )) (12.2.6) |ξξ |2+2p p=0 1 1  −6 2  1 . + + O |ξξ | k = −((ξj ξ )) 2 k |ξξ |2 |ξξ |4 Then

ξ ξ −6 k 4 )Fy→ξ f (y) (N1 − H1 )f = Fξ−1 →x ψ(|ξ |)O(|ξ |  is a smoothing operator, where ψ(η) = 1 − χ(y) is a cut–off function with ψ(η) ≡ 1 for small η ≥ 0 and ψ(η) ≡ 0 for 1 ≤ η (see (7.1.5)). Similarly, ξ ξ −6 k 2 )Fy−1 (N2 − H2 )f = Fξ−1 →x ψ(|ξ |)O(|ξ |) →ξ f (y). Based on (12.2.3), the fundamental solution can be recovered by using the Fourier inverse transform which leads to   1 Γ (x, y) = I + 2 grad x divx Gk (x − y) k (12.2.7) 1 eik|x−y| Gk (x − y) = I. 4π |x − y|

12.3 Representation formulae In order to find the surface potentials on the boundary Γ = ∂Ω of a bounded domain Ω, we follow Section 10.2 and begin with the representation formula which is obtained by using the second Green’s formula (6.9.2) and Appendix B, II(8),  {Q · curl curl E − E · curl curl Q}dy Ω

 (E × curl Q − Q × curl E) · ndsy

= Γ

722

12. Remarks on Pseudodifferential Operators for Maxwell Equations

with E and Q = Γ from (12.2.7):  E(x) = Γ (x − y)f (y)dy − Γ (x − y), (n × curl E)Γ Ω



 − curl y Γ (x − y), n × E(y) Γ , Γ (x − y)f (y)dy − I, Γ (x − y)(n × curl E)Γ

= Ω

+ I, curl y Γ (x − y)(n × E(y))Γ . Or, since n × E = n × πt E, we obtain   E(x) = Γ (x − y)f (y)dy − Γ (x − y)(n × curl E)dsy Ω

 +

Γ

 curl y Γ (x − y) n × πt E(y) dsy

(12.3.1)

Γ

in which we may consider n × curl E and πt E as the Neumann and Dirichlet Cauchy data of PE in (12.1.3), respectively. Alternatively, Equation (12.3.1) may be written in the form:  (12.3.2) E(x) = Γ (x − y)f (y)dy Ω

 − Γ

+

1 k2 

+

 Gk (x − y) n(y) × curl E(y) dsy 



 grad y Gk (x − y) divΓ n × curl E(y) dsy

Γ



grad y Gk (x − y) × (n × E)(y)dsy for x ∈ Ω,

Γ

where we can show that   1 Γ (x − y)f (y)dy = {Gk (x − y)(iωμ)J0 (y) + (y)grad y Gk (x − y)}dy ε Ω Ω  +(σ − iωε)−1 {grad y Gk (x − y) n · J0 (y)}dsy . Γ

Hence, Equation (12.3.2) is exactly the equation (6.9.4) with iωμ(n × H) replaced by n × curl E and ! " n · E = − k −2 divΓ (n × curl E) + (σ − iωε)−1 n · J0 . Thus, for f = 0 the electric field E(x) in Ω is completely determined if the corresponding Cauchy data on Γ are known.

12.4 Symbols of the Electromagnetic Boundary Potentials

723

12.4 Symbols of the Electromagnetic Boundary Potentials In the following, for simplicity, we consider only the principal symbols, i.e., the surface Γ is considered as a two–dimensional plane with a local orthonormal positively oriented basis e1 , e2 , n. The simple layer potential has the form  λ(x) := − Γ (x − y)λ(y)dsy , x ∈ Ω (12.4.1) Vλ Γ

where λ is a tangential field on Γ and can be written as a Newton potential operating on the distribution γ0∗ λ = λ ⊗ δΓ which has the form   Vλ(x) = − Γ (x − y) λ(y) ⊗ δΓ dy , x ∈ R3 \ Γ R3



=−



I+

R3



=− R3

1 − 2 k

  1 grad x divx Gk (x − y) λ(y) ⊗ δΓ dy k2

 Gk (x − y) λ(y) ⊗ δΓ dy 

(12.4.2)

   grad x divx Gk (x − y) λ(y) ⊗ δΓ dy.

R3

The first potential considered as a Newton potential   V 1 λ(x) = − Gk (x − y) λ(y) ⊗ δΓ dy

(12.4.3)

R3 3 is a pseudodifferential operator in L −2 c (R ), hence,  Gk (x − y  )λ(y  )dsy for x ∈ Γ lim  V 1 λ(x) = − x→x

y  ∈Γ

which defines a pseudodifferential operator of order m + 1 = −1 ; Vk ∈ L−1 c (Γ ) on Γ where Vk is the boundary integral operator  Gk (x − y  )λ(y  )dsy for x ∈ Γ. Vk λ(x ) = y  ∈Γ

The second potential can also be considered as a Newton potential applied to γ0∗ λ = λ(y  ) ⊗ δΓ :

724

12. Remarks on Pseudodifferential Operators for Maxwell Equations

1 V 2 λ(x) := − 2 k



  grad x divx Gk (x − y)(λ(y) ⊗ δΓ dy

(12.4.4)

R3

for x ∈ R3 \ Γ which is a pseudodifferential operator in L0c (R3 ) with the symbol e2 n e1 ⎛ 2 ⎞ ∞ ξ1 ξ1 ξ2 ξ1 ξ3 e1 1 1    k 2   σV 2 (ξ) = − e2 ⎝ξ1 ξ2 ξ22 ξ2 ξ3 ⎠ 2 2 |ξ| k |ξ|2 n ξ1 ξ3 ξ2 ξ3 ξ32 =0  1 = a00 (ξ) + O |ξ|−2 2 , k the principal part of which is given explicitly as ⎞ ⎛ 2 ξ1 ξ2 ξ1 ξ3 ξ1 1 1 a00 (ξ) = − ⎝ξ1 ξ2 ξ22 ξ2 ξ3 ⎠ 2 2 . |ξ| k ξ1 ξ3 ξ2 ξ3 ξ32

(12.4.5)

(12.4.6)

According to Theorem 7.1.12, the remainder defines a pseudodifferential operator of order −2 whose limit Ω  x → x ∈ Γ exists and defines a boundary integral operator or pseudodifferential operator on Γ in L−1 c (Γ ). ormander’s For the principal part a00 (ξ) Theorem 9.5.1 can also be applied, if H¨ transmission condition is satisfied. Indeed, if we choose the coordinates such that the direction of the normal vector n coincides with the third component. H¨ormander’s transmission condition (9.3.40) in Theorem 9.3.11 with m = 0 , j = 0 , α = 0 reads ⎛ ⎞ ⎛ ⎞  α  α 0 0 0 0 0 0 ∂ ⎝0 0 0⎠ = ∂ a0 ( , 0; 0, +1) = a0 ( , 0; 0, −1) = ⎝0 0 0⎠ . ∂ξ ∂ξ 0 0 1 0 0 1 Hence, the transmission and extension conditions are satisfied. Next, we observe that the parity condition (8.1.75) for the only case −3 − q = 0 for q ∈ Z is obviously satisfied for σ = 0 since all entries in a00 (ξ) in (12.4.6) satisfy (−ξj )(−ξ ) = ξj ξ . Lemma 9.4.1 implies that also the Tricomi conditions are satisfied. As a consequence we conclude that the operator V2 is a pseudodifferential operator in L 0c (R3 ) and its restriction to Γ defines a pseudodifferential operator on Γ in L 1c (Γ ). However, the boundary integral operator of order 1 on Γ might contain a surface tangential differential operator T (see Theorem 9.4.6). Hence,

12.4 Symbols of the Electromagnetic Boundary Potentials

725

(−k 2V 2 λ)(x) = T λ(x )    + p.f. n(x ) · grad x divx Gk (x − y  ) λ1 (y  )e1 (y  ) lim

Ω x→x ∈Γ

Γ

 + λ2 (y  )e2 (y  ) x=x dsy n(x )     + p.f. e1 (x ) · grad x divx Gk (x − y  ) λ1 (y  )e1 (y  ) Γ

 + λ2 (y  )e2 (y  ) x=x dsy e1 (x )    + p.f. e2 (x ) · grad x divx Gk (x − y  ) λ1 (y  )e1 (y  ) Γ

+ λ2 (y  )e2 (y  )

 

x=x

dsy e2 (x ), x ∈ Γ,

where e1 (x ) , e2 (x ) and n(x ) form a basis of a local orthogonal coordinate system. The tangential differential operator TV 2 λ = − k12 T λ and T λ is explicitly given by (9.4.26): T =

1  2



(−i)|α|+1

0≤j≤1 |α|=1−j

 1 ∂α a00−j   , 0, −(0, 1) Dα .  α α! (∂ξ )

a00 (ξ) is given in (12.4.6) and a0−1 = 0. With ξ = (ξ1 , ξ2 , 0) we find ⎛ ⎛ ⎞ ⎞ 2ξ1 ξ2 ξ3  0 0 1  ∂  ∂ ∂ ∂ a0 λ = ⎝ ξ2 0 0 ⎠ ( , 0), (0, 1) λ = ⎝0 0 0⎠ λ1 ∂ξ1 0 ∂x1 ∂x1 ∂x1 ξ3 0 0 1 0 0 ⎛ ⎛ ⎞ ⎞ 0 ξ1 0  0 0 0  ∂  ∂ ∂ ∂ a00 λ = ⎝ξ1 2ξ2 ξ3 ⎠ ( , 0); (0, 1) λ = ⎝0 0 1⎠ λ2 ∂ξ2 ∂x2 ∂x2 ∂x2 1 1 0 0 ξ3 0 and T V2 λ = −

1 1  ∂λ1 ∂λ2  n(x ). + ( 2 k 2 ∂x1 ∂x2

Then 1 1 V2 λ)(x) = − divΓ λ(x )n(x ) lim (V 2 k2    1 − 2 p.f. grad x divx Gk (x − y  )λ(y  ) x=x dsy . k

x→x ∈Γ

(12.4.7)

Γ

This result is in agreement with the relation (5.5) of Section 6.5 in the book by E. Meister [293]. For the double layer potential in (12.3.1), i.e.,

726

12. Remarks on Pseudodifferential Operators for Maxwell Equations

 Wϕ)(x) := − (W

ϕ(y) × grad y Gk (x − y)dsy Γ



= −curl x

Gk (x − y)ϕ(y)dsy for x ∈ Ω,

(12.4.8)

Γ

we consider it as a double layer Newton potential in the form   Wϕ)(x) = −curl x Gk (x − y) ϕ(y) ⊗ δΓ )dy , x ∈ R3 \ Γ. (W R3

Then the symbol of W , σW (ξ) = −iξ ×

1 I, |ξ|2 − k 2

has the asymptotic expansion ξ    k 2 2  ×I (12.4.9) |ξ|2 |ξ|2 =0 ⎞ ⎛ ∞ 0 iξ3 −iξ2 1    k 2   ⎠ ⎝ 0 iξ1 for |ξ| > |k|, = −iξ3 |ξ|2 |ξ|2 iξ2 −iξ1 0 =0 ⎛ ⎞ 0 iξ3 −iξ2 1 0 iξ1 ⎠ 2 + O(|ξ|−3 k 2 ) = a0−1 (ξ) + O(|ξ|−3 k 2 ). = ⎝−iξ3 |ξ| iξ2 −iξ1 0 ∞

σW (ξ) = −i

3 The principal part defines a pseudodifferential operator in L−1 c (R ) and the −3 3 0 remainder is in L c (R ). The principal symbol a−1 (ξ) satisfies H¨ormander’s transmission condition (9.3.40) in Theorem 9.3.11 and extension is guaranteed. a0−1 (ξ) also satisfies the parity condition (9.4.2) in Lemma 9.4.1 with parity σ = 1. Consequently, the Tricomi condition (9.3.18) is satisfied in view of Lemma 9.3.5 and W on Γ defines a pseudodifferential boundary integral operator in L 0c (Γ ). According to Theorem 9.4.6, the tangential differential operator TW is given by Equation (9.4.26).

 1 (−i)a0−1  , 0, ; (0, 1) ϕ(x ) (12.4.10) 2 e1 e2 n ⎛ ⎞ ⎛ ⎞ ϕ1 (x ) e1 0 i 0 1 = e2 ⎝−i 0 0⎠ (−i) ⎝ϕ2 (x )⎠ 2 n 0 0 0 0  1 1 = ϕ1 (x )e2 − ϕ2 (x )e1 = − n × (e1 ϕ1 + e2 ϕ2 ) 2 2

TW ϕ =

12.4 Symbols of the Electromagnetic Boundary Potentials

727

since j = 0 and α = 0, Hence, 1 Wϕ)(x) = − n(x ) × ϕ(x ) − W k ϕ(x ) (W 2   where W k ϕ(x ) = p.v. ϕ(y) × grad y Gk (x − y)dsy . lim

Ω x→x ∈Γ

(12.4.11)

Γ

Without loss of generality we consider the representation formula (12.3.2) for the solution of the homogeneous equation with f = 0:  E(x) = − Gk (x − y)λ(y)dsy Γ

 1 grad Gk (x − y)divΓ λ(y)dsy x k2  Γ − curl x Gk (x − y)ϕ(y)dsy −

Γ

= V 1 (λ) + V 2 (λ) + W (ϕ)

(12.4.12)

where λ(y) = (n × curl E)(y) and ϕ(y) = (n × E)(y) are the Cauchy data on Γ for PE in Ω, with potentials V 1 (12.4.3) V 2 (12.4.4) and W (12.4.8). In what follows, the limit is taken as Ω  x → x ∈ Γ of (12.4.12) and we will arrive at a first boundary integral equation  V 1 λ(x) + V 2 λ(x) + W ϕ(x } πt E(x ) = πt lim x→x ∈Γ  1 1   div λ · n(x ) = −πt Vk λ(x ) + πt − Γ 2 k2  1 1 − 2 grad Γ Vk divΓ λ(x ) + πt − n(x ) × ϕ(x ) − (Wk ϕ)(x ) k 2   1 = − πt Vk λ(x ) + 2 grad Γ Vk divΓ λ(x ) k 1  − (n × ϕ)(x ) − πt W k ϕ(x ) for x ∈ Γ (12.4.13) 2 where Vk g(x ) :=

Gk (x − y)g(y)dsy , x ∈ Γ.

Γ

By applying n × curl E to (12.4.11) and taking the limit, we obtain a second boundary integral equation,   γt curl Γ E = γt lim curl V 1 λ + lim curl V 2 λ + lim curl W ϕ . Again, the limit is taken as Ω  x → x ∈ Γ . The first term is

728

12. Remarks on Pseudodifferential Operators for Maxwell Equations

 γt lim curl V 1 λ = γt lim



grad y Gk (x − y) × λ(y)dsy = γt lim W λ

Γ

1 = − γt (n × λ)(x ) − γt W k λ(x ), 2 1 = πt λ(x ) − γt W k λ(x ). 2 The second term is πtV 2 = γt lim curl V 2 λ  1 = 2 γt lim curl x grad y Gk (x − y)divΓ λ(y)dsy k Γ  1 = − 2 γt lim curl x grad x Gk (x − y)divΓ λ(y)dsy = 0. k Γ

For the third term we have the relation



γt lim curl xW ϕ(x) = −γt lim curl x curl x  = γt lim curl x p.v.  = γt p.f.



Gk (x − y)ϕ(y)dsy Γ

grad y Gk (x − y) × ϕ(y)dsy

Γ



 curl x grad y Gk (x − y) × ϕ(y) |x=x dsy

Γ

= −γt curl W k ϕ(x ). Collecting terms, we obtain the second boundary integral equation: γt curl E(x ) =

1 πt λ(x ) − γt W k λ(x ) − γt curl W k ϕ(x ). 2

(12.4.14)

In both boundary integral equations, we need to substitute λ and ϕ by the Cauchy data λ = γt curl E and ϕ = n × πt E, respectively. In the same manner as for the Laplace equation we are able to define a Calderon projector for the differential equation (12.1.4), curl curl E − k 2 E = 0 in terms of the Cauchy data πt E and γt curl E in the form   (   ' 1  1 2 πt E πt E 2 I − πt W k , − πt Vk + k2 grad x Vk divΓ = 1 2 k, γt curl E γt curl E −γt curl W 2 I − γt W k (12.4.15)

12.5 Symbols of boundary integral operators

where



u(y) × grad y Gk (x − y)dsy ,

W k u := p.v.

729

(12.4.16)

Γ

2 k u = W k (n × u) for x ∈ Γ. W

(12.4.17)

Equation (12.4.15) defines an alternative Calderon projector which is similar to the Calderon projector which is given by (6.10.6), see also (6.4.1)–(6.4.4). With the availability of the two boundary integral equations (12.4.13) and (12.4.14) we now compute their symbols in the next section.

12.5 Symbols of boundary integral operators For

 Vk λ = −γ0V 1 λ =

Gk (x − y)λ(y)dsy

Γ

p   ∞  ik(|x − y|) 1 λ(y)dsy with x ∈ Γ. = 4π p=0 p!|x − y| Γ

Fourier transform of the Schwartz kernel’s pseudohomogeneous expansion defines the symbol expansion σVk (ξ  ) = where αp = 21−p Γ

∞ 1  αp |ξ  |−p−1 4π p=0

(12.5.1)

 1−p 7  p+1 Γ 2 p!. Its principal symbol is 2 a00 (ξ  ) =

1  −1 |ξ | . 2

For the second simple layer potential operator we find the pseudohomogeneous kernel expansion 1 1 1 n(x )divΓ λ(x ) − 2 grad Γ,x Vk divΓ λ 2 k2 k 1 1 = − 2 n(x )divΓ λ(x ) k 2   ∞  (ik|x − y|)p  1 1 + 2 λ(y)dsy grad Γ,y k 4π p!|x − y| p=0

lim V 2 λ = −

Γ

whose two–dimensional Fourier transform yields

(12.5.2)

730

12. Remarks on Pseudodifferential Operators for Maxwell Equations

1 1  E *1 + iξ2 λ *2 ) n(x )(iξ1 λ lim V 2 λ(ξ ) = 2 k 2   ∞ 1 1 iξ1  *1 , λ *2 ). + 2 αp |ξ  |−p−1 (λ k 4π iξ2 p=0 Hence, the symbol reads σγ0V 2 (x , ξ  ) =

1 1 n(x )(iξ1 , iξ2 ) k2 2 ∞  2 1 1  ξ1  −p−1 − 2 αp |ξ | o k 4π p=0

 0 . ξ22

(12.5.3)

Finally, the double layer potential  lim

x→x ∈Γ

W ϕ(x) = −

lim

x→x ∈Γ

1 = − n(x ) × ϕ(x ) + p.v. 2



Gk (x − y)ϕ(y)dsy

curl x



Γ

grad y Gk (x − y) × ϕ(y)dsy

Γ

which has the pseudohomogeneous expansion 1 lim W ϕ(x) = n(x ) × ϕ(x ) 2   ∞  ∂ (ikr)p  1  1 + p.v. (y − x ) × ϕ(y)dsy  4π ∂r p!r r r=|x −y| p=0

x→x ∈Γ

Γ

  ∞ ∂ (ikr)  1  1 1    = − n(x ) × ϕ(x ) + p.v. ×  2 4π p=0 ∂r p!r r r=|x −y| Γ   (y1 − x1 )ϕ2 − (y2 − x2 )ϕ1 n(x )dsy since ϕ(y) is a tangential field. The symbol of γ 0W : 1 ϕ(ξ  ) = − n(x ) × ϕ(ξ * ) σγ0W (x , ξ  )* 2  ∞  1  −p−1 −iξ2 + αp |ξ | 0 4π p=0

(12.5.4) 

0 * (ξ  )n(x ). ϕ iξ1

With the symbols of γ0V 1 , γ0 V2 and γ0W available we return to the capacity operators defined by the Calderon projector in (6.10.6) and in (12.4.15).

12.6 Symbols of the Capacity Operators

731

12.6 Symbols of the Capacity Operators For the interior domain Ω, the operator TE in (6.10.2) on the boundary Γ is given by  TE (n × H)(x ) = −(iωμ)n(x ) × lim V 1 (n × H)(x) x→x ∈Γ  + lim V 2 (n × H)(x) x→x ∈Γ  = −(iωμ) n(x ) × Vk (n × H)  1 + n(x ) × 2 grad Γ,x Vk divΓ (n × H) . k        σTE (x , ξ  ) = iωμ n(x ) × σγV V1 (x , ξ ) + n(x ) × σγV V2 (x , ξ )   = iωμ n(x ) × − σVk (x )    ξ12 0  1 I + n(x ) × − 2 σVk (ξ  ) 0 ξ22 k    2  1 0 −1  ξ 0  = iωμ σVk (ξ  ) + 2 σVk (ξ  ) 1 (12.6.1) 1 0 0 ξ22 k   ∞ 0 −1 1  = −iωμ αp |ξ  |p−1 1 0 4π p=0  2    ∞ 0 −1 1 1  0  p−1 ξ1 − iωμ α |ξ | p 1 0 k 2 4π 0 ξ22 p=0

For TH (n × E) (see (6.9.5)) the symbol can be computed in the same manner   ∞ 0 −1 1  αp |ξ  |p−1 (12.6.2) σTH (x , ξ  ) = −(σ − iωε) 1 0 4π p=0    2  ∞ 0 −1 1 1  0  p−1 ξ1 − (σ − iωε) αp |ξ | . 1 0 k 2 4π 0 ξ22 p=0 For the double layer Newton potential’s boundary values one obtains for Ω  x → x ∈ Γ :  1 n(x ) × lim W (n × E)(x) = − n(x ) × n(x ) × n(x ) × E 2   + n(x ) × grad y Gk (x − y) × (n × E)dsy Γ

1 = n(x ) × E(x ) − R(n × E). 2

732

12. Remarks on Pseudodifferential Operators for Maxwell Equations

From (12.5.4) we find that the principal symbol of R vanishes: 0 σR (x , ξ )ϕ * = n(x ) ×

∞  1 αΓ |ξ  |p−1 (−iξ2 ϕ *1 + iξ1 ϕ *2 )n(x ) = 0. (12.6.3) 4π p=0

On the other hand, note that the operator R in (6.9.3) is for x ∈ Γ given by R(n × E)(x ) = p.v. n(x ) × curl x Vk (n × E). It has the pseudohomogeneous kernel expansion ∞ 1  (ik|x − y|)p , x ∈ Γ, −n(x ) × curl x Vk = −n(x ) × curl x 4π p=0 p!|x − y| 



with the symbol expansion 

0 σR (x , ξ ) = −n(x ) × −iξ2 





 ∞ iξ1  1 αp |ξ  |p−1 . 0 4π p=0

The symbol is of order 0 and the inverse Fourier transform on the two– dimensional manifold Γ seems to define ψdo0 on Γ . However, the operator can be written as  ∂Gk (x − y) R(n × E)(x ) = −n(x ) × (n × E)(y)dsy for x ∈ Γ, ∂nx Γ \{x }



= −n(x ) × K k (n × E)(x )

(12.6.4)

with the double layer potential operator K k of the Helmholtz equation (see Section 2.1). For smooth Γ , the Schwartz kernel has a pseudohomogenous expansion of order −1, and on the two–dimensional manifold Γ it defines (Γ ) which is compactly imbedded in a ψdo−1 mapping Hst (Γ ) into Hs+1 t Hst (Γ ). In summery, the symbols of the capacity operators in the Calderon projectors in Ω, (6.9.6) are given in Equations (12.6.1), (12.6.2), (12.6.3). In the exterior domain Ω c = R3 \ Ω, the boundary integral operators have a similar form. The Calderon projector in (12.4.15) will provide two boundary integral equations for the missing parts of the Cauchy data, i.e. πt E or γt curl E. If the corresponding integral equation is solved, the solution of P E = 0 in Ω, as the boundary value problem is then obtained by the representation formula (12.3.2).

12.7 Boundary Integral Operators for the Fundamental Problems

733

12.7 Boundary Integral Operators for the Fundamental Boundary Value Problems The symbols of the boundary integral operators in (12.4.12) are given by     σπt E (x , ξ ) = σ−πt Vk (x , ξ ) + σπt γV V2 (x , ξ ) + σπt γW (x , ξ )   ∞  1 0 1  =− αp |ξ  |−p−1 0 1 4π p=0     ∞ 1 1    −p−1  ξ12 0 1 0 −1 − 2 − . |ξ | 0 ξ22 k 4π 2 1 0 p=0

For equation (12.4.14), then one finds σγt curl E (x , ξ ) = σγt γW (x , ξ  ) + σγt curl γW (x , ξ  )     ∞ 1 1 0 1   1−p 1 0 = . − αp |ξ | 0 1 2 0 1 4π p=0 For the first boundary integral equation (12.4.14), namely πt E =

! " 1 2 k ◦ πt E − πt Vk + 1 grad x Vk divΓ (γt curl E), πt E − πt W 2 k2

it is a boundary integral equation of the second kind for πt E if γt curl E is given on Γ . In terms of symbols, the second kind integral equation for πt E reads   1E 1 curl E) πt E = −πt σVk (x , ξ ) + 2 σgrad x Vk (x , ξ  )divΓ (n × 2 k + O(|ξ |−2 )(πE E, γt curl E) t for given γt curl E on Γ . This corresponds to a classical Fredholm integral equation of the second kind. On the other hand, in the second boundary integral equation (12.4.14), 2 k (πt E) = ( 1 I + γt W k )(γt curl E) −γt curl W 2 the unknown πt E satisfies a boundary integral equation with the corresponding symbols:   1 0  1 0 E a0 |ξ | πE − 0 1 t 4π  1  I + γt σW k (x , ξ  ) γt curl E + O|ξ  |−1 (πE = t E, γt curl E). 2

734

12. Remarks on Pseudodifferential Operators for Maxwell Equations

This is a hypersingular integral equation for πt E with a coercive principal part and is a Fredholm mapping of index zero. For the boundary value problem with πt E on Γ given, the second equation in (12.4.15) will be a classical Fredholm integral equation for the unknown (γt curl E), whereas the first equation (12.4.13) is a Fredholm equation of the first kind with Fredholm index zero. Either one of the boundary integral equations in (12.4.14) can be used for finding the missing Cauchy datum. A similar analysis can be obtained for the system (12.4.15). For the EFIE equations in (6.12.10) πt Vk J +

1 grad Γ {Vk divΓ J} = g for x ∈ Γ, k2

the Hodge decomposition of J has the form J = grad Γ ϕ + curl Γ ψ where ϕ ∈ {ϕ ∈ H∗1 (Γ ) such that ΔΓ ϕ ∈ H −1/2 (Γ ) with ϕ, 1 = 0} and ψ ∈ H 1/2 (Γ ). In terms of ϕ and ψ we have the equation as πt Vk (grad Γ ϕ + curl Γ ψ) +

1 (grad Γ Vk (ΔΓ ϕ) = g. k2

(12.7.1)

In local coordinates in directions of e1 and e2 in the tangential plane, equation (12.7.1) can be written as   1 πt Vk (grad Γ ϕ + curl Γ ψ) − 2 grad y Gk (x, y) ΔΓ ϕdsy = g. (12.7.2) k Γ

An existence proof for (12.7.2) has been established in Theorem 6.12.6 which will be revisited in Section 12.10 below.

12.8 Coerciveness and Strong Ellipticity From the symbol σP (x , ξ ) in (12.2.1) of the differential operator P with the determinant (12.2.2) we see that the principal part of the operator has vanishing determinant. Hence, P is not elliptic in the sense of Petrovski. For k = 0, the determinant of the complete symbol vanishes, thus P is not elliptic either in the sense of Agmon, Douglis, Nirenberg. However, for k = 0 the inverse symbol matrix of P exists and determines a fundamental solution Γ (x, y), see (12.2.7). Moreover, if solutions of the Maxwell system are sought in the subspace of H1 (Ω) satisfying div E = 0 i.e., as a solenoidal field, then in that subspace G˚ arding’s inequality is valid and now the system is strongly elliptic.

12.8 Coerciveness and Strong Ellipticity

735

According to M.S.Agranovich (see [5]) the Maxwell system has ”hidden ellipticity”. For the Maxwell equations (12.1.1) in Ω and div E = 0 in Ω, it motivates us to consider the Vektor–Helmholtz equation curl curl E − grad div E − k 2 E = f in Ω. Costabel considers in [83] (see also [63, p. 60], [96]) the weak variational formulation,  a0 (E, Et ) = {curl E · curl Et + div E div Et − k 2 E · Et }dsx Ω



 (div E)n · E ds −

= Γ

(12.8.1) E · (n × curl E)ds

t

t

Γ

for all test functions Et ∈ H1n or Et ∈ H1t (Ω). Then a0 (E, E) satisfies G˚ arding’s inequality  a0 (E.E) = {|curl E|2 + (div E)2 − k 2 |E|2 }dx Ω

≥ E2H1 (Ω) − (k 2 + 1)E2L2 (Ω) n

for all E in the subspace H1n (Ω) = {u ∈ H1 (Ω) with γt (u) = 0}, and also satisfies G˚ arding’s inequality a0 (E, E) ≥ E2H1 (Ω) − (k 2 + 1)E2L2 (Ω) t

on the second subspace H1t (Ω) := {u ∈ H1 with γn (u) = 0} of H1 (Γ ). Hence, the boundary value problems P E = 0 for E ∈ H1t (Ω) with the natural boundary condition div E|Γ = g ∈ H−1/2 (Γ ) on Γ and P E = 0 in Ω on the subspace H1n (Ω) with the second natural boundary condition

(12.8.2)

736

12. Remarks on Pseudodifferential Operators for Maxwell Equations

n × curl E = g ∈ H−1/2 (Γ )

(12.8.3)

are uniquely solvable if k 2 is not an eigenvalue of the corresponding homogeneous problems in the subspaces. Therefore Costabel modifies the bilinear forms, where the new natural boundary conditions are not the original standard ones, but the modified bilinear forms now satisfy G˚ arding’s inequalities on the whole space H1 (Γ ). Solenoidal solution fields are also considered by R.C. MacCamy and E.P. Stephan in [276], [277], [278] [166]. For the scattering problem by a perfect conductor from the Stratton–Chu representation formula in Ω c , Vk (J)(x) − grad xV k (M )(x) for x ∈ Ω c (12.8.4) E(x) = iωμV  with V k λ(x) = Gk (x − y)λ(y)dsy , EFIE in the form of the boundary Γ

condition n × E = −n × Einc on Γ , leads to the Equation iωμπt Vk J − grad Γ Vk M = −πt Einc on Γ,

(12.8.5)

where J = n × H and M = E · n are considered as unknowns. Since  = 0 in Ω c and μ and ε are constant, E is solenoidal, i.e. div E = 0 in Ω c which can be used as an additional equation in Ω c on Γ . Applying div to equation (12.8.4) yields iωμ div Vk (J)(x) + k 2Vk (M )(x) = 0 for x ∈ Ω c in view of (Δx + k 2 )Gk (x − y) = 0 for x ∈ Ω c . Taking the limit Ω c  x → x ∈ Γ then yields iωμVk (divΓ J) + k 2 Vk (M ) = 0 on Γ.

(12.8.6)

This is an additional boundary integral equation for J and M . In principle, Equations (12.8.5) and (12.8.6) on Γ provide us a system of boundary integral equations for J and M . However, we may consider a modified system of equations as follows: Taking the first equation (12.8.5) and applying the surface divergence divΓ to it yields iωμdivΓ πt Vk J − ΔΓ Vk M = −divΓ πt Einc

(12.8.7)

with ΔΓ the Laplace–Beltrami operator. Subtracting (12.8.6) from (12.8.7) gives iωμ{divΓ Vk J − Vk divΓ J} − (ΔΓ + k 2 )Vk M = −divΓ (πt Einc ).

(12.8.8)

12.9 G˚ arding’s inequality for the sesquilinear form A in (6.12.23)

737

This modified equation (12.8.8) together with the original equation (12.8.5), i.e. (12.8.9) iωμπt Vk (J) − grad Γ Vk (M ) = −πt Einc now forms a strongly elliptic system of boundary integral equations on Γ with the triangular principal symbol (see MacCamy and Stephan [277]) ⎛ 2 i ⎞ |ξ | |ξ | , 0 0 ⎜ ⎟ ⎜ ⎟ 0 ⎟ σ(ξ ) = iωμ ⎜iξ1 |ξ  |−1 , |ξ  |−1 (12.8.10) ⎝ ⎠ iξ2 |ξ  |−1 ,

0

|ξ  |−1

since the commutator {divΓ Vk (J) − Vk divΓ J} has the principal symbol 0, and therefore is of order −1 on Γ mapping Hs (Γ ) continuously into Hs+1 (Γ ) and as a mapping Hs (Γ ) into Hs (Γ ) defines a compact operator for s > 0. As a consequence, the original EFIE (12.8.5) together with the modified 2  equation (12.8.5) for the unknowns J ∈ H−1/2 (Γ ) and M ∈ H 1/2 (Γ ) is Fredholm with index zero. Hence, uniqueness implies existence of a solution. Uniqueness is guaranteed if k 2 is not an eigenvalue of the interior Dirichlet problem of the Maxwell system. The triangular form of the symbols in (12.8.10) is more amenable than solving the original system from the numerical point of view.

12.9 G˚ arding’s inequality for the sesquilinear form A in (6.12.23) Motivated by the work of Buffa, Costabel and Schwab [51] who proved the solvability of Equation (6.12.12) by using a variational saddle point problem for four unknown functions, based on the result by Hildebrandt and Wienholtz [180], we now use our concept of strong ellipticity, Theorem 7.2.7 (see also i.e. Stephan and Wendland [405]), which also refers to Hildebrandt and Wienholtz [180] and tacitly takes care of the invariance of linear systems of differential equations with respect to their isomorphic linear transformations, in order to establish G˚ arding’s inequality for the sesquilinear form A in (6.12.23). Now, we are using local orthonormal coordinates near Γ with the base (e1 , e2 , n) and grad Γ ϕ = (∂1 ϕ)e1 + (∂2 ϕe2 ) , curl Γ ψ = (∂2 ψ)e1 − (∂1 ψ)e2 .

738

12. Remarks on Pseudodifferential Operators for Maxwell Equations

Theorem 12.9.1. The principal part of the sesquilinear form A in (6.12.23), i.e. 1 A0 (ϕ, ψ; ϕt , ψ t ) = − 2 V0 ΔΓ ϕΔΓ ϕt Γ k 0 1 + πt V0 (grad Γ ϕ + curl Γ ψ), (grad Γ ϕt + curl Γ ψ t ) Γ with ϕ, ϕt ∈ H(Γ ) = {α ∈ H∗1 (Γ ) : ΔΓ α ∈ H −1/2 (Γ ) with α, 1Γ = 0} and (ϕ, ψ; ϕt , ψ t ) with ϕ, ϕt ∈ H(Γ ) and ψ, ψ t ∈ H 1/2 (Γ ) is strongly elliptic satisfying (7.2.62) with ⎛ ⎞ −1 0 0 Θ = ⎝ 0 1 0⎠ , ξ ∈ R2 and ζ ∈ C3 : 0 0 1 ⎛

−1 ζT ⎝ 0 0

⎞⎛ 1 ⎞ 0 0 − k2 |ξ| 0 0 1 ⎠: 0 1 0⎠ ⎝ 0 |ξ| (iξ1 e1 + iξ2 e2 ) 1 0 0 0 0 |ξ| (iξ2 e1 − iξ1 e2 ) ⎞ ⎛ 2 0 0 |ξ| ⎠ ζ (12.9.1) ⎝ 0 0 −(iξ1 e1 + iξ2 e2 ) 0 0 −(iξ2 e1 − iξ1 e2 ) = ζ1 |ξ|3

and with c0 = min



1 k2 , 1}

1 ζ1 + ζ2 |ξ|ζ2 + ζ3 |ξ|ζ3 ≥ c0 |ζ|2 for |ξ| = 1 k2

> 0.

The system of differential operators on Γ : 1 V0 ΔΓ ϕ + πt V0 (grad Γ ϕ + curl Γ ψ) k2 ⎛ ⎞ −1 0 0 is strongly elliptic with Θ = ⎝ 0 1 0⎠ and σ = 0 according to 0 0 1 Theorem 7.2.7. 1 L (ϕ, ψ) = −

Proof: For the proof we use the sesquilinear form A defined by (6.12.23), which includes explicitly the principal part of A . The difference R := A − B0 + k −2 C 0 −1/2

is generated by operators of lower order and continuous from H×H −1/2 H (Γ )2

(Γ )×

into {ϕ ∈ H (Γ ) : ΔΓ ϕ ∈ H (Γ ), ϕ, 1 = This image is compactly imbedded into the space of arguments due to the 1

3/2

1

0}×H1 (Γ )×H1 (Γ ).

In Theorem 7.2.7, the operator L is denoted by A and α0 by γ.

12.9 G˚ arding’s inequality for the sesquilinear form A in (6.12.23)

739

Rellich lemma [362], Theorem 4.1.6. Hence, R is represented by a compact linear operator. Therefore it suffices to consider the principal part of A given in (6.12.23). With the tangential local base (e1 , e2 , n), corresponding local coordinates and the matrix Θ , the principal symbol is given in (12.9.1). Hence, the principal part is strongly elliptic as defined by Stephan and Wendland [405], a simplified version of the conditions by Hildebrandt and Wienholtz [180]. Moreover, the modified sesquilinear form A0+ of the principal part of A, ⎞> ⎛ −2 = k V Δ ϕt 0

A0+ :=

Γ

Θ ⎝πt V0 grad Γ ϕt ⎠ (−ΔΓ ϕ, grad Γ ϕ, curl Γ ψ)Θ t πt V0 curl Γ ψ

Γ

satisfies the coerciveness inequality ⎛ −2 ⎞> = k V0 ΔΓ ϕ Θ ⎝πt V0 grad Γ ϕ⎠ (−ΔΓ ϕ, grad Γ ϕ, curl Γ ψ)Θ (12.9.2) πt V0 curl Γ ψ Γ   ≥ cV1 k −2 ΔΓ ϕ2H −1/2 (Γ ) + grad Γ ϕ2H −1/2 (Γ ) + curl Γ ψ2H −1/2 (Γ ) 



with the coerciveness constant cV1 > 0 of the scalar single layer potential operator V0 in (5.6.32) which is independent of k. 0  1 Re ΔΓ ϕ, (grad Γ ϕ + curl Γ ψ) , k −2 V0 ΔΓ ϕ, πt V0 (grad Γ ϕ + curl Γ ψ  0 + (grad Γ ϕ + curl Γ ψ) , C 1 (grad Γ ϕ + curl Γ ψ) ≥ c0 ΔΓ ϕ2H −1/2 (Γ ) + grad Γ ϕ2H −1/2 (Γ ) + curl Γ ψ|2H −1/2 (Γ ) 



with the compact mapping C 1 (•, •) = C(•, •) − πt V0 grad Γ ϕ. This completes the proof of Theorem 12.9.1.  Remark 12.9.1: As shown in Lemma 6.12.3, in the space ϕ, ϕt ∈ H(Γ ) ∧ −1/2 grad Γ ϕ ∈ L2 (Γ ), and grad Γ ϕt ∈ H (divΓ , Γ ), the sesquilinear form πt Vk grad Γ ϕt  is compact and therefore can be neglected in (12.9.2). However, (12.9.2) can still be used for computational purposes. The saddle point formulation by Buffa, Costabel and Schwab [51]. For the EFIE saddle point formulation (6.12.25) in −1/2

X X = H 1 (Γ )/C × H 1/2 (Γ )/C × H∗ with

⎛

0 ⎜0 Θ=⎜ ⎝0 −1

0 1 0 0

0 0 −1 0

⎞ −1 0⎟ ⎟ 0⎠ 0

× H 1 (Γ/C)

740

12. Remarks on Pseudodifferential Operators for Maxwell Equations

the principal part of the system (6.12.26)  B 0 (ϕ, ψ, m, λ) , Θ (ϕ, ψ, m, λ) i = m, πt , V0 m + iωμcurl Γ ψ, πt V0 curl Γ ψ εω − grad Γ λ, grad Γ λ − grad Γ ϕ, grad Γ ϕ has the symbol   1 1 1 ζ1  ζ 1 − ωμζ3 (iξ2 − iξ1 )  (iξ2 − iξ1 )ζ 3 εω |ξξ | |ξξ |       − ζ4 (iξ1 + iξ2 ) · (iξ1 + iξ2 )ζ 4 + ζ2 (iξ1 + iξ2 ) · (iξ1 + iξ2 )ζ 2     1 1 |ζ1 |2 + ωμ|ξξ | |ζ3 |2 + |ξξ |2 |ζ4 |2 + |ξξ |2 |ζ2 |2 =i  %ω |ξξ |

σ (ξξ , ζ ) = i

Hence real and imaginary parts of B 0 are both strongly elliptic and satisfy the coerciveness inequalities (6.12.28).

12.10 Existence Theorem 6.12.6 revisited Motivated by the work of Buffa, Costabel and Schwab [51] we now use the pseudodifferential equations corresponding to (6.12.18) for defining a system of boundary integral equations which is strongly elliptic and admits a unique solution if k 2 is not an eigenvalue of the interior Dirichlet problem. Applying Fourier transform we obtain to the system of boundary integral equations (6.12.18), the equations 1  |ξ |iξ1 ϕ * = g*1 , k2 1 * − |ξ  |−1 iξ1 ψ* − 2 |ξ  |iξ2 ϕ * = g*2 . |ξ  |−1 iξ2 ϕ k * + |ξ  |−1 iξ2 ψ* − |ξ  |−1 iξ1 ϕ

(12.10.1)

This system also can be written in terms of the symbol matrix as ' (    |ξ |−1 iξ1 − k12 |ξ  |iξ1 , |ξ |−1 iξ2 ϕ * g*1 = .  −1   −1 1 g*2 ψ* |ξ | iξ2 − k2 |ξ |iξ2 , −|ξ | iξ1 The determinant of the symbol matrix is given by   1 − 1 − 2 |ξ  |2 > 0 for |ξ  |2 > k 2 . k Hence, the system defines a strongly elliptic system of boundary integral operators on Γ . The principal part is given by

12.10 Existence Theorem 6.12.6 revisited

'

− k12 |ξ  |iξ1 ,

− k12 |ξ  |iξ2 ,

741

(    ϕ * g*1  0  −1 * * + O(|ξ | )ψ = + O(|ξ | )ϕ . g*2 ψ* −|ξ  |−1 iξ1 |ξ |−1 iξ2

and multiplying the first row with k 2 iξ1 and the second row with k 2 iξ2 yields after summation * * = k 2 (iξ1 g*1 + iξ2 g*2 ) + k 2 O(|ξ  |)ϕ * + k 2 O(|ξ  |0 )ψ. |ξ  |3 ϕ Similarly, by multiplying the first row with −iξ2 , the second row with iξ1 and summation one finds * + O(|ξ  |0 )ψ* |ξ  |ψ* = (−iξ2 g*1 + iξ1 g*2 + O(|ξ  |1 )ϕ Hence, * * + k 2 O(i|ξ|−3 )ψ, ϕ * = k 2 |ξ  |−3 (iξ1 g*1 + iξ2 g*2 ) + k 2 O(|ξ  |−2 )ϕ ψ* = |ξ |−1 (−iξ2 g*1 + iξ1 g*2 ) + O(|ξ  |0 )ϕ * + O(|ξ  |−1 ψ* which indicates how to solve the boundary integral equations (6.12.18) and (6.12.21) or (12.10.1) for ϕ and ψ. By using a local parametrization in the vicinity of x , we have J = grad Γ ϕ + curl Γ ψ and EFIE takes the form

  πt V grad Γ ϕ + curl Γ ψ

(12.10.2)

 1 + 2 grad Γ V divΓ (grad Γ ϕ + curl Γ ψ) = g k Applying divΓ to (12.10.2) and then testing with ϕt gives 0

1 1 1 ϕt , divΓ V{grad Γ ϕ+curl Γ ψ} + 2 ϕt , divΓ grad Γ V ΔΓ ϕ = ϕt , divΓ g. k

Integration by parts yields 0 1 1 − grad Γ ϕt , V {grad Γ ϕ + curl Γ ψ} + 2 ΔΓ ϕt , V ΔΓ ϕ = ϕt , divΓ g. k (12.10.3)    Combining Appendix V(7) with IV(5) and V(4) gives curl Γ ψ t , v = ψ t , curlΓ v. Hence, applying curlΓ to (12.10.2) and then testing with ψ t gives with curlΓ grad Γ = 0 the equation curl Γ ψ t , Vcurl Γ ψ + curl Γ ψ t , Vgrad Γ ϕ = ψ t , curl g. Γ

(12.10.4)

742

12. Remarks on Pseudodifferential Operators for Maxwell Equations

After multiplication of (12.10.3) with k 2 and replacing ϕt by ϕt we obtain the sesquilinear form ΔΓ , ϕt , V ΔΓ ϕ − k 2 grad Γ ϕt , V grad Γ ϕ − k 2 grad Γ ϕt , V curl Γ ψ = k 2 ϕt , divΓ g.

(12.10.5)

It is easy to see that the sesquilinear form on the left is continuous for ϕ and ϕt ∈ H 3/2 (Γ ) and ψ ∈ H 1/2 (Γ ). Moreover, due to (5.6.32) the principal part is coercive: Re ΔΓ ϕ, VΓ ΔΓ ϕ ≥ cv  ΔΓ ϕ 2H −1/2 (Γ ) and for

ϕ ∈ H 3/2 (Γ ) , grad Γ ϕ ∈ H1/2 (Γ )  (H −1/2 (Γ ))3

which is, due to Rellich’s theorem, compactly imbedded into H−1/2 (Γ ). Finally, the commutator V ◦ curl Γ − curl Γ ◦ V is a pseudodifferential operator of degree −1 on Γ , hence, ψ ∈ H 1/2 (Γ ) is mapped into H 3/2 (Γ ) which, due to Rellich, is compactly imbedded into H 1/2 (Γ ). Moreover; curl Γ ◦ grad Γ = 0. For the second equation (12.10.4), in the sesquilinear form, i.e. t

t

t

curl Γ ψ , V curl Γ ψ + curl Γ ψ , Vgrad Γ ϕ = ψ , curlΓ g

(12.10.6)

we also use the coerciveness of V, Re curl Γ ψ, V curl Γ ψ ≥ cv  curl Γ ψ 2H−1/2 (Γ ) . Again, the commutator curl Γ ◦V−V◦curl Γ is a pseudodifferential operator of degree −1 on Γ whereas curlΓ ◦grad Γ = 0. In conclusion we see that the system of equations (12.10.3) and (12.10.4) satisfies G˚ arding’s inequality  Re ΔΓ ϕ, V ΔΓ ϕ − k 2 grad Γ ϕ, V grad Γ ϕ − k 2 grad Γ ϕ, Vcurl Γ ψ  + curl Γ ψ, V curl Γ ψ + curl Γ ψ, Vgrad Γ ϕ   = Re ΔΓ ϕ, V ΔΓ ϕ + curl Γ ψ, V curl Γ ψ + C k2 (ϕ, ϕ) ; (ψ, ψ)    ≥ c(k) ϕ2H 3/2 (Γ )/C + ψ2H 3/2 (Γ )/C + Ck 2 (ϕ, ϕ); (ψ, ψ)  where Ck2 (ϕ, ϕ) ; (ψ, ψ) denotes a sesquilinear form generated by a com 2 pact linear operator from H 3/2 (Γ )/C × H 1/2 (Γ )/C into H 1/2 (Γ ) . Hence, for given g ∈ H1/2 (Γ ), the variational equations (12.10.5), (12.10.6) with arbitrary test functions ϕt ∈ H 3/2 (Γ ) , ψ t ∈ H 1/2 (Γ ) are Fredholm with index zero. Since for ϕ = c1 , ψ = c2 the homogeneous equation (12.10.2) (g = 0) is satisfied, ϕ0 = c1 ∈ C and ψ0 = c2 ∈ C define the null space of (12.10.2) whereas the right hand side divΓ g in (12.10.3) and curlΓ g in (12.10.4) automatically satisfy the orthogonality conditions c1 , divΓ g = −grad Γ c1 , g = 0 and c2 , curlΓ g = curlΓ c2 , g = 0.

12.11 Concluding Remarks

743

To conclude this section, as long as k 2 is not an eigenvalue of the interior Dirichlet problem in Ω, the sesquilinear equation (12.10.6) admits a unique solution (ϕ, ψ) ∈ H 3/2 (Γ )/C × H 1/2 (Γ )/C and the scattered solution in Ω c is given by (12.10.5) as  E(x) = Ei (x) + iωμ+ Gk (x − y){grad Γ ϕ + curl Γ ψ}dsy i − ωε



Γ

grad y Gk (x − y)ΔΓ ϕdsy , x ∈ Ω c . Γ

12.11 Concluding Remarks The class of pseudodifferential operators is essentially the smallest algebra of operators which contains all differential operators, all fundamental solutions of elliptic differential operators and all integral operators with pseudohomogeneous kernel expansions. The linear pseudodifferential operators can be characterized by generalized Fourier multipliers, known as symbols. The development of the theory of pseudodifferential operators has made it possible to provide a unified treatment of differential and integral operators. These materials are subjects of Chapter 7 and Chapter 8. However, for a quick glance of the materials, we refer the reader to our paper [206] with emphasis on boundary integral equations recast as pseudodifferential equations. Chapter 12 is a newly added chapter in this second edition. This chapter together with Chapters 10 and 11 are dealing with boundary integral operators arising in potential flow, acoustics, elasticity, Stokes flow, and electromagnetics, which serve as concrete examples illustrating the basic ideas how one may apply the theory of pseudodifferential operators and their calculus to obtain basic properties for the corresponding boundary integral operators.

A. Differential Operators in Local Coordinates with Minimal Differentiability

As we mentioned earlier, the special coordinate transformation (3.3.7) contains n(σ  ) which is defined by derivations of Γ involving one order of differentiability more than that of Γ . Therefore, here we use a slightly different coordinate–transformation which requires no more differentiability than that of the surface Γ as was introduced via the representation in Section 3.3, namely x = x(r) + T(r) (τ  , a(r) (τ  ) + τn ) = Ψ(r) (τ ) (A.1) and its inverse

τ := (τ  , τn ) = Φ(x)

where τ

=

−1 (T(r) (x − x(r) )) ,

τn

=

−1 Φn (x) := (T(x) (x − x(r) ))n − a(r)



−1 (x − x(r) ) T(r)

 

(A.2)

for x ∈ B(r) . In what follows, without loss of generality we assume that T(r) is an orthonormal matrix ((Tjk )) associated with the point x(r) by the column vectors T•n = n(x(r) ) and T•λ , λ = 1, . . . , n − 1, where the latter form an orthonormal basis of the tangent plane to Γ at x(r) . Then the inverse of T(r) is given by its transpose with entries −1 )jk = Tkj . (A.3) (T(r) For the special choice of the local coordinates in (A.1), the parameters σ from the previous sections and the parameters τ are related by a diffeomorphism given by the equations τκ

=

n 

Tκ (y (σ  ) − x(r) + σn n (σ  )) for κ = 1, . . . , n − 1 ,

=1

τn

=

−a(r) (τ  ) +

n 

 Tn y (σ  ) − x(r) + σn n (σ  ) .

=1

© Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6

746

A. Appendix A: Local Coordinates

For the local coordinates (A.1), the Jacobian of the corresponding transformation is given by ∂a(r) ∂xk = Tkμ + Tkn ∂τμ ∂τμ

for μ = 1, . . . , n − 1 ,

∂xk = Tkn ∂τn

(A.4)

and, hence, the coefficients of the Riemannian metric are given by gμλ

gμn

∂a(r) ∂a(r) ∂τμ ∂τλ

=

δμλ +

=

∂a(r) , ∂τμ

for λ, μ = 1, . . . , n − 1 ,

(A.5)

gnn = 1 .

Since the inverse transformation is explicitly given by (A.2), we find with (A.3) ∂τλ = Tqλ , ∂xq n−1 (A.6)  ∂a(r) ∂τn = Tqn − Tq ∂xq ∂τ =1 for λ = 1, . . . , n − 1 and q = 1, . . . , n. Therefore, n  ∂τ ∂τq ∂x j ∂xj j=1

(A.7)

∂a(r) for λ, ν = 1, . . . , n − 1 , ∂τλ n−1   ∂a(r) 2 2 2 1 + b where b = ; and g = 1 . ∂τ =1

(A.8)

g q = and we obtain with (A.6) g λν

=

g nn

=

δ λν , g λn = −

For the Riemannian metric of Γ , we find γμλ = δμλ +

∂a(r) ∂a(r) for λ, μ = 1, . . . , n − 1 ; ∂τμ ∂τλ

(A.9)

and with elementary algebra γ

=

1 + b2

γ λν

=

δ λν −

(A.10)

and 1 ∂a(r) ∂a(r) . 1 + b2 ∂τλ ∂τν

The transformation of differential operators can be obtained from

(A.11)

A. Appendix A: Local Coordinates

747

 ∂τ ∂u ∂u = ∂xq ∂xq ∂τ n

=1

by using (A.6) or Lemma 3.4.1; in either case we find ' ( n−1 n−1   ∂a(r) ∂u ∂u ∂u = Tqμ + Tqn − Tqλ ∂xq ∂τμ ∂τλ ∂τn μ=1 =

λ=1

n−1 

∂u ∂Φn ∂u Tqμ + . ∂τμ ∂xq ∂τn μ=1

(A.12)

Since Γ is given implicitly by Φn (x) = 0, the normal vector n(x) can be extended to a vector–field n (x) in B(r) ⊂ IRn by ∼ ? n (x) := ∇Φn (x) |∇Φn (x)| . (A.13) ∼ n (x), then (A.12) can be written as In terms of ∼ n−1  ∂u ∂u ∂u n (x)|∇Φn (x)| = Tqμ +∼ . q ∂xq ∂τ ∂τ μ n μ=1

(A.14)

Correspondingly, we define ∂u n ∂∼

:=

n  q=1

∂u n (x) ; ∼q ∂xq

and we find with (A.14) the relation ∂u n ∂∼

:=

n n−1   =1 μ=1

  ∂u ∂u n (x)Tμ + ∇Φn (x) . ∼ ∂τμ ∂τn

∂u can be inserted into (A.14), and we finally obtain Hence, |∇Φn (x)| ∂τ n

∂u ∂xq

= =:

n−1 

n  ∂u  ∂u n (x)Tλ Tqλ − n (x) + n (x) ∼q ∼ n ∂τλ ∼q ∂∼ λ=1 =1  ∂  D +n u ∼ q ∼q ∂ n ∼

where D := ∼q

n−1 

n 

λ=1

=1

Tqλ −

 ∂ n (x)Tλ ∼ ∂τλ

(A.15)

(A.16)

is again a tangential differential operator as in (3.4.33). We proceed in the ∂ α ) in (3.4.41) by defining same manner as for the operator (D + n ∂n

748

A. Appendix A: Local Coordinates |α|   ∂ku ∂ α n (x) C Dα = D + u = ∼ ∼ ∂n ∼α,k ∂ n k ∼ k=0 ∼

where C ∼α,k u =



 |γ|≤|α|−k

c (x) ∼α,γ,k

∂ ∂τ 

(A.17)

γ u.

(A.18)

c and operators C can be defined It turns out that the coefficients ∼ ∼α,k α,γ,k by the same recursion formulae as for cα,γ,k and Cα,k in (3.4.43)–(3.4.46), (3.4.48). However, instead of (3.4.47), we now have c ∼β,,k (x)

=

n−1 

n 

λ=1

q=1

n +∼

Tλ − n (x) ∼

n (x)Tqλ ∼q

 ∂ c  ∼α,,k +c ∼α,(−(δjλ )),k ∂τλ

∂ c  ∼α,,k c +∼ .  α,,k−1 n ∂∼

(A.19)

The covering of Γ by B(r) ⊂ IRn and the local representation (A.1) in B(r) is associated with the partition of unity α(r) (x) introduced in Section 3.3 which satisfies p  α(r) (x) = 1 in some neighbourhood of Γ r=1

and supp α(r)  B(r) , r = 1, . . . , p. Then all of our local transformations are valid in B(r) having corresponding tangential and normal differential operators which are different for different r. This now will be indicated by the additional index r. The general differential operator P in (3.6.1) admits the form P u(x)

= =

p  r=1 p 

P (α(r) u)(x) =

p 

P(r) u(x)

r=1



a(r)β (x)Dβ u(x)

r=1 |β|≤2m

with coefficients a(r)β (x) defined by the Leibniz product rule in P (α(r) u) and the original coefficients aβ (x) of P . In particular, we have supp(a(r)β )  B(r) . Then P again can be partitioned along Γ : P u(x)

=

=

p  2m 

∂ku P ∼(r)k (∂ n )k r=1 k=0 ∼(r)

|α| p   

∂ku C . ∼(r)α,k (∂ n )k r=1 |α|≤2m k=0 ∼(r)

A. Appendix A: Local Coordinates

749

For x ∈ Γ we have τn = 0 and ∂ku ∂ k u  = . k n ) (∂ ∼ ∂nk Γ (r) Hence, P u(x) =

2m  k=0

∂ku P ∼k ∂nk

for x ∈ Γ ,

(A.20)

where now P ∼k

=

p   r=1 |β|≤2m

=

p 

a(r)β (x) C (x) ∼(r)β,k





r=1 |γ|≤2m−k |α|≤2m

' a(r)α (x) c (x) ∼(r)α,γ,k

∂  ∂τ(r)

(γ for x ∈ Γ

are tangential operators. In particular, we again have  P (x) = aα (x)nα (x) for x ∈ Γ . ∼2m |α|=2m

The operator P = ∼0

p  



r=1 |γ|≤2m |α|≤2m

' c a(r)α (y)∼

(r)α,γ,0

(y)

∂  ∂τ(r)

(γ

  for y = x(r) + T(r) (τ(r) , a(r) (τ(r) )) ∈ Γ corresponds to the Beltrami operator  . associated with P on Γ ; however, with respect to local parameters τ(r) If supp(α(r) ) ∩ supp(α(r ) ) = ∅, then we have two different sets of param  and τ(r eters τ(r)  ) belonging to the same point y on Γ , which are related by the transformation   −1   (τ(r  ) , a(r  ) (τ(r  ) )) = T  (x(r) − x(r  ) + T(r) τ(r) , a(r) (τ(r) )) . (r )

Hence, the differential operators in P ∼k can eventually be expressed solely in   or τ(r either terms depending on τ(r)  ) , respectively, and we find: P u(x) =

2m 

∂ku Pk , ≈ ∂ nk k=0 ∼

where this expansion is to be understood locally. Note, that with this approach one only needs Γ ∈ C 2m , but not more. In particular, one finds jump relations corresponding to those in Lemma 3.7.1, namely

750

A. Appendix A: Local Coordinates

∂k (uχΩ ) = nk ∂∼

'

∂ku nk ∂∼

( χΩ −

k=1  =0

'

∂ n ∂∼

(k−−1 

 ∂  u  ⊗ δ  Γ ∂n Γ

(A.21)

for k ≥ 1. Again, we find the second Green formula, now in the form,    uP ϕdy − ϕP udy (A.22) Ω

Ω

=

−

2m−1  2m−−1   =0

p=0

 ∂  p ∂n

P

≈ p++1

 ϕ (γ u)ds

Γ

which is slightly different from formula (3.7.8). In exactly the same manner as in Section 3.7 one may derive representation formulae as (3.7.10)–(3.7.14) by choosing for ϕ the fundamental solution E(x, •).

B. Vector Identities, Integration Formulae and Surface Differential Operators

In this appendix we summary some of the frequently used vector Identities, Gauss’s and Green’s formulae, surface differential operators as well as Stokes’ Formula and its consequences. I. Vector and vector field identities (1) (2) (3) (4) (5) (6) (7)

A × (B × C) = (A · C)B − (A · B)C ∇ × (B × C) = (∇ · C)B − (∇ · B)C + (C · ∇)B − (B · ∇)C A · (B × C) = C · (A × B) = B · (C × A) ∇ · (B × C) = C · (∇ × B) − B · (∇ × C) ∇(A · B) = B × (∇ × A) + A × (∇ × B) + (A · ∇)B + (B · ∇)A ∇ · (∇ × A) = 0 ∇ × (∇ × A) = −ΔA + ∇(∇ · A)

II. Integration formulae. In the following, Ω ⊂ R3 is a bounded domain with Γ := ∂Ω, the boundary of Ω.   (1) ∇ · udx = u · n ds. Ω Γ  (2) ∇ × u dx = n × u ds. Ω   Γ (3) ∇ϕdx = ϕn ds. Ω  Γ   (4) − Δu · v dx = ∇u : ∇v dx − ∂u ∂n · n ds Ω Ω Γ   (5) − Δu · v dx = {curl u · curl v + (div u)(div v)} dx Ω Ω − {curl u · (n × v) + (div u)(n · v)}ds Γ  (6) {curl u · v − u · curl v}dx = (n × u) · v ds Ω Γ    (7) v · (curl curl u) dx = (curl v) · (curl u) dx + n · (curl u × v) ds. Ω Ω Γ  (8) {v · (curl curl u) − u · (curl curl v)} dx Ω = {u × curl v − v × curl u} · n ds. Γ

(9) Helmholtz indentity: Given A with compact support in L2 (R3 )3 then there exits a scalar field ϕ and a vector field v such that A(x) = grad ϕ + curl v © Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6

752

B. Appendix B: Vector Field Identities, Integration Formulae

1 A(x) = curl 4π

 R3

1 1 curl A(y) dy − grad |x − y| 4π



R3

1 div A(y) dy. |x − y|

(See Dautray and Lions [96, p.313].) III. Surface differential operators. The differential operators on the surface Γ can be defined either by restriction or by extension. For scalar v and vector v functions in R3 defined on the tubular neighborhood of Γ , the surface operators are defined as follows: ∂v (1) grad Γ v := grad v − n ∂n = −n × (n × ∇v) = πt (grad v) πt u := −n × (n × u) ∂v = −(n × (n × ∇)) · v (2) divΓ v := div v − n · ∂n ∂v = −(n × (n × ∇)) × v (3) curl Γ v := ∇ × v − n × ∂n

Now for scalar function ϕ and vector function u in R3 defined on the surface ˜ defined on Γ , we associate with ϕ and u the extensions (or lifting) ϕ˜ and u the tubular neighborhood of Γ in Ω such that ϕ| ˜ Γ = ϕ and˜|Γ = . Then we define (4) (5) (6) (7)

grad Γ ϕ := grad ϕ| ˜ Γ (Surface gradient) curl Γ ϕ := −γt (grad Γ ϕ) (Surface vector curl) ; γt u := n × u ˜ |Γ (Surface divergence) divΓ u := div u ˜ |Γ (Surface scalar curl) curlΓ u := n · curl u

(See [331, (2.5.145), (2.5.149), (2.5.189), (2.5.190)] for (4) –(7)). We remark that surface divergence divΓ and surface scalar curlΓ can also be defined by duality. IV. Stokes Theorem and its consequences. In the following, S denotes a surface in R3 with boundary curve C = ∂S, and Γ = ∂Ω denotes the closed surface in R3  F (1) · ds = · d (Stokes Theorem) . S

C

As consequences, we have  F (2) − curl Γ ϕ ds = ϕdr (= 0 for a closed surface)  S F C (3) divΓ uds = (n × u) · dr (= 0 for a closed surface, if u is tangential ) S  FC (4) curlΓ ds = u · dr (= 0 for a closed surface, see e.g. [33, p.503]) S C   (5) grad Γ ϕ · v ds + ϕ divΓ v ds = 0. Γ  Γ (6) curl Γ ϕ · v ds − ϕ curlΓ v ds = 0. Γ  Γ  (7) − (ΔΓ w)u ds = grad Γ w · grad Γ u ds = curl Γ w · curl Γ u ds Γ

Γ

Γ

B. Appendix B: Vector Field Identities, Integration Formulae

(8) −



ΔΓ (w · v) ds =

Γ



divΓ w divΓ v ds +

Γ



curl Γ w · curl Γ v ds

Γ

( See, e.g. [331, Theorem 2.5.19] for (5) - (8) ).   (9) (curlΓ u)ϕ ds = (n × u) · grad Γ ϕds, which implies Γ

Γ

−divΓ (n × u) = curlΓ u V. Surface differential identities ΔΓ u = divΓ grad Γ u = −curlΓ curl Γ u ΔΓ v = grad Γ divΓ v − curl Γ curlΓ v divΓ curl Γ u = 0 curlΓ grad Γ u = 0 divΓ (n × v) = −curlΓ v divΓ (πt v) = curlΓ (n × v) and hence divΓ v = curlΓ (n × v), if v ∈ L2t (Γ ). (7) curl Γ ϕ = grad Γ ϕ × n.

(1) (2) (3) (4) (5) (6)

753

References

1. Adams, R.A.: Sobolev Spaces. Academic Press, New York 1975. 2. Adrian, S.: Rapidly converging boundary integral equation solvers in compu´ tational Electromagnetics. Doctorial Thesis, Bretagne–Pays de la Loire Ecole doctorale Math STIC and Technische Universit¨ at M¨ unchen 2018. 3. Agmon, S.: Lectures on Elliptic Boundary Value Problems. D.van Nostrand, Princeton, NJ. 1965. 4. Agranovich, M.S.: Spectral properties of elliptic pseudo–differential operators on a closed curve. Functional Analysis Appl. 13 (1979) 279–281. 5. Agranovich, M.S.: Sobolev spaces, their Generalization, and Elliptic Problems in Smooth and in Lipschitz Dmains. Springer 2015 6. Agranovich, M.S.: Elliptic operators on closed manifolds. In: Encyclopaedia of Mathematical Sciences. Vol. 63, Partial Differential Equations VI. (Yu.V. Egorov, M.A. Shubin eds.) Springer–Verlag, Berlin 1994, pp. 1–130. 7. Ahner, J.F. and Hsiao, G.C.: On the two–domensional exterior boundary value problems of elasticity. SIAM J. Appl. Math. 31 (1976) 677–685. 8. Ammari, H.: Propagation d’ondes ´electromagn´etiques. Habilitation a ` Diriger des Recherches Mathematiques, Univ. de Paris VI, 1999. 9. Ammari, H.and Nedelec, J.C.: Coupling of finite and boundary element methods for time–harmonic Maxwell equations. Part I in: Mathematical Aspects of Boundary Element Methods (Bonnet et al. eds.) CRC Press UK 1999. pp.11–22. PartII: A symmetric Formulation. Operator Theory: Advances and Applications 110 (1999), 23–32. 10. Amrouche, C., Bernardi, C., Dauge, M. and Girault, V: Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci. 21: 823-864 (1998). 11. Amrouche,C., , Ciarlet, P.G. and Mardare, C.: On a lemma of Jacques–Louis Lions and its relation to othe fundamental results. J. Math. Pures Appl. 104 (2015) 207–226. 12. Ammari, H., Buffa, A. and Nedelec, J.–C. : A justification of eddy current problems, SIAM J. Appl. Math.60 (2000) 1805–1823. 13. Angell, T.S., Kleinman, R.E and Kral, J.: Layer potentials on boundaries with ˇ corners and edges. Cas. Peˇst. Mat, 113 (1988) 387–401. 14. Angell, T.S. and Kirsch, A.: Optimization Methods in Electromagnetic Radiation, Springer, New York 2004. 15. Angell, T.S., Hsiao, G.C. and Wen, L.: On the two–dimensional inverse scattering problems in electromagnetics, Appl. Anal. 82 (2007) 483–497. 16. Atkinson, K.E.: The Numerical Solutions of Integral Equations of the Second Kind. Cambridge University Press 1997. 17. Aubin, J.–P.: Approximation of Elliptic Boundary Value Problems. John Wiley & Sons, New York 1972.

© Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6

756

References

18. Avantaggiati, A.: Sulle matrici fondamentali principali per una classe di sistemi ellitici e ipoellitici. Ann. Mat. Pura Appl. 65 (1964) 191–238. 19. Bach, M.: On the treatment of propagating mode-1 cracks by variational inequalities. In: The Mathematics of Finite Elements and Applications X (J. Whiteman ed.) Elsevier, Oxford 2000, pp. 377–386. 20. Bach,M., Nazarov, N. and Wendland, W.: Stable propagation of a mode-1 planar crack in an isotropic space. Comparison of the Irwin and the Griffith approaches. In: Current Problems of Analysis of Analysis and Mathematical Physics. Aracne, Rome 2000, pp. 167–189. 21. Balanis, C.A.: Advanced Engineering Electromagnetics, Wiley, New York, 1989. 22. Balanis, C.A.: Antenna Theory: Analysis and Design, 3rd edition, Wiley, 2005. 23. Balaˇs, J., Sladek J. and Sladek, V.: Stress Analysis by Boundary Element Methods. Elsevier, Amsterdam 1989. 24. Barkeshli, K.: Advanced Electromagnetics and Scattering Theory, Springer, 2015. 25. Bear, J.: Hydraulic of Groundwater. Mac Graw Hill, London 1979. 26. Bellman, R.: Introduction to Matrix Analysis, McGraw–Hill, New York 1960. 27. Bendali, A.: Numerical analysis of the exterior boundary value problem for the time–harmonic Maxwell equations by a boundary finite element method. Part I: The continuous problem. Math. Comp. 43 (1984) 29–46. 28. Bendali, A.: Numerical analysis of the exterior boundary value problem for the time–harmonic Maxwell equations by a boundary finite element method. Part II: The discrete problem. Math. Comp. 43 (1984) 47–68. 29. Benitez, F.G. (ed.): Fundamental Solutions in Boundary Elements. SAND (Camas) Sevilla 1997. 30. Berger, M. and Gostiaux, B.: G´eom´etrie differentielle: Vari´et´es, Courbes et Surfaces. Presses Universitaires de France, Paris 1987. 31. Bir´ o, O. and Preis, K.: On the use of the magnetic potential in the finite element analysis of the three–dimensionaleddy currents. IEEE Trans. Magn. 25 (1988) 32. Bishop, R.L. and Goldberg, S.I.: Tensor Analysis on Manifolds. Dover Publ. Inc., New York 1968. 33. Bladel, J.: Electromagnetic Fields, Second Edition. Wiley Intersience 2007. 34. Boffi, D., Brezzi, F. and Fortin, M.: Mixed Finite Element Methods and Applications. Vol. 44 of Springer Series in Computaional Mathematics, Springer, Heidelberg, 2013. 35. Bogovskii, M.E-: Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Sov. Math. Dokl. 20 (1979) 1094– 1098. 36. Bonnemay, P.: Equations int´egrales pur l’´elasticit´e plane. Th`ese de 3`eme cycle, Univ. de Paris VI, 1979. 37. Bonnet, M.: Boundary Integral Equation Methods for Solids and Fluids. John Wiley & Sons, Chichester 1995. 38. Bossavit, A: Computational Electromagnetism. Academic Press, London 1998. 39. Boutet de Monvel, L.: Comportement d’un op´erateur pseudo–differentiel sur une vari´et´e a ` bord. J. Analyse Fonct. 17 (1966) 241–304. 40. Boutet de Monvel, L.: Op´erateurs pseudo–differentiels et probl`emes aux limites elliptiques. Ann. Inst. Fourier 19 (1969) 169–268. 41. Boutet de Monvel, L.: Boundary problems for pseudo–differential operators. Acta Math. 126 (1971) 11–51. ¨ 42. Brakhage, H. and Werner, P.: Uber des Dirichletsche Aussenraumproblem f¨ ur die Helmholtzsche Schwingungsgleichung. Arch. Math. 16 (1965) 325–329. 43. Brebbia, C.A.: Boundary Element Methods. Springer-Verlag, New York 1983.

References

757

44. Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C.: Boundary Element Techniques. Springer–Verlag, Berlin 1984. 45. Brezzi, F. and Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer–Verlag New York 1991. ¨ 46. Bruhn, G. and Wendland, W.L.: Uber die n¨ aherungsweise L¨ osung von linearen Funktionalgleichungen. Internat. Schriftenreihe Numerische Mathematik (L. Collatz et al eds.) Birkh¨ auser–Verlag, Basel 7 (1967) 136–164. 47. Buffa, A.: Some Numerical and Theoretical Problems in Computational Electromagnetism. Doctoral Thesis, Univ. of Pavia 2000. 48. Buffa, A.: Trace theorems on non-smooth boundaries for functional spaces related to Maxwell equations: an overview. In: Computational Electromagnetics, T.L. Bath et al. (Eds), pp.23-34, Springer 2003. 49. Buffa, A. and Ciarlet, P. Jr.: On traces for functional spaces related to Maxwell’s equations I. An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24 (2001) 1–30. 50. Buffa, A. and Ciarlet, P. Jr.: On traces for functional spaces related to Maxwell’s equations II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Methods Appl. Sci. 24 (2001) 31–48. 51. Buffa, A., Costabel, M. and Schwab, C.: Boundary element methods for Maxwell equations on non–smooth domains. Numer. Math. 92 (2002) 679–710. 52. Buffa, A., Costabel, M. and Sheen, D.: On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl. 276, 845-867 (2002). 53. Buffa, A., Hiptmair, R., von Petersdorff, T. and Schwab,C.: Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math. 95 (2003) 459–496. 54. Buffa, A. and Hiptmair, R.: Galerkin boundary element methods for electromagnetic scattering. In: Computational Methods in Wave Propagation. Springer–Verlag, 2003. 55. Burago, Yu. and Maz’ya, V.G.: Potential theory and function theory for irregular regions. Zap. Nau˘cn. Sem. LOMI 3 (1967) 1–152; Seminars in Mathematics, V.A. Steklov Inst. Leningrad; Consultants Bureau, New York 1969. 56. Burago, Yu., Maz’ya, V.G. and Sapozhnikova, V.D.: On the double layer potential for nonregular domains. Dokl. Akad. Nauk SSSR 147 (1962) 523–525; Sov. Math. Dokl. 3 (1962) 1640–1642. 57. Bureau, F.: Divergent integrals and partial differential equations. Commun. Pure Appl. Math. 8 (1955) 143–202. 58. Cakoni, F., Hsiao, G.C. and Wendland, W.L.: On the boundary integral equation method for a mixed boundary value problem of the biharmonic equation. Complex Variables 50 (2005) 681–696. 59. Calder´ on, A.P.: Boundary value problems for elliptic equations. Outlines of the Joint Soviet–American Symposium on Partial Differential Equations. August, 1963, Novosibirsk. 60. Calderon, A.P. and Zygmund A.: On singular integrals. Amer. J. Math. 78 (1956) 289–309. 61. Calderon, A.P. and Zygmund A.: Singular integral operators and differential equations. Amer. J. Math. 79 (1957) 901–921. ¨ 62. Carleman, T.: Uber das Neumann–Poincar´esche Problem f¨ ur ein Gebiet mit Ecken. Doctoral Dissertation, Uppsala 1916. 63. Cessenat, M.: Mathematical Methods in Electromagnetism. World Scientific, Singapore 1996. 64. Chazarain, J. and Piriou, A.: Introduction to the Theory of Linear Partial Differential Equations. North–Holland, Amsterdam 1982.

758

References

65. Chen, G. and Zhou, J.: Boundary Element Methods. Academic Press, London 1992. 66. Chenais, D: Un r´esultat de compasit´e d’un ensemble de partie de IRn . Comp. Rend. Acad. Sci. Paris 277 (1973) 905–907. 67. Christiansen, S.H.: Discrete Fredholm properties and convergence estimates for the EFIE, Tech. Report CMAP Ecole Polytechnique Palaiseau 2001. 68. Ciarlet, P.G.: Mathematical Elasticity, Vol.I: Three–Dimensional Elasticity. North–Holland, Amsterdam 1988. 69. Ciarlet, P.G. and Ciarlet, P.Jr.: Another approach to linearized elasticity and a new proof of Korn’s inequality. Math. Models and Appl. Sci. 15 (2005) 259–271. 70. Ciavaldini, J.F., Pogu, M. and Tournemine, G.: Existence and regularity of stream functions for subsonic flows past profiles with a sharp trailing edge. Arch. Rat. Mech. Anal. 93 (1986) 1–14. 71. Claeys, X. and and Hiptmair, R.: First kind boundary integral equations for the Hodge–Helmholtz equation. SIAM J. Appl. Math. (2019). 72. Claeys, X. and and Hiptmair, R.: First–kind Galerkin boundary element methods for the Hodge–Laplacian in three dimensions. Math. Meth. Appl. Sci., submitted 2019. 73. Clements, D.L.: Boundary Value Problems Governed by Second Order Systems. Pitman Publ. London 1981. 74. Coclici, C.A. and Wendland, W.L.: On the treatment of the Kutta–Joukowski condition in transonic flow computations. Z. Angew. Math. Mech. 79 (1999) 507–534. 75. Colton, D. and Kress, R.: Integral Equation Methods in Scattering Theorie. John Wiley & Sons, New York 1983. 76. Colton, D. and Kress, R: Karp’s theorem in electromagnetic scattering theory. Proc. AMS 104 (3) (1988) 764–769. 77. Colton, D. and Kress, R: Inverse Acoustic and Electromagnetic Scattering Theory. Springer–Verlag, Berlin 1992, 3rd. Edn. Springer–Verlag, New York 2013. 78. Computational Electromagnetism. (de Castro, A.B. and Valli, A. eds.) Springer, Lecture Notes in Math. 2148, Springer New York 2015. 79. Costabel, M.: Starke Elliptizit¨ at von Randintegralgleichungen erster Art. Habilitation Thesis, Technical University Darmstadt 1984. 80. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19 (1988) 613–626. 81. Costabel, M.: Coupling of FEM and BEM for the numerical solution of the problem of electromagnetic scattering from an inhomogeneous body: In: Computational Mechanics 1988 (S.N. Atluri, G. Yagawa eds.) Vol.1 5v.1-5v.4, Springer– Verlag Berlin. 82. Costabel, M.: A remark on the regularity of solutions of Maxwell equations in Lipschitz domains. Math. Methods Appl. Sci. 12 (4) (1990) 365–368. 83. Costabel, M.: A coercive bilinear form for Maxwell’s equations- J. Math. Anal. Appl. 157 (1991), 527–541. 84. Costabel, M.: Some historical remarks on the positivity of boundary integral operators. In: Boundary Element Analysis (M. Schanz, O. Steinbach eds.) Springer Berlin 2007, pp. 1–27. 85. Costabel, M., Dauge, M. and Duduchava, R.: Asymptotics without logarithmic terms for crack problems. Comm. PDE 28 (2003) 1673–1687. 86. Costabel, M. and McIntosh, A.: On Bogovskii and regularized Poincar´e integral operators for the Rham complexes on Lipschitz domains. Math. Z. 265 (2010) 297–320.

References

759

87. Costabel, M. and Saranen, J.: Boundary element analysis of a direct method for the biharmonic equation. Operator Theory: Advances and Applications 41 (1989) 77-95. 88. Costabel, M. and Stephan, E.: A direct boundary integral method for transmission problems. JMAA. 106 (1986) 367–413. 89. Costabel, M. and Stephan, E.: Strongly elliptic integral equations for electromagnetic transmission problems. Proc. Royal Soc. Edinburgh 109A (1988) 271–296. 90. Costabel, M. and Wendland, W.L.: Strong ellipticity of boundary integral operators. J. Reine Angew. Mathematik 372 (1986) 34–63. 91. Costabel, M., Stephan, E. and Wendland, W.L.: On boundary integral equations of the first kind for the bi–Laplacian in a polygonal domain. Ann. Scuola Norm. Sup. Pisa, Ser. IV, 10 (1983) 197–242. 92. Crouch, S.L. and Starfield, A.M.: Boundary Element Methods in Solid Mechanics. George Allen & Unwin, London 1983. 93. Cruse, T.A.: Boundary Element Analysis in Computational Mechanics. Kluwer Academic Publ., Dordrecht 1988. 94. Dassios, G. and Kleinman, R.: Low Frequency Scattering, Oxford Science Publications, 2000 95. Dautray, R. and Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1: Physical Origins and Classical Methods. Springer–Verlag, Berlin 1990. 96. Dautray, R. and Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3: Springer–Verlag, Berlin 1990. 97. Dautray, R. and Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 4: Integral Equations and Numerical Methods. Springer–Verlag, Berlin 1990. −1/2 98. De la Bourdonnaye, A.: Decomposition de T Hdiv (Γ ) et nature de l’op´erateur de Steklov–Poincar´e du probl`eme ext´erieur de l’´electromagn´etisme. C.R. Acad. Sci. Paris, S´erie I.t.316 (1993) 369–372. 99. De La Bourdonnaye, A.: Some formulations coupling finite element and integral equation and electromagnetism. Numer. Math. 69(1995) 257–268. 100. Dieudonn´e, J.: El´ements d’Analyse. Vol. 8, Gauthier-Villars, Paris 1978. 101. Dindos, M. and Mitrea, M.: The stationary Navier–Stokes system in nonsmooth manifolds: The Poisson problem in Lipschitz and C 1 domains. Ration. Mach. Anal. 174 (2004) 1–47. 102. Ding, Z.: A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. AMS 124 (1996) 591–600. 103. Douglis, A. and Nirenberg, L.: Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8 (1955) 503–538. 104. Duduchava, R.: The Green formula and layer potentials. Integral Eqns. Operator Theory 41 (2001) 127–178. 105. Duduchava, R. and Speck, F.–O.: Pseudodifferential operators on compact manifolds with Lipschitz boundary. Math. Nachr. 160 (1993) 149–191. 106. Duduchava, R., S¨ andig, A.M. and Wendland, W.L.: Interface cracks in anisotropic composites. Math. Methods Appl. Sci. 22 (1999) 1413–1446. 107. Duduchava, R. and Wendland, W.L.: The Wiener–Hopf method for systems of pseudodifferential equations with an application to crack problems. Integral Eqns. Operator Theory 23 (1995) 294–335. 108. Duffin, R.J. and Noll, W.: On exterior boundary value problems in elasticity. Archive Rat. Mech. Anal. 2 (1958/1959) 191–196.

760

References

109. Egorov, Yu.V. and Shubin, M.A.: Linear partial differential equations. Elements of modern theory. In: Encyclopaedia of Mathematical Sciences, Partial Differential Equations II, Vol.31. (Yu. V. Egorov, M.A. Shubin eds.) Springer– Verlag Berlin 1994, pp. 1–120. 110. Ejdel’man, S.D.: Parabolic Systems. North–Holland 1969. 111. Ehrenpreis, L.: Solution of some problems of division. Part I. Division by a polynomial of derivation. Amer. J. Math. 76 (1954) 883–903. 112. Eskin, G.I.: Boundary Value Problems for Elliptic Pseudodifferential Equations. Amer. Math. Soc., Providence, Rhode Island 1981. 113. Essential Numerical Methods in Electromagnetics (Ciarlet, P.G. et al. eds.) Elevier, Amsterdam 2011. 114. Fabes, E.B., Kenig, C.E. and Verchota, G.C.: The Dirichlet problem for the Stokes system in Lipschitz domains. Duke Mathematical J. 57 (1988) 769–793. 115. Fabes, E.B., Kenig, C.E. and Verchota, G.C.: Boundary value problems for the system of elastostatics and Lipschitz domains. Duke Mathematical J. 57 (1988) 795–818. 116. Farwig, R. Simader, C., Sohr, H. and Varnhorn, W.: General properties of the Helmholtz decomposition in spaces of Lq –type. Contemporary Mathematics 666 Amer. Math. Soc. (2016) 163–177. 117. Feng Kang: Finite element method and natural boundary reduction. In: Proc. International Congress of Mathematicians. Warszawa 1983, pp.1439–1453. 118. Fichera, G.: Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anisotropic inhomogeneous elasticity. 1961 Partial Differential Equations and Continuum Mechanics, pp.50–80. Univ. of Wisconsin Press, Madison, Wis. 119. Fichera G.: Existence theorems in elasticity. In: Handbuch der Physik, Vol.VIa-2 (S. Fl¨ ugge, C. Truesdell eds.) Springer–Verlag, Berlin 1972, pp. 347– 389. 120. Fichera, G.: Sul problema misto per le equazioni lineari alle derivate parziali del secondo ordine di tipo elletico. Rev. Roumaine Pure Appl. 9 (1964) 3–9. 121. Fichera, G.: Linear Elliptic Differential Systems and Eigenvalue Problems. Lecture Notes in Mathematics 8, Springer-Verlag, Berlin-New York 1965. 122. Filippi, P.: Integral equations in acoustics. In: Theoretical Acoustics and Numerical Techniques. CISM Courses and Lectures 277. Springer–Verlag, Wien– New York 1983, pp. 1–49. 123. Fischer, T., Hsiao, G.C. and Wendland, W.L.: Singular perturbations for the exterior three–dimensional slow viscous flow problem. J. Math. Anal. Appl. 110 (1985) 583–603. 124. Fischer, T., Hsiao, G.C. and Wendland, W.L.: On two–dimensional slow viscous flows past obstacles in a half–plane. Proc. Royal Soc. Edinburgh A 104 (1986) 205–215. 125. Folland, G.B.: Lectures on Partial Differential Differential Equations. Tata Institute Fundamental Research, Bombay, Springer-Verlag, Berlin 1983. 126. Folland, G.B.: Introduction to Partial Differential Equations. Princeton Univ. Press, New Jersey 1995. 127. Freeden, W. and Michel, V.: Multiscale Potential Theory. Birkh¨ auser, Boston 2004. 128. Friedlander, F.G.: Introduction to the Theory of Distributions. Cambridge Univ. Press, Cambridge 1982. 129. Friedman, A.: Partial Differential Equations. Holt, Rinehardt and Winston, Inc., New York 1969.

References

761

130. Fulling, S.A. and Kennedy, G.: The resolvent parametrix of a general elliptic linear differential operator: A closed form for the intrinsic symbol. Transac. Amer. Math. Soc. 310 (1988) 583–617. 131. Gagliardo, E.: Caratterizzazioni delle trace sulla frontiera relative ad alcune di funzioni in n variabili. Rend. Seminario Mat. Universit´ a di Padova (1957) 27 284–305). 132. Gagliardo,E.: Propriet` a di alcune classi di funzioni in piu variabili. Ricerche Mat. 7 (1958) 102–137. 133. Gaier, D.: Konstruktive Methoden der konformen Abbildung. Springer– Verlag, Berlin 1964. 134. Gaier, D.: Integralgleichungen erster Art und konforme Abbildungen. Math. Zeitschr. 147 (1976) 113–139. 135. Gakhov, F.D.: Boundary Value Problems. Pergamon Press, New York 1966. 136. Galdi, C.P: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Vol. I. Springer–Verlag, New York 1994. 137. G˚ arding, L.: The Dirichlet problem. Math. Intelligencer 2 (1979) 43–53. 138. Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method Theory and Applications. Springer New York 2014. 139. Gatica, G. and Hsiao, G.C.: Boundary–Field Equation Method for a Class of Nonlinear Problems. Pitman Res. Notes in Math. 331. Addison–Wesley Longman, Edinburgh, Gate Harlow 1995. 140. Gaul, L., K¨ ogel, M. and Wagner, M.: Boundary Element Methods for Engineers and Scientists. Springer, Berlin 2003. 141. Gauss, C.F.: Allgemeine Lehrs¨ atze in Beziehung auf die im verkehrten Verh¨ altnisse des Quadrats der Entfernung wirkenden Anziehungs– und Abstoßungs–Kr¨ afte. (1839), Werke 5, 2nd Ed. G¨ ottingen 1877, 194–242. 142. Gauss, C.F.: Atlas des Erdmagnetismus. (1840), Werke 12, G¨ ottingen 1929, 326–408. 143. Gelfand, I.M. and Shilov, G.E.: Generalized Functions, Vol.1. Academic Press, New York 1964. 144. Giraud, G.: Sur certains probl`emes concernant des syst`emes d’´equations du type elliptique. Comp. Rendus Acad. Sci. Paris 192 (1931) 471–473. 145. Giraud, G.: Equations a ` int´egrales principales, ´etude suivie d’une application. Ann. Ec. Norm. Super. (3) 51 fasc.3 (1934) and fasc.4 (1936) 251–372. 146. Girauld, V. and Raviart, P.–A.: Finite Element Approximation of the Navier– Stokes Equations. Lecture Notes in Mathematics Nr. 749, Springer–Verlag Berlin 1981. 147. Girault, V. and Raviart, P. A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer, Berlin 1986. 148. Giroire, M.J.: Etudes de Quelques Probl`emes aux Limites Ext´erieurs et R´esolution par Equations Int´egrales. Thesis Dr. d’Etat. Paris VI, 1987. 149. Giroire, J. and Nedelec, J.C.: Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comp. 32 (1978) 973–990. 150. Glauert, H.: The Elements of Aerofoil and Airscrew Theory. Cambridge University Press, Cambridge 1926. 151. Gohberg, I. and Feld’man, I.A.: Convolution Equations and Projection Mathods for their Solution. Amer. Math. Soc., Providence, Rhode Island 1974. 152. Gohberg, I. and Krupnik, N.: One–dimensional Linear Integral Equations. Birkh¨ auser–Verlag, Basel 1992. 153. Goldstein, S.: Lectures on Fluid Mechanics. AMS, Providence, Rhode Island 1976. 154. Greco, D.: Le matrici fondamentali di alcuni sistemi di equazioni lineari a derivate parziali di tipo ellitico. Ricerche di Mat. Napoli 8 (1959) 197–221.

762

References

155. Green, G.: An Essay on the Applicability of Mathematical Analysis of the Theories of Elasticity and Magnetism. Nottingham, U.K. 1828. 156. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, London 1985. 157. Große, N. and Schneider, C.: Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces . Math. Nachr. 1–28 (2013) 158. Grubb, G.: Properties of normal boundary problems for elliptic even–order system. Ann. Sc. Sup. Pisa (4) 1 (1974) 1–61. 159. Grubb, G.: Functional Calculus of Pseudo-Differential Boundary Problems. Birkh¨ auser, Boston 1986. 160. Grubb, G.: Pseudodifferential boundary problems and applications. Jber. d. Dt. Math.–Verein. 99 (1997) 110–121. 161. Grusin, V.V.: Fundamental solutions of hypoelliptic equations. Usp. Mat. Nauk 16 n. 4 (1961) 147–153 (Russian). 162. G¨ unter, N.M.: Potential Theory and its Application to Basic Problems of Mathematical Physics. Gestekhizdat, Moscow 1953. 163. Guiggiani, M.: Hypersingular formulation for boundary stress evaluation. Engrg. Analysis with Boundary Elements 13 (1994) 169–179. 164. Guo,H.: Surface Crouzeix–Raviart element for the Laplace–Beltrami equation. Numer. Math. 144 (2020) 527–551. 165. Gurtin, M.E.: The linear theory of elasticity. In: Handbuch der Physik Vol.VIa/2 (S. Fl¨ ugge ed.), Springer–Verlag, Berlin 1972, pp. 1–295. 166. Gwinner, J. and Stephan, E.P.: Advanced Boundary Element Methods. Springer Nature 2018. 167. Hackbusch, W.: Integralgleichungen. B.G. Teubner, Stuttgart 1989. 168. Hadamard, J.: Recherches sur les solutions fondamentales e l’int´egration des ´ Norm. ´equations lin´eaires aux d´eriv´ees partielles (deuxi`em m´emoire). Ann. Ec. Sup. S´er. 3, 22 (1905) 101–141 (Œuvre 3, 1195–1235). 169. Han, Houde: Boundary integro differential equations of elliptic boundary value problems and their numerical solution. Sci. Sin. Ser. A31 (1988) 1153–1165. 170. Han, Houde: The boundary–integro–differential equations of three– dimensional Neumann problem in linear elasticity. Numer. Math. 68 (1994) 269–281. 171. Harriharan, S.I. and MacCamy, R.C.: Integral equation procedures for eddy current problems. J. Compt. Physics 45 (1982) 80–99. 172. Harrington, R.F.: Time-harmonic Electromagnetic Fields, McGraw-Hill, 1961. 173. Hartmann, F.: Introduction to Boundary Elements. Springer-Verlag, Berlin 1989. 174. Hartmann, F. and Zotemantel, R.: The direct boundary element method in plate bending. Intern. J. Numer. Methods Engrg. 23 (1986) 2049–2069. 175. Hellwig, G.: Remarks on a group of Hilbert theorems on systems of elliptic partial differential equations. Math. Zeitschr. 55 (1952) 276–283. 176. Hellwig, G.: The boundary value problem of a linear elliptic system. Math. Zeitschr. 56 (1952) 388–408. 177. Hellwig, G.: Partial Differential Equations. Blaisdell Publ. Comp., New York 1964. 178. Hess, J.L. and Smith, A.M.O.: Calculation of potential flow about arbitrary bodies. In: Progress in Aeronautical Sciences 8 (D. Kuchemann ed.) Pergamon Press, Oxford 1966, pp. 1–138. 179. Hilbert, D.: Grundz¨ uge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig 1912. 180. Hildebrandt, S.H. and Wienholtz, E.: Constructive proofs of representation theorems in seperable Hilbert spaces. Comm. Pure Appl. Maths. XVII (1964) 369–373.

References

763

181. Hiptmair, R.: Symmetric coupling for eddy current problems, SIAM J. Numer. Anal. 40 (1) (2002) 41–65. 182. Hiptmair, R.: Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM J. Numer. Anal. 41 (2003) 919–944. 183. Hiptmair, R. and Schwab, C.: Natural BEM for the electric field integral equation on polyhedra. Res. Report 2001–04, Sem. on Appl. Math. ETH Z¨ urich 2001. 184. H¨ ormander, L.: Pseudo–Differential Operators. Comm. Pure Appl. Math. XVIII (1965) 501–517. 185. H¨ ormander, L.: Pseudo–differential operators and non–elliptic boundary problems. Ann. of Math. 83 (1966) 129–209. 186. H¨ ormander, L.: Fourier integral operators I. Acta Math. 127 (1971) 79–183. 187. H¨ ormander, L: Linear Partial Differential Operators, Springer–Verlag, Berlin 1976. 188. H¨ ormander, L: The Analysis of Linear Partial Differential Operators, I–IV, Springer–Verlag Berlin 1985. 189. Hofmann, St., Mitrea, M. and Taylor, M.: Singular Integrals and Elliptic Boundary Value Problems on Regular Semmes–Kenig–Toro Domains. Internat. Math. Research Notes. Oxford Univ. Press, Oxford 2009. 190. Hofmann, S., Mitrea, M. and Taylor, M.: Symbol calculus for operators of layer potential type on Lipschitz surface with VMO normals and related pseudodifferential operator calculus. Anal. PDE 8 (1) (2015) 115–181. 191. Hsiao, G.C.: A Neumann series representation for solutions to the exterior boundary value problem of elasticity. In: Function Theoretic Methods for Partial Differential Equations, Lecture Notes in Mathematics 561 Springer–Verlag Berlin 1976, pp. 252–260. 192. Hsiao, G.C.: On the stability of integral equations of the first kind with logarithmic kernels. Archive Rat. Mech. Anal. 94 (1986) 179–192. 193. Hsiao, G.C.: Mathematical foundations for the boundary–field equations methods in acoustics and electromagnetic scattering. In: Analytical and Computational Methods in Scattering and Applied Mathematics. Notes in Math. 47 (2000) Chapman & Hall 149–163. 194. Hsiao, G.C. and Kleinman, R.E.: Mathematical foundation for the error estimation in numerical solutions of integral equations in electromagnetics, IEEE Transactions on Antennas and Propagation , vol. 45, No.3, 316–328, (1977) 195. Hsiao, G.C., Kleinman, R.E. and D.Q. Wang, D.Q.: Applications of boundary integral equation methods in 3D, J. Computational and Applied Mathematics, vol. 104, pp. 89-110, 1999. 196. Hsiao, G.C., Kopp, P. and Wendland, W.L.: A Galerkin–collocation method for some integral equations of the first kind. Computing 25 (1980) 89–130. 197. Hsiao, G.C., Kopp, P. and Wendland, W.L.: Some applications of a Galerkin– collocation method for boundary integral equations of the first kind. Math. Methods Appl. Sci. 6 (1984) 280–325. 198. Hsiao, G.C. and MacCamy, R.C.: Solution of boundary value problems by integral equations of the first kind. SIAM Rev.15 (1973) 687–705. 199. Hsiao, G.C. and MacCamy, R.C.: Singular perturbations for two–dimensional viscous flow problems. In: Lecture Notes in Math. 942 (W. Eckhaus, E.J. de Jager eds.) Springer–Verlag, Berlin 1982, pp.229–244. 200. Hsiao, G.C., Monk, P. and Nigam, N.: Error analysis of a finite element– integral equation scheme for approximate the time–harmonic Maxwell system, SIAM J. Numer. Anal. 40 (2002) 198–219. 201. Hsiao, G.C. and Wendland, W.L.: A finite element method for an integral equation of the first kind. J. Math. Anal. Appl. 58 (1977) 449–481.

764

References

202. Hsiao, G.C. and Wendland, W.L.: Boundary element methods for exterior problems in elasticity and fluid mechanics. In: The Mathematics of Finite Elements and Applications IV (J. Whiteman ed.) Academic Press, London 1988, pp.323–341. 203. Hsiao, G.C. and Wendland, W.L.: Boundary integral equations in low frequency acoustics. J. Chinese Inst. Engineers 23 (2000) 369–375. 204. Hsiao, G.C. and Wendland, W.L.: Boundary element methods: foundation and error analysis. In: Encyclopaedia of Computational Mechanics, Vol.1 (E. Stein et al eds.) John Wiley & Sons Publ., Chichester 2004, pp. 339–373. 205. Hsiao, G.C. and Wendland, W.L.: On a boundary integral method for some exterior problems in elasticity. Proc. Tbilissi Univ. (Trudy Tbiliskogo Ordena Trud. Krasn. Znam. Gosud. Univ.) UDK 539.3, Mat. Mech. Astron. 257, 18 (1985) 31–60. 206. Hsiao, G. C. and Wendland, W. L.: Boundary integral equations recast as pseudodifferential equations. Rendiconti, Accademia Nazionale delle Scienze detta dei XL, Memorie de Matematica e Applicazioni 124o (2006), Vol. XXX, fasc. 1, pagg. 29-48 207. Hsiao, G.C. and Wendland, W.L.: The boundary integral equation method for almost incompressible elastic materials, revisited. Rendiconti Acad. Naz. Science XL. Memorie di Matematicae Applicazione 124 (2006) XXX(1) 41–64. 208. Hsiao, G.C. and Wendland, W.L.: A characterization of the Calderon projector for the biharmonic equation. Proc. ISAAC Conference 2005. WSPC (2009) 23– 38. 209. Hsiao, G.C., Stephan, E.P. and Wendland, W.L.: On the integral equation method for the plane mixed boundary value problem with the Laplacian. Math. Methods Appl. Sci. 1 (1979) 265–321. 210. Hsiao, G.C., Stephan, E.P. and Wendland, W.L.: On the Dirichlet problem in elasticity for a domain exterior to an arc. J. Comp. Appl. Math. 34 (1991) 1–19. 211. Ilyinsky, A.S and Smirnow, Y.G.: Electromagnetic Wave Diffraction by Conducting Screens. VSP, Utrecht, The Netherlands, 1998. (Russian Original IPRZR 1996) 212. Jameson, A.: Aerodynamics. In: Encyclopaedia of Computational Mechanics, Vol 3 (E. Stein et al eds.) John Wiley & Sons Publ., Chichester 2004, pp. 325– 406. 213. Jaswon, M.A, and Symm, G.T.: Integral Eduation Methods in Potential Theory and Elastostatics. Academic Press, New York 1977. 214. Jeggle, H.–G.: Konvergenz und Fehlerabsch¨ atzung f¨ ur die diskretisierte Integralgleichung des Neumannschen Problems. Doctoral Thesis, TH Darmstadt 1966. 215. Jeggle, H–G.: Nichtlineare Funktionalanalysis. B.G. Teubner, Stuttgart 1979. 216. Jerison, D.S. and Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Fund. Anal. 130 (1995)-219. 217. Jiang, R. and Kauranen, A.: Korn’s inequality and John domains. Calculus of Variations and Partial Differential Equations 56 (2017) 109. 218. Jin, J.M.: The Finite Element Method in Electromagnetics, John Wiley and Sons. IEEE–Press 1993. 219. John, F.: Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience Publ., New York 1955. 220. Jones, D.S.: Methods in Electromagnetic Wave Propagation, Clarendon Press 1979 (Second Edition, Oxford, 1994).

References

765

221. Jost, G.: Integral equations with modified fundamental solution in timeharmonic electromagnetic scattering , IMA J. Applicat. Mathem., vol. 40 pp. 129-143, 1988. 222. Kantorowitsch, L.V and Akilow, G.P.: Functional Analysis in Normed Spaces. Pergamon Press, New York 1964. 223. Kato, T.: Perturbation Theory for Linear Operators. Springer–Verlag, Berlin 1966. 224. Kellog, O.D.: Foundations of Potential Theory. Springer–Verlag, Berlin 1929. ¨ 225. Kieser, R.: Uber einseitige Sprungrelationen und hypersingul¨ are Operatoren in der Methode der Randelemente. Doctoral Thesis, Universit¨ at Stuttgart 1990. 226. Kirchhoff, G.: Zur Theorie freier Fl¨ ussigkeitsstrahlen. Crelles J. f. d. Reine u. Angewandte Mathematik 70 (1869) 295–307. 227. Kirsch, A. and Hettlich, F.: The Mathematical Theory of Time–Harmonic Maxwell’s Equations. Springer International Publishing, Switzerland 2015. 228. Kleinman, R. E. and Kress, R.: On the condition number of integral equations in acoustics using modified fundamental solutions. IMA. J. Appl. Math. 31 (1983), 79–90. 229. Klingenberg, W.: A Course in Differential Geometry. Springer–Verlag, Heidelberg 1978. 230. Kn¨ opke, B.: The hypersingular integral equation for bending moments mxx , myy and mxy of the Kirchhoff plate. Comp. Mechanics 14 (1996) 1–12. 231. K¨ onig, H.: An explicit formula for fundamental solutions of linear partial differential equations with constant coefficients. Proc. Amer. Math. Soc. 120 (1994) 1315–1318. 232. Kohr, M.: The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in IRn . Math. Nachr. 280 (2007) 534–559. 233. Kohr, M. and Pop, I.: Viscous Incompressible Flow. WIT Press, Southampton 2004. 234. Kohr, M. and Wendland, W.L.: Variational boundary integral equations for the Stokes system. Applicable Anal. 85 (2006) 1343–1372. 235. Komech, A.I.: Linear partial differential equations. In: Linear Partial Differential Equations II. Encyclopaedia of Mathematical Sciences Vol.31 (Yu.V. Egorov, M.A. Shubin eds.) Springer–Verlag, Berlin 1994, pp. 121–255. 236. Kondratiev, V.A. and Oleinik, O.A.: On Korn’s inequalities. Comp. Rendus Acad. Sci., Paris 308 (1989) 483–487. 237. Kong, J.A.: Theory of Electromagnetic Waves, John Wiley & Sons, New York, 1975. 238. Koshlyakov, N.S., Smirnov, M.M. and Gliner, E.N.: Differential Equations of Mathematical Physics. North Holland, Amsterdam 1964. 239. Kr´ al, J.: The Fredholm method in potential theory. Trans. Amer. Math. Soc. 125 (1966) 511–547. 240. Kr´ al, J.: Boundary regularity and normal derivatives of logarithmic potentials. Proc. Roy. Soc. Edinburgh A 106 (1987), 241–258. 241. Kr´ al, J.: Integral Operators in Potential Theory. Lecture Notes in Mathematics, 823, Springer–Verlag, Berlin 1980. 242. Kr´ al, J. and Wendland, W.L.: Some examples concerning applicability of the Fredholm-Radon method in potential theory. Apl. Mat. 31 (1986), 293–308. 243. Kr´ al, J. and Wendland, W.L.: On the applicability of the Fredholm–Radon method in potential theory and the panel method. In: Panel Methods in Fluid Mechanics with Emphasis in Aerodynamics (eds. J. Ballmann et al.) Notes in Fluid Mechanics, Vieweg, Braunschweig 21 (1988) 120–136. 244. Kress, R.: Grundz¨ uge einer Theorie der verallgemeinerten harmonischen Vektorfelder. Meth. Verf. Math. Physik 2 (1069) 49–83.

766

References

245. Kress, R.: Die Behandlung zweier Randwertprobleme f¨ ur die vektorielle Poissongleichung nach einer Integralgleichngsmethode. Archive Rat. Mech. Analysis 39 (1070) 206–226. 246. Kress, R.: Potentialtheoretische Randwertprobleme bei Tensorfeldern beliebigeer Dimension und beliebigen Ranges. Methoden Verf. Math. Physik 2 (1969) and Archive Rat. Mech. Analysis 47 (1972) 59–80. 247. Kress, R.: Linear Integral Equations. Springer–Verlag, Berlin 1998. 248. Kriezis, E.E., Tsiboukis, T.D. and Panas, S.M.: Eddy currents: theory and applications, Proceedings of IEEE , 80, (10) (1992) 1559-1589. 249. Kufner, A., John, O. and Fuˇcik, S.: Function Spaces. Academia, Praha, Noordhoff, Groningen 1977. 250. Kufner, A. and Kadlec, J.: Fourier Series. Iliffe Books, London 1971. 251. Kupradze, V.D.: Randwertaufgaben der Schwingungstheorie und Integralgleichungen. Dt. Verlag d. Wissenschaften, Berlin 1956. 252. Kupradze, V.D.: Potential Methods in the Theory of Elasticity. Israel Program Scientific Transl., Jerusalem 1965. 253. Kupradze, V.D., Gegelia, T.G. Basheleishvili, M.O. and Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North Holland, Amsterdam 1979. 254. Kurz and Auchmann: Differential forms and boundary integral equations for Maxwell–type problems. In: U. Langer et al. (Eds.): Fast Boundary Element Methods, LNACM 63, pp.1-62. Springer-Verlag Berlin 2012 255. Kythe, P.K: Fundamental Soluitons for Differential Operators and Applications. Birkh¨ auser, Boston 1996. 256. Ladyˇzenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach Science Publishers, New York 1969. 257. Ladyˇzenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Springer–Verlag, Berlin 1985. 258. Lamb, H.: Hydronamics. University Press, Cambridge 1932. 259. Lamp, U., Schleicher, K. and Wendland, W.L.: The fast Fourier transform and the numerical solution of one–dimensional boundary integral equations. Numer. Math. 47 (1985) 15–38. 260. Langenberg, K.J.: A thorough look at the nonuniqueness of the electromagnetic scattering integral equation solutions as compared to the scalar acoustic ones, Radio Sci. 38 (2) 8021, doi: 10.1029/2001RS002558, 2003. 261. Lehman, R.S.: Developments at an analytic corner of solutions of elliptic partial differential equations. J. Mathematics and Mechanics 8 (1959) 727–760. 262. Leis, R.: Vorlesungen u ¨ber partielle Differentialgleichungen zweiter Ordnung. BI–Verlag, Mannheim 1967. 263. Leis, R.: Initial Boundary Value Problems in Mathematical Physics. B.G. Teubner, Stuttgart and John Wiley & Sons, Chichester 1986. 264. Leonard, P.J. and Rodger, D.: Finite element scheme for transient 3D eddy current. IEEE Trans. Magn. 24 1988. 265. Levi, E.E.: Sulle equazioni lineari totalmente ellittiche alle derivate parziali. Rend. Circ. Mat. Palermo 24 (1907) 275–317. 266. Levi, E.E.: I problemi dei valori al contorno per le equazioni lineari totalmente ellitiche alle derivate parziali. Mem. Soc. It. dei Sc. XL 16 (1909) 1–112. 267. Liggett, J.A. and Liu, P.L.F.: The Boundary Integral Equation Method for Porous Media Flow. George Allen & Unwin, London 1983. 268. Lions, J.L.: Espaces interm´edaires entre espaces hilbertiens et applications. Bull. Math. Soc. Sc. Math. Phys. R.P. Roumaine 2 (1958) 419–432. 269. Lions, J.L.: Equations Differentielles. Operationuelles et Problem`es aux Limites. Springer–Verlag, Berlin 1961.

References

767

270. Lions, J.L. and Magenes, E.: Non–Homogeneous Boundary Value Problems and Applications, Vol. I. Springer–Verlag, Berlin 1972. 271. Ljubiˇc, Ju. I.: On the existence “in the large” of fundamental solutions of second order elliptic equations. Math. Sbornik 57 (1962) 45–58 (Russian). 272. Lopatinskii, Ya. B.: Fundamental solutions of a system of differential equations ˇ 3 (1951) 290–316. of elliptic type. Ukrain. Mat. Z. 273. Lopatinskii, Ya. B.: A fundamental system of solutions of an elliptic system of differential equations. Ukrain. Mat. Z. 3 (1951) 3–38 (Russian). 274. MacCamy, R.C.: Low frequency oscillations. Quart. Appl. Math. 23 (1965) 247–255. 275. MacCamy, R.C.: Low frequency expansions for two–dimensional interface scattering problems. SIAM J. Appl. Mathematics 57 (1997) 1687–1701. 276. MacCamy, R.C. and Stephan, E.P.: A simple layer potential method for three– dimensional eddy current problems. In: Lecture Notes in Mathematics (A. Dold, B. Eckmann eds.) 964 (1982) Springer–Verlag Berlin, 477–484. 277. MacCamy, R.C. and Stephan, E.P.: A boundary element method for an exterior problem for three–dimensional Maxwell’s equations. Appl. Analysis 16 (2) (1983) 141–163. 278. MacCamy, R.C. and Stephan, E.P.: Solution procedures for the three– dimensional eddy–current problems, J. Math. Anal. Appl. 101 (1984), 348–379. 279. MacCamy, R.C. and Stephan, E.P.: A skin effect approximation for eddy current problems. Arch. Rational Mech. Anal. 90 (1985) 87–98. 280. Malgrange, B.: Existence et approximation des sulutions des ´equations aux d´eriv´ees partielles et des ´equations de convolution. Ann. Inst. Fourier (Grenoble) 6 (1955/56) 271–355. 281. Manolis, G.D. and Beskos, D.E.: Boundary Element Methods in Elastodynamics. Unwin Hyman Ltd., London 1988. 282. Martensen, E.: Berechnung der Druckverteilung an Gitterprofilen in ebener Potentialstr¨ omung mit einer Fredholmschen Integralgleichung. Archive Rat. Mech. Anal. 3 (1959) 235–270. 283. Martensen, E.: Potentialtheorie. B.G.Teubner, Stuttgart 1968. ¨ 284. Maue, A.W.: Uber die Formulierung eines allgemeinen Diffraktionsproblems mit Hilfe einer Integralgleichung. Zeitschr. f. Physik 126 (1949) 601–618. 285. Mautz, J. R. and Harrington, R. F.: H-field, E–field, and combined–field so¨ lutions for conducting bodies of revolutions, Arch. Elektron Ubertragungstech, Electron, Commun, 32 (4) (1978) 159-164, . 286. Maxwell, J.C.: A dynamical theory of electromagnetic field. Phil. Transactions Royal Soc. of London 155 (1865) 459–512. 287. Maxwell Equations: Analysis and Numerics. (Langer, U., Pauly, D. and Repin, S. eds.) Series on Comp. and Appl. Mathematics. De Gruyter, Berlin 2019. 288. Maz‘ya, V.G.: Sobolev Spaces. Springer–Verlag, Berlin 1985 289. Maz‘ya, V.G.: Boundary Integral Equations. In Encyclopaedia of Mathematical Sciences. Vol 27, Analysis IV. (Maz’ya, V.G., Nikolskiˇi eds.). SpringerVerlag, Berlin 1991, pp. 127–228. 290. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Univ. Press, Cambridge UK 2000. 291. McLean, W. and Wendland, W.L.: Trigonometric approximation of solutions of periodic pseudodifferential operators. Operator Theory: Advances and Appl. 41 (1989) 359–383. 292. Meddahi, S. and Selgas, V: A mixed FEM and BEM coupling for a three– dimensional eddy current problem. M2AN 37 (2) (2003) 291–318. 293. Meister, E.: Partielle Differentialgleichungen. Akademie–Verlag, Berlin 1996.

768

References

294. Meyers, N. and Serrin, J.: H = W . Proc. Nat. Acad. Sci. USA 51 (1964) 1055–1056. 295. Mikhailov, S.E.: Analysis of united boundary–domain integro–differential and integral equations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences 29 (2006) 715–739. 296. Mikhailov, S.E.: About traces, extensions and co–normal derivative operators on Lipschitz domains. In: Integral Methods in Science and Egineering: Techniques And Applications. (C. Constanda, S. Potapenko eds.) Birkh¨ auser, Boston 2007, pp. 151–162. 297. Mikhailov, S.E.: Traces, extensions and co–normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378 (2011) 324–342. 298. Mikhlin, S.G.: Le probl`eme fondamental biharmonique a ` deux dimensions. Comp. Rendus Acad. Sci., Paris 197 (1933) 608. 299. Mikhlin, S.G.: Plane problem of the theory of elasticity. Trudy Seismol. Inst. Akad. Nauk SSSR 65 (1935), pp. 372–396 (Russian). 300. Mikhlin, S.G.: Problems of equivalence in the theory of singular integral equations. Matem. Sbornik 3 (45) (1938) 121–140. 301. Mikhlin, S.G.: Integral Equations and their Applications. Pergamon Press, Oxford 1957. 302. Mikhlin, S.G.: Multidimensional Singular Integrals and Integral Equations. Pergamon Press, Oxford 1965. 303. Mikhlin, S.G.: Mathematical Physics, an Advanced Course. North Holland, Amsterdam 1970. 304. Mikhlin, S.G., Morozov, N.F. and Paukshto, M.V.: The Integral Equations of the Theory of Elasticity. B.G.Teubner, Stuttgart 1995. 305. Mikhlin, S.G. and Pr¨ ossdorf, S.: Singular Integral Operators. Springer–Verlag, Berlin 1986. 306. Millman, R.S. and Parker, G.D.: Elements of Differential Geometry. Prentice Hall Inc., Englewood Cliffs N.J. 1977. 307. Miranda, C: Partial Differential Equations of Elliptic Type. Springer–Verlag, Berlin 1970. 308. Mitrea, D., Mitrea, M. and Pipher, J.: Vector potential theory on non–smooth domains in R3 and applications to electromagnetic scattering. J. Fourier Anal. Appl. 3 (2) (1967) 131–192. 309. Mitrea, D., Mitrea, M. and Taylor, M.: Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds. Memoirs of the Amer. Math. Soc. 150 Nr. 173, 2001. 310. Mitrea, M. and Wright, M.: Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains. Ast´erisque 344, 2012. 311. Mitrea, D., Mitrea,I., Mitrea, M. and Taylor, M.: The Hodge–Laplacian, Vol. 64 of De Gruyter, Studies in Mathematics, De Gruyter, Berlin 2016. 312. Miyazaki, Y.: New proofs of the trace theorem of Sobolev spaces. Proc. Japan Acad. Ser. A. 84 (2008) 112–116. 313. Moioda, A. and Perugia, I.: A space–time Trefftz discontinuous Galerkin method of the acoustic wave equation in the first–order formulation. Numer. Math. 138 (2018) 389–435. 314. Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon Press, Oxford 2003. 315. Morrey, C.B.: Second order elliptic systems of differential equations. Proc. Nat. Acad. USA 39 (1953) 201–206. 316. Morrey, C.B.: Second order elliptic systems of differential equations. Ann. Math. Studies 33 (1954) 101–159.

References

769

317. M¨ uller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer–Verlag, Berlin 1969. 318. Murat, F.: Compacit´e par compensation II. In:Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (E. Giorgi, E. Magenes, U. Mosco eds.), Pitagora Editrice, Bologna, 1979, pp. 245–256. 319. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen 1953. 320. Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen 1958. 321. Natroshvili, D.: Estimation of Green’s Tensors of the Theory of Elasticity and some of their Applications. Izdat. Tbiliskogo Universiteta, Tbilisi 1978 (Russian). 322. Natroshvili, D.: Investigations of Boundary Value and Initial–Boundary Value Problems of the Mathematical Theory of Elasticity and Thermoelasticity for Homogeneous Anisotropic Media Using the Potential Methods. Dr. of Science Thesis, Tbilisi 1984 (Russian). ˇ Some Problems in 323. Natroshvili, D., Dˇzagmaidze, A.Ja. and Svanadze, M.Z.: the Linear Theory of Elastic Mixtures. Izd. Tbiliskogo Universiteta, Tbilisi 1986 (Russian). 324. Natroshvili, D. and Wendland, W.L.: Boundary variational inequalities in the theory of interface cracks. In: Operator Theory: Advances and Appl. 147 (2004) 387–402. 325. Nazarov, S.A.: The polynomial property of self–adjoint elliptic boundary value problems and algebraic description of their attributes. Uspekhi Mat. Nauk 54:5 (1999) 77–142. 326. Neˇcas, J.: Les M´ethodes Directes en Th´eorie des Equations Elliptiques. Masson, Paris 1967. English Translation: Direct Methods in the Theory of Elliptic Equations. Springer–Verlag Berlin–Heidelberg 2011. 327. Nedelec, J.C.: Curved finite element methods for solutions of singular integral equations on surfaces in IR3 . Comput. Methods Appl. Mech. Engrg. 9 (1976) 191–216. 328. Nedelec, J.C.: Approximation des Equationes Int´egrales en M´ecanique et en Physique. Lecture Notes, Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, Palaiseau, France, 1977. 329. Nedelec, J.C.: Approximation par potential de double cuche du probl`eme de Neumann ext´erieur. Comp. Rendus Acad. Sci. Paris, S´er. I. Math. 286 (1978) 616-619. 330. Nedelec, J.C.: Integral equations with non integrable kernels. Integral Eqs. Operator Theory 5 (1982) 562–572. 331. Nedelec, J.C.: Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems. Springer–Verlag, New York 2001. 332. Nedelec, J.C. and Planchard, J.: Une m´ethode variationelle d’´elements pour la r´esolution num´erique d’un probl`eme ext´erieur dans IR3 . RAIRO, Anal. Numer. 7 (1973) 105–129. 333. Neittaanm¨ aki, P. and Roach, F. G.: Weighted Sobolev spaces and exterior problems for the Helmholtz equation. Proc. Roy. Soc. London Ser. A 410 (1987), No. 1839, 373–383. 334. Netuka, I.: Smooth surfaces with infinite cyclic variation. Casopis pro pˇestov´ ani matematiky 96 (1971) 86–101. 335. Neumann, C.: Zur Theorie des logarithmischen und des Newtonschen Potentials. Ber. Verh. K¨ onigl. S¨ achsische Ges. d. Wiss. zu Leipzig 22 (1870) 45–56, 264–321. 336. Neumann, C.: Untersuchungen u ¨ber das logarithmische und Newtonsche Potential. Teubner–Verlag, Leipzig 1877.

770

References

¨ 337. Neumann, C.: Uber die Methode des arithmetischen Mittels. Hirzl, Leipzig 1887 (erste Abhandlung), 1888 (zweite Abhandlung). 338. Nitsche, J.: On Korn’s second inequality. RAIRO, Anal. Numer. 15 (1981) 237–248. 339. Of, G.: BETI–Gebietszerlegungsmethoden mit schnellen Randelementverfahren und Anwendungen. Doctoral Thesis, Universit¨ at Stuttgart 2006. 340. Of, G., Steinbach, O. and Wendland, W.L.: Applications of a fast multipole Galerkin boundary element method in linear elastostatics. Computing and Visualization in Sciences 8 (2005) 201–209. 341. Ortner, V.N.: Regularisierte Faltung von Distributionen, Teil 1: Zur Berechnung von Fundamentall¨ osungen. ZAMP 31 (1980) 133–154. 342. Ortner, V.N.: Regularisierte Faltung von Distributionen, Teil 2: Eine Tabelle von Fundamentall¨ osungen. ZAMP 31 (1980) 155–173. 343. Parasjuk, L.S.: Fundamental solutions of elliptic systems of differential equations with discontinuous coefficients. Dopovidi Akad. Nauk Ukrain. RSR (1963) 986–989 (Ukrainian). 344. Park, B.-J.: On the boundedness of pseudo–differential operators on Triebel– Lizorkin and Besov spaces. J. Math. Anal. Appl. 461 (2018) 544–576. 345. Peetre, J,: Another approach to elliptic boundary value problems Comm. Pure Appl. Math. 14, 711–731, 1961. 346. Petersen, B.E.: Introduction to the Fourier Transform and Pseudodifferential Operators. Pitman, London 1983. 347. Picard, R. and Des McGhee: Partial Differential Equations. Walter de Gruyter, Berlin 2011. 348. Picard, R., Weck, N. and Witsch, k.-J.: Time–harmonic Maxwell equations in the exterior of perfectly conducting irregular obstacles. Analysis (Munich) 21(3), (2001) 231–263. 349. Plemelj, J.: Potentialtheoretische Untersuchungen. B.G. Teubner–Verlag, Leipzig 1911. 350. Poincar´e, H.: La m´ethode de Neumann et le probl`eme de Dirichlet. Acta Math. 20 (1896) 59–142. 351. Pomp, A.: The Boundary-Domain Integral Method for Elliptic Systems with an Application to Shells. Lecture Notes in Mathematics, 1683. Springer-Verlag, Berlin 1998. 352. Power, H. and Wrobel, L.C.: Boundary Integral Methods in Fluid Mechanics. Comp. Mech. Publ., Southampton 1995. 353. Pozrikidis, C.: Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge Univ. Press, Cambridge 1992. 354. Pr¨ ossdorf, S.: Linear integral equations. In: Encyclopaedia of Mathematical Sciences. Vol 27, Analysis IV, Maz’ya, V.G., Nikolskii (eds.). Springer–Verlag, Berlin 1991, pp.1–125. 355. Pr¨ ossdorf, S. and Rathsfeld, A.: On strongly elliptic singular integral operators with piecewise continuous coefficients. Integral Equations and Operator Theory 8 (1985) 825–841. 356. Pr¨ ossdorf, S. and Schmidt, G.: A finite element collocation method for singular integral equations. Math. Nachr. 100 (1981) 33–60. 357. Pr¨ ossdorf, S. and Schmidt, G.: A finite element collocation method for systems of singular integral equations. In: Complex Analysis and Applications ’81, Sofia 1984, pp. 428–439. 358. Pr¨ ossdorf, S. and Silbermann, B.: Projektionsverfahren und die n¨ aherungsweise L¨ osung singul¨ arer Gleichungen. Teubner–Verlag, Leipzig 1977. 359. Pr¨ ossdorf, S. and Silbermann, B.: Numerical Analysis for Integral and Related Operators. Birkh¨ auser–Verlag, Basel 1991.

References

771

¨ 360. Radon, J.: Uber die Randwertaufgabe beim logarithmischen Potential. Sitzungsber. Akad. Wiss. Wien, Math.–nat. Kl. Abt. IIa 128 (1919) 1123–1167. 361. Reidinger, B. and Steinbach, O.: A symmetric boundary element method for the Stokes problem in multiple connected domains. Math. Methods Appl. Sci. 26 (2003) 77–93. 362. Rellich, F.: Ein Satz u ¨ber mittlere Konvergenz. G¨ ottinger Nachr. Math. Phys. Klasse (1930) 30–35. 363. Rellich, F.: Darstellungen der Eigenwerte von Δu + λu = 0 durch ein Randintegral. Math. Z. 46 (1940) 635–636. ¨ 364. Rellich, F.: Uber das asymptotische Verhalten der L¨ osungen von Δu + λu = 0 in unendlichen Gebieten. Jber. Deutsch. Math. Verein 53 (1943) 57–65. 365. Rodriguez, R.: Numerical approximation of Maxwell equations in low– frequency regime. In: Computational Electromagnetism (A. Bermudez de Castro, A. Valli eds.) Lecture Notes in Mathematics 2148 Springer (2014) pp. 59–110. 366. Rodriguez, A. A. and Valli, A: Eddy Current Approximation of Maxwell Equations, Theory, Algorithms and Applications, Springer–Verlag Italia 2010. 367. Rudin, W.: Functional Analysis. Mc Graw–Hill, New York 1973. 368. Rumsey, V.H.: Reaction concept in electromagnetic theory. Physical Review 94 (1954) 1438–1491. 369. Saranen, J. and Vainikko, G.: Integral and Pseudodifferential Equations with Numerical Approximation. Springer–Verlag, Berlin 2002. 370. Saranen, J. and Wendland, W.L.: The Fourier series representation of pseudo– differential operators on closed curves. Complex Variables 8 (1987) 55–64. 371. Sauter, St. and Schwab, Ch.: Randelementmethoden. B.G. Teubner, Stuttgart 2004. 372. Sayas, F.–J., Brown, Th.S. and Hassell, M.E.: Variational Techniques for Elliptic Partial Differential Equations. CRC Press, Boca Raton, London, New York 2019. 373. Schanz, M. and Steinbach, O. (Eds.): Boundary Element Analysis. Springer– Verlag, Berlin 2006. 374. Schatz, A., Thom´ee, V. and Wendland, W.L.: Mathematical Theory of Finite and Boundary Elements. Birkh¨ auser Verlag, Basel 1990. 375. Schechter, M.: Integral inequalities for partial differential operators and functions satisfying general boundary conditions. Comm. Pure Appl. Math. 12 (1959) 37–88. 376. Schechter, M.: Principles of Functional Analysis. Academic Press, New York 1971. 377. Schechter, M.: Modern Methods in Partial Differential Equations. McGraw– Hill, New York 1977. 378. Schrohe, E. and Schulze, B.–W.: Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities I and II. In: Advances in Partial Differential Equations, Pseudodifferential Calculus and Mathematical Physics. Vol.I, pp.97–209 (1994) and Vol.II, pp.70–205 (1995) Akademie–Verlag Berlin. 379. Schulze, B.–W.: Boundary Value Problems and Singular Pseudo–Differential Operators. John Wiley & Sons, Chichester 1998. 380. Schwab, C. and Wendland, W.L.: Kernel properties and representations of boundary integral operators. Math. Nachr. 156 (1992) 187–218. 381. Schwab, C. and Wendland,W.L.: On the extraction technique in boundary integral equations. Math. Comp. 68 (1999) 91–121. 382. Schwartz, L.: Th´eorie des Distributions. Hermann, Paris 1950.

772

References

383. Seeley, R.T.: Extension of C ∞ functions defined in a half–space. Proc. Amer. Math. Soc.15 (1964) 625–626. 384. Seeley, R.T.: Singular integrals and boundary value problems. Amer. J. Math. 88 (1966) 781–809. 385. Seeley, R.T.: Topics in pseudo–differential operators. In: Pseudodifferential Operators (L. Nirenberg ed.), Centro Internazionale Mathematico Estivo (C.I.M.E.), Edizioni Cremonese , Roma 1969, pp. 169–305. 386. Sellier, A.: Hadamard’s finite part concept in dimension n ≥ 2, distributional definition, regularization forms and distributional derivatives. Proc. R. Soc. London A 445 (1994) 69–98. 387. Sellier, A.: Hadamard’s finite part concept in dimension n ≥ 2; definition and change of variables, associated Fubini’s theorem, derivation. Math. Proc. Camb. Phil. Soc. 122 (1997) 131–148. 388. P.P. Silvester and R.L. Ferrari: Finite Elements for Electrical Engineers, Third Edition, Cambridge Univeristy Press, 1996. 389. Simader, C.G. and Sohr, M.: Necessary and sufficiant conditions for the existence of Helmholtz decompositions in general domains. Annali dell’ Unicersit´ a di Ferrara, 60 (2014): Scienzematematiche, 245–262. 390. Sloan, I. and Wendland, W.L.: Qualocation methods for elliptic boundary integral equations. Numer. Math. 79 (1998) 451–483. 391. Sloan, I. and Wendland, W.L.: Spline qualocation methods for variable– coefficient elliptic equations on curves. Numer. Math. 83 (1999) 497–533. 392. Smirnov, V.I.: A Course of Higher Mathematics. Vol IV, Pergamon and Addison-Wesley, New York 1964. 393. Sobolev, S.L.: Equations of Mathematical Physics. Gostokhizdat. Moscow 1954. (Russian) 394. Sokolnikoff, I.S.: Tensor Analysis, Theory and Applications. Mac Graw Hill, New York 1951. 395. Sommerfeld, A.: Partial Differential Equations in Physics. Academic Press, New York 1969. 396. Somigliana, C.: Sui sistemi simmetrici di equazioni a derivate parziali. Ann. Mat. Pura Appl. 22 (1894) 143–156. 397. Staelin, D.H., Morgenthaler, A.W. and Kong, J.A.: Electromagnetic Waves. Prentice Hall, New Jersey 1994. 398. Steinbach, O.: A robust boundary element method for nearly incompressible linear elasticity. Numer. Math. 95 (2003) 553–562. 399. Steinbach, O.: Numerische Verfahren f¨ ur elliptische Randwertprobleme. B.G. Teubner, Stuttgart 2003. 400. Steinbach, O.: A note on the ellipticity of the single layer potential in two– dimensional linear elastostatics. J. Math. Anal. Appl. 294 (2004) 1–6. 401. Steinbach, O. and Wendland, W.L.: On C. Neumann’s method for second order elliptic systems in domains with non–smooth boundaries. J. Math. Anal. Appl. 262 (2001) 733–748. 402. Stephan, E.P.: Boundary integral equations for magnetic screens in IR3 . Proc. Royal Soc. Edinburgh 102A (1986) 189–210. 403. Stephan, E.P.: Boundary Integral Equations for Mixed Boundary Value Problems, Screen and Transmission Problems in IR3 . Habilitationsschrift, TH Darmstadt 1984. 404. Stephan, E.P.: A boundary integral equation procedure for the mixed boundary value problem of the vector Helmholtz equation. Appl. Anal. 35 (1990) 59–72.

References

773

405. Stephan, E.P. and Wendland, W.L.: Remarks to Galerkin and least squares methods with finite elements for general elliptic problems. Manuscripta Geodaetica 1 (1976) 93–123. 406. Stephan, E.P. and Wendland, W.L.: An augmented Galerkin procedure for the boundary integral method applied to two–dimensional screen– and crack problems. Applicable Analysis 18 (1984) 183–220. 407. Sterz, O. and Schwab, C.: A scalar boundary integrodifferential equation for eddy current problem using impedance boundary condition. Comput. Visual. Sci. 3 (2001) 129–136, 408. Stratton, J.A.: Electromagnetic Theory. McGraw–Hill, New York 1941. 409. Stratton, J.A. and Chu, I.J.: Diffraction theory of electromagnetic waves. Physical Review 56 (1939) 99–107. 410. Stummel, F.: Rand– und Eigenwertaufgaben in Sobolewschen R¨ aumen. Lecture Notes in Mathematics 102, Springer–Verlag, Berlin 1969. 411. Szabo, I.: H¨ ohere Technische Mechanik. Springer–Verlag, Berlin 1956. 412. Taira, K: Diffusion Processes and Partial Differential Equations. Academic Press, Boston 1988. 413. Tartar, L.: On the characterization of traces of a Sobolov space used for Maxwell’s equations. In: Proc. Conf. in honour of M. Artota, Bordeaux 1997. 414. Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer–Verlag Berlin 2007. 415. Taylor, M.E.: Pseudodifferential Operators. Princeton Univ. Press, Princeton 1981. 416. Taylor, M.E.: Partial Differential EquationsI. Basic Theory. Springer, New York 1996. 417. Temam, R.: Navier–Stokes Equations. North–Holland Publ., Amsterdam 1979. 418. Treves, F.: Linear Partial Differential Equations with Constant Coefficients. Gordon and Breach, New York 1966. 419. Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York 1967. 420. Treves, F.: Pseudodifferential and Fourier Integral Operators. Vol. 1: Pseudodifferential Operators. Vol. 2: Fourier Integral Operators. Plenum Press, New York 1980. 421. Triebel, H.: H¨ ohere Analysis. Dt. Verlag d. Wissenschaften, Berlin 1972. 422. Tychonoff, A.N. and Samarski, A.A.: Partial Differential Equations of Mathematical Physics. Vol I and II, Holden–Day, San Francisco 1964 and 1967. 423. Ursell, F.: On the exterior problems of acoustics. Proc. Cambridge Philos. Soc., I 74 (1973) 117–125; II 84 (1978) 545–548. 424. Varnhorn, W.: The Stokes Equations. Akademie Verlag, Berlin 1994. 425. Vekua, I.N.: New Methods for Solving Elliptic Equations. North Holland, Amsterdam 1968 (Russian 1948, Moscow). 426. Vishik, M.J.: On strongly elliptic systems of differential equations. Mat. Sb. 25 (1951) 615–676 (Russian). 427. Vishik, M.J. and Eskin, G.T.: Elliptic equations in convolution in a bounded domain and their applications. Russian Math. Surveys 22 (1967) 13–75 (Uspekhi Mat. Nauk 22 (1967) 15–76). 428. Wagner, P.: Parameterintegration zur Berechnung von Fundamentall¨ osungen. Polska Akademia Nauk, Inst. Matematyczny, Dissertationes Mathematicae, CCXXX, Warszawa 1984. 429. Wagner, P.: On the explicit calculation of fundamental solutions. J. Math. Anal. Appl. 297 (2004) 404–418.

774

References

430. Weber, Ch. and Werner, P.: A local compactness theorem for Maxwell‘s equations. Math. Meth. in the Appl. Sci. 2 (1980) 12–25. 431. Weggler, L.: Stabilized boundary element methods for low–frequency electromagnetic scattering. Math. Meth. Appl. Sci. 35 (2012) 574–597. 432. Wendland, W.L.: Die Behandlung von Randwertaufgaben im R3 mit Hilfe von Einfach– und Doppelschichtpotentialen. Numer. Math. 11 (1968) 380–404. 433. Wendland, W.L.: Martin Costabel’s version of the trace theorem revisited. Math. Methods Appl. Sci. 37 (2014) 1924–1955. 434. Wendland, W.L. and Zhu, J.: The boundary element method for three– dimensional Stokes flows exterior to an open surface. Math. Comput. Modelling 15 (1991) 19–41. 435. Werner, P.: Zur mathematischen Theorie akustischer Wellenfelder. Archive Rat. Mech. Anal. 6 (1960) 321-260. 436. Werner, P.: On the behaviour of stationary electromagnetic wave fields for small frequencies. J. Math. Anal. Appl. 15 (1966) 447–496. 437. Wielandt, H.: Das Iterationsverfahren bei nicht selbstadjungierten linearen Eigenwertaufgaben. Math. Z. 50 (1944) 93–143. 438. Wilcox, C.H.: Scattering Theory for the d’Alembert Equation in Exterior Domains. Lecture Notes in Mathematics 442, Springer-Verlag, Berlin 1975. 439. Wloka, Y.: Partielle Differentialgleichungen. B.G. Teubner, Stuttgart 1982. 440. Wloka, J.T., Rowley, B. and Lawruk, B.: Boundary Value Problems for Elliptic Systems. Cambridge Univ. Press, Cambridge 1995. 441. Yaghjian, A. D. : Augmented electric–and magnetic integral equations, Radio Sci., 16 (1981) 987-1001 . 442. Yu, De–hao: Natural Boundary Integral Method and its Applications. Kluwer Acad. Publ., Dordrecht 2002. 443. Zeidler, E.: Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems. Springer–Verlag Berlin 1985. 444. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Vol. II/A: Linear Monotone Operators. Springer–Verlag, Berlin 1990.

Index

β  , 96 α S m (Ω × Rn ), 425 c →, 165 C0∞ (Ω), 96 C k,κ , 108 C m,α (Ω), 97 C m,α (Ω), 97 C m (Ω), 96 C m (Ω), 97 C0∞ (Ω), 97 Dα , 96, 99 H0m (Ω c ; λ), 192 H∗−1 (Γ ), 198 H s (Ω), 160 H0m (Ω) , 167 s (Ω), 167 H00 s Hcomp (Ω), 169 s Hloc (Ω), 168 Km , 492 L2 (Γ ), 170 L2 (Ω), 159 L∞ (Ω), 160 Lp (Ω), 159 OP S m (Ω × Rn ), 431 P , 130 P  , 130 Q, 378 [s], 174 ΔHL , 390 Ψ hfq , 101, 479 Ψ hkq (Ω), 480 , 96 α”, 96 H(div , Ω), 194 H1n (Ω), 204 H1t (Ω), 204 1/2 H⊥ (Γ ), 197 H0 (div , Ω), 196 L2t (Γ ), 196 curl Γ , 198

grad Γ , 198 1/2 H⊥ (Γ ), 197 curlΓ , 198 (ν) δΓ , 128 δy (x), 131 divΓ , 379 divΓ , 198 γ0 u, 171 γ0 , 171 γ0∗ , 178, 196 γn , 196 γt , 199 N0 , 95 B, 317 D, 317 E, 317 H, 317 J , 317 Lm (Ω), 440 n Lm c (Ω × R ), 445 ∗ T (Γ ), 543 B, 322 D, 322 E, 322 H(Γ ), 201 H, 322 J , 322 J0 , 319 V, 382 VHL , 385 W, 384 WHL , 385 GradΓ u, 117 μ, 322 πt , 196, 199 ε, 322  charge density, 317, 322  584 Q, u ⊗ δΓ , 548 H( div , Ω), 194

© Springer Nature Switzerland AG 2021 G. C. Hsiao und W. L. Wendland, Boundary Integral Equations, Applied Mathematical Sciences 164, https://doi.org/10.1007/978-3-030-71127-6

776

Index

H( curl , Ω), 194 Q, 325 grad Γ , 198 D(Ω), 97, 98 D := D (Rn ), 98 E(Ω), 98 E  (Ω), 98 Hs (Γ ), 175 HE (Ω c ), 221 QΩ , 142 QΩ c , 142 S  , 99 Hs (Γ ), 175 Ker (γn ), 196 a priori estimate, 307 adjoint, 178 – equation, 270, 608 – operator, 236, 443 – system, 63 – trace operator, 178 Agmon–Douglis–Nirenberg system, 454, 455, 466, 476 Airy stress function, 79 almost incompressible materials, 75 Ampere’s law, 321 – –Maxwell equation, 325 – –Maxwell law, 318 amplitude function, 440 angle of rotation, 81 associated pressure, 64 asymptotic – expansion – – of equations, 31 – – of kernels, 508, 629, 703 – – of symbols, 435, 442, 443 – representation, 525 – asymptotic , 724 atlas A, 540, 541, 544, 547 Banach – space, 97 – fixed point principle, 234 Beltrami operator, 119, 749 bending moment, 81 Bessel potential operator, 469, 689 Betti–Somigliana formula, 45 biharmonic equation, 79, 85, 305, 709, 716 bijective transformation, 529 bilinear form, 80, 261, 314 360, 691 Bogovskii operator, 195, 320 bound current, 322

boundary – conditions, 146 – normal, 146 – integral – – capacity operators, 344 – – equation, 418, 420, 725 – – equations of the first kind, 92, 411 – – equation of the second kind, – – generated by domain pseudodifferential operators, – – mapping properties, 544 – – operator, 140, 381, 547 – sesquilinear forms, 294 – surface, 108 – traction, 257 Boutet de Monvel – algebra, 620, 621 – operator, 620 Calder´ on projector, 9, 14, 67, 85, 141, 280, 386 709, 713 Calderon projector, 386 canonical – extension, 496, 599 – representation of Γ , 680 capacity – identities, 349 – operators, 349 Carl Neumann’s method, 298, 302 Cauchy – data, 3, 65, 81, 149, 388 – principal value integral, 47, 105 – problem, 319 – singular integral equation, 690, 695 – singular integral equation, 684, 689 cavity problem, 348 change of coordinates, 446, 447, 520, 524 characteristic determinant, 455 charge density, 322 choice of gauge, 328 Christoffel symbols, 112, 115 circulation, 17 classical Sobolev spaces, 167 closed range, 237 coercive, 306 coerciveness estimate, 364, 374 combined Dirichlet–Neumann problem, 213, 225 commutative diagram, 447 commutator, 663 compact – imbedding, 165

Index – operator, 229 – support, 96 compatibility – conditions, 66, 71, 81, 86, 142, 186, 224, 485, 498 – relations, 541 complementary – boundary operator, 210, 211 – tangential boundary operators, 146 complete symbol, 442, 443, 449, 492, 629, 646, 707 – of the elastic double layer operator, 706 composition of operators, 443 computation of stress and strain, 669 conduction current density, 317 conductivity, 319 conormal derivative, 206 conservation of charge, 321 constitutive relations, 319 continuous – extension, 579 – functional, 98, 100 – operator, 431, 584 contour integral, 575 contraction, 298, 300 contravariant coordinates, 543 convergence – in C0∞ , 97 – in S  , 99 convergent iterations, 302 cotangent bundle, 543 Coulomb gauge condition, 322, 368 coupled system of integral equations, 622 current, 322 current density, 316, 319, 322 curved polygon, 185 deflection, 81 derivatives of boundary potentials, 667, 673 dielectric, 321 – constant, 322 – domain, 383 diffeomorphism, 111, 525, 536, 540, 572, 745 direct – formulation, 11, 66 – method, 395 direction of propagation, 393 Dirichlet – Cauchy data, 720

777

– problem, 27, 66, 81 – to Neumann map, 299 – type boundary conditions, 387 displacement problem, 47 distribution, 97, 431 – with compact support, 98 – homogeneous, 496 div–operator, 195 domain integral equations, 468 double – biharmonic, 81 – layer boundary integral operator, 385 – hydrodynamic, 65 – layer Maxwell potential, 352 – layer Newton potential, 724 – layer potential, 4, 635, 639, 702, 6703 dual – norm, 162 – operator, 98 – spaces, 197 duality, 162, 175, 185 – pairing, 198 202, 359 dynamic viscosity, 62 Eddy current – configuration, 356 – equations, 358 – Maxwell equations, 331 – problems, 330, 355 – problem in R3 , 332 – problem in the dielectric domain, 377 – problem in the unbounded domain, 366, 371 EFIE, 394 eigensolutions, 271, 608 eigenvalue problem, 333 elasticity tensor, 256 electric – displacement, 322 – displacement field or flux, 317 – field, 317, 322 – induction, 322 – permittivity, 319 – scalar potential, 325 electromagnetic – energy, 320 – waves, 323 elliptic, 452, 453, 455, 466, 625, 689 – formally positive, 256, 258 – in the sense – – of Agmon–Douglis–Nirenberg, 694 – – of Petrovski, 476 – on Γ , 545

778

Index

– regular, 147 energy – estimate, 319 – space, 268, 287, 300 enforced constraints, 227 equivalence – between boundary value problems and integral equations, 285 – classes of Cauchy sequences, 160 – norms, 168 – theorem, 151 essential – boundary condition, 210 – bounded, 159 exceptional or irregular frequencies, 28 existence for EFIE, 399 explicit formulae for the symbol, 684 extension – conditions, 571, 594 – operator, 166 exterior – boundary value problem, 152, 274 – capacity operator, 348 – exterior Dirichlet problem, 377 – combined Dirichlet–Neumann problem, 224 – Dirichlet problem, 14, 28, 86, 218, 329 – displacement problem, 56 – energy space, 226 – Green’s formula, 208 – Neumann problem, 222, 223 – problems, 65 – representation formula, 138 – traction problem, 60 Faraday’s law, 318, 325 Faraday/meter, 321 finite – atlas, 109 – energy, 221 – part integral, 481, 523, 531, 600 first Green’s formula, 206, 613 fluid density, 62 Fourier – integral operators, 440 – series – – expansion, 676, 704, 707 – – norm, 183 – – representation, 681, 683, 6702, 703, 708 – – representation of the single layer potential operator V , 676

– transform, 99, 164, 185, 430 Frechet space, 169 Fredholm – alternative, 236, 250, 608 – index, 252, 689, 690, 693, 694, 698 – integral equation, 463, 468 – – of the first kind, 11, 15, 86 – – of the second kind, 12, 15 – operator, 252, 689 – theorem, 236 – mapping, 355 free space Green’s operator, 482 Frenet’s formulae, 680 frequency ω, 323 Friedrichs inequality, 168 function – spaces, 194 – – for Maxwell’s equations, 194 fundamental – assumption, 146, 149 – matrix, 632 – solution, 2, 45, 63, 81, 131, 217, 326, 457, 471, 719 – tensor, 328 – velocity tensor, 64 G¨ unter derivative, 116, 117, 662 gauge condition, 327 Gauss’ law – for electric fields, 317 – for magnetic fields, 317 Gaussian – bracket, 101, 174 – curvature, 114 general – exterior – – Dirichlet problem, 222 – – Neumann–Robin problem, 223 – Green representation formula, 130 – interior – – Dirichlet problem, 210 – – Neumann–Robin problem, 211, 213 generalized – current concept, 321 – first Green’s formula, 305 – Newton potential, 625, 626 – plane stress, 45 – Plemelj–Sochozki jump relations, 600 – polynomials, 221 – representation formula, 277, 621 – second Green’s formula, 272 global parametric representation, 181 graph norm, 262

Index Green’s – formula, 80, 195, 198, 719 – identity, 128 – operator, 457, 620 growth conditions at infinity, 65 G˚ arding’s – inequality, 229, 254, 255, 259, 264, 266, 269, 287, 294, 297, 303, 350, 355, 403, 469, 546, 607, 693, 734 – theorem, 255 H¨ older – continuity, 49 – modulus, 97 – norm, 97 – spaces, 5 Hadamard – coordinates, 111, 113 – finite part integral, 103, 104 – – operator, 480, 492 Helmholtz – decomposition, 349 – equation, 326, 480, 628, 701 – operator, 475 Henry/meter, 321 Hilbert – space, 160, 170, 174, 221 – – formulation, 219 – transform, 690, 693 Hodge – decomposition, 200 – – of tangential vector fields, 201 – – of vector fields, 200 – –Laplace equation, 384 homogeneous, 498 – degree, 436 – distribution, 495 – equation, 270 – function, 479 – polynomial, 495 – principal symbol, 445, 452 – symbol, 515, 701 – – expansion, 631 hydrodynamic boundary – integral operator, 303 – traction, 66 hypersingular – boundary – – integral equation, 13 – – operator, 386 – – potential, 661 – elastic operator, 707 – integral equation of the first kind, 16

779

– operator, 49, 68, 681, 704 – surface potential, 651 impedance, 324 – boundary condition, 420 incident fields, 392 incompressible viscous fluid, 62 indirect method, 395 induced mapping, 541 inhomogeneous Helmholtz equations, 323 inner product, 160, 164, 173, 221 insertion problem, 93 insulator, 321 integration by parts formula, 202 interface, 328 interior – capacity operator, 347 – problems, 329 – – boundary value , 270 – – Dirichlet, 86, 210, 716 – – displacement, 75 – – Neumann, 211, 264, 717 intersection space, 194, 204 invariance property, 531, 532 invariant parity conditions, 537 inverse Fourier transform, 100, 164 isomorphism, 100, 163 iterated Laplacian, 475 jump relation, 136, 145, 280, 308, 580, 639, 643, 656 – for the derivatives of boundary potentials, 661 kernel – function, 480 – of the double layer potential, 680 Kieser’s theorem, 650 kinematic viscosity, 62 Korn’s inequality, 258, 261 Kutta–Joukowski condition, 16 Lam´e – constants, 45 – system, 631, 705 Laplace–Beltrami operator, 122 Laplacian, 1, 122 large–time behavior, 31 Lax–Milgram theorem, 205, 230, 233 Levi function, 460, 462, 466, 622, 630 linear isotropic homogeneous media, 319 Lipschitz

780

Index

– boundary, 110 – continuous, 97 – – boundary, 194 – norm, 97 local – chart, 540, 547, 593 – fundamental solutions, 471 – operator, 540 – pseudodifferential operator, 542 – spaces, 168 locally – convex topological vector space, 97 – H¨ older continuous, 97 Lopatinski–Shapiro condition, 147 Lorentz’s gauge, 328 lossless and lossy media, 324 Lyapounov boundaries, 19, 110 magnetic – constant, 322 – displacement current, 322 – field, 317, 322 – flux density, 317, 322 – induction, 322 – media, 321 – permeability, 319 magnetizing field, 322 mapping properties of potentials, 278, 293, 623 Maue formula, 704 Maxwell, 321 – –Ampere’s law, 324 – –Calderon projector, 346 – differential equations, 318 – –Dirichlet problem, 348 – equations, 318, 323, 327, 356, 719 mean curvature, 114 MFIE, 394, 407, 412 mixed boundary conditions, 91 modified Calder´ on projector, 150 multi–index, 95 multiple layer potentials, 139, 142 multiplication by ϕ, 169 multiply connected domains, 698 natural – boundary condition, 211 – trace space, 170 necessary compatibility condition, 141, 387 Neumann – boundary condition, 81, 387 – Cauchy data, 720

– problem, 66 – series, 461 Newton potential, 46, 626, 629, 631, 632, 720 Nikolski’s Theorem, 248 non–uniqueness of scattering, 413 nonstandard potential, 378 norm, 175, 176 normal trace map γn , 196 oddly elliptic, 691, 693, 696 Ohm’s law, 324 one scalar second order equation, 253 order of a symbol, 430 ordinary differential operator, 399 orthogonality conditions, 28, 71, 76, 271, 608 orthonormal matrix, 745 oscillatory integrals, 433 Paley–Wiener–Schwartz theorem, 100, 427, 483 parametric 1–periodic representation, 675 parametrix, 453, 456, 458, 482, 545, 719 parity condition, 515, 516, 525, 530–532, 571, 573 Parseval’s formula, 100 Parseval–Plancherel formula, 164 partial differential equation, 130 partie finie integral, 481, 531 partition of unity, 169, 544 penetration depth, 325 perfect – electric conductor, 329, 394 – electric media, 321 periodic – functions, 181 – pseudodifferential operators, 684 permeability, 322 permittivity, 322 plane strain, 45 plane waves, 324 Pocklington’s integral equation, 396, 399 Poincar´e inequality, 168, 375 Poisson – equation, 1 – operator, 539, 549, 567, 574, 620 – ratio, 45, 80 polar coordinates, 525 polarization vectors, 393 polyhedral domains, 194

Index polyhomogeneous, 436 positively homogeneous – function, 430, 506 – principal symbol, 495 positivity, 298 pressure operator, 708 principal – fundamental solutions, 474 – symbol, 442, 446, 542, 543, 700 – – of the acoustic double layer boundary integral operator, 645 – – of the hypersingular boundary integral operator of linear elasticity, 659 – – of the operators Kk and Kk , 644 – – of the single layer boundary integral operator, 647 – symbol, 735 product with a distribution, 98 projection theorem, 230 propagation vector, 324 properly supported, 436, 438, 439, 719 pseudodifferential – classical, 436, 445, 481, 506 – general representation, 481 – integro–differential operator, 432 – mapping properties, 442, 544 – operator, 429, 620, 719 – standard, 426 – transposed, 443 pseudohomogeneous – distribution, 498 – expansion, 480, 684, 703, 713 – function, 101, 485, 498, 501, 507, 515 – kernel, 480, 520 – – expansion, 701 – – under the change of coordinates, 520 pullback, 446, 541 pushforward, 446, 541 radiation condition, 56, 134, 142, 214, 218, 219 rank condition, 259 rapidly decreasing, 99 rational function, 572 regular – diffeomorphism, 112 – elliptic boundary value problem, 147 regularizer, 441 Rellich Lemma, 262 representation – formula, 80, 82, 93, 131, 135, 138, 275, 393

781

– of pseudodifferential operators, 509 – of E, 720 Riemannian metric, 112, 523, 746 Riesz – representation theorem, 232 – –Schauder system, 337 – –Schauder Theorem, 246 right inverse, 180, 383 rigid motion, 264 Rumsey principle, 354, 400 saddle point problem, 262, 391 scalar – differential equation, 119, 474, 636 – product, 174, 176 scattering – by a dielectric body, 413, 416 – problems, 329, 392 Schwartz – kernel, 427, 441, 482, 507, 508, 546 – space S, 99 second – Betti number, 390 – boundary integral equation, 727 – fundamental form, 114 – Green’s formula, 63, 130, 272 – order system, 148, 205 sectional trace, 592, 604 semiconductor, 322 sesquilinear form, 206, 209, 233, 257, 269, 402 shear force, 81 shift–theorem, 608 Silver–M¨ uller radiation conditions, 332 single layer – boundary integral operator, 385 – hydrodynamic potential, 65 – Maxwell potential, 352 – potential, 3, 81, 634, 640, 676, 701, 705, 721 singular – Green’s operator, 621 – perturbation, 33 skew–symmetric bilinear form, 212 skin – depth, 325, 420 – effect, 329 Slobodetskii norm, 161 smoothing – operator, 441, 541 – property, 541 Sobolev – imbedding theorems, 166

782

Index

– spaces of negative order, 162 Sommerfeld – matrix, 256 – radiation conditions, 26 special parity conditions, 598 speed of light, 321 spurious resonances, 409 Steklov–Poincar´e operator, 299 Stokes system, 61, 62, 455 Stratton–Chu formula – for E in the interior, 342 – for H in the interior, 342 stream function, 79 stress – operator, 63 – tensor, 63 strong – ellipticity, 229, 355, 640, 682, 690, 695, 736 – Lipschitz domain, 110 strongly elliptic, 452, 469, 545, 606, 639, 641, 691, 696, 736, 739 – strongly elliptic system, 636 739 – system of pseudodifferential operators, 607 supplementary transmission problem, 281 support of a distribution, 98 surface – gradient, 117 – integral, 170 – potential in the half space, 593 symbol, 439, 442, 572, 708, 719 – of capacity operators, 730 – class, 425 – of double layer potential, 729 – expansion, 713, 728 – of the hypersingular integral operator, 653 – matrix, 456, 650 – of the Newton potential, 720 – of rational type, 570, 573, 601, 604 – of second order Maxwell, 720 – of V, 722 symmetric part, 221 system – of pseudodifferential equations, 694, 717 – of periodic integral equations, 699 system of volume integral equations, 415 tangent

– bundle, 543 – space, 543 tangential differential operator, 116, 198, 581, 597, 723, 724 – projection, 196 – trace operator, 196 – trace spaces, 197 tempered distribution, 99, 215 tensor product, 135 Theorem by Rellich, 337 thin plate, 80, 81 thin–wire antenna, 396 total – currents, 322 – fields, 392 trace – trace adjoint, 177 – of Newton potential, 628 – operator, 170, 177, 604, 620 – sectional, 592, 604 – spaces, 169, 170, 178, 180, 181 – – on an open surface, 189 – theorem, 177 – theorem for H(div , Ω), 196 traction problem, 55 traditional transformation formula, 520 transformed kernel, 520 transmission – condition, 226, 227, 328, 332, 369, 413, 417, 539, 550, 559, 570, 571, 573, 620, 724 – problems, 225, 275, 330 transposed differential operator, 130 Tricomi conditions, 483, 507, 509, 538, 571, 594 tubular neighbourhood, 547 two–dimensional – Laplacian, 676 – potential flow, 16 uniform cone property, 166 uniformly strongly elliptic, 130, 469 unique – continuation property, 215 – solution, 272, 275 uniqueness – of the cavity problem, 332 – of the homogeneous exterior problem, 337 – of the transmission problem, 339 variational – equation, 349, 360, 384

Index – formulation, 205, 210, 214, 218, 225, 293, 294, 606 vector potential, 325, 377 vectorial – double layer potential, 378 – Newton potential, 378 – single layer potential, 378 very strongly elliptic, 255 volumetric EFIE, 416 volume potential, 46, 457, 602 vorticity, 80

wave – amplitudes, 324 – number, 25 weak – formulation, 400 – variational equation, 353 weakly singular, 4, 68 weighted Sobolev spaces, 191, 381

783