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Applied Mathematical Sciences
EditorsS. S. Antman J. E. Marsden L. Sirovich
AdvisorsJ. K. Hale P. Holmes J. KeenerJ. Keller B. J. Matkowsky A. MielkeC. S. Peskin K. R. Sreenivasan
Volume 164
Applied Mathematical Sciences
1. John: Partial Differential Equations, 4th ed.
2. Sirovich: Techniques of Asymptotic Analysis
3. Hale: Theory of Functional Differential Equations,2nd ed.
4. Percus: Combinatorial Methods
5. von Mises/Friedrichs: Fluid Dynamics
6. Freiberger/Grenander: A Short Course inComputational Probability and Statistics.
7. Pipkin: Lectures on Viscoelasticity Theory
8. Giacaglia: Perturbation Methods in Non-linearSystems
9. Friedrichs: Spectral Theory of Operators inHilbert Space
10. Stroud: Numerical Quadrature and Solution ofOrdinary Differential Equations
11. Wolovich: Linear Multivariable Systems
12. Berkovitz: Optimal Control Theory
13. Bluman/Cole: Similarity Methods for DifferentialEquations
14. Yoshizawa: Stability Theory and the Existence ofPeriodic Solution and Almost Periodic Solutions
15. Braun: Differential Equations and TheirApplications, 4th ed.
16. Lefschetz: Applications of Algebraic Topology
17. Collatz/Wetterling: Optimization Problems
18. Grenander: Pattern Synthesis: Lectures in PatternTheory, Vol. I
19. Marsden/McCracken: Hopf Bifurcation and ItsApplications
20. Driver: Ordinary and Delay Differential Equations
21. Courant/Friedrichs: Supersonic Flow and ShockWaves
22. Rouche/Habets/Laloy: Stability Theory byLiapunov’s Direct Method
23. Lamperti: Stochastic Processes: A Survey of theMathematical Theory
24. Grenander: Pattern Analysis: Lectures in PatternTheory, Vol. II
25. Davies: Integral Transforms and TheirApplications, 3rd ed.
26. Kushner/Clark: Stochastic Approximation Methodsfor Constrained and Unconstrained Systems
27. de Boor: A Practical Guide to Splines, RevisedEdition
28. Keilson: Markov Chain Models-Rarity andExponentiality
29. de Veubeke: A Course in Elasticity
30. Sniatycki: Geometric Quantization and QuantumMechanics
31. Reid: Sturmian Theory for Ordinary DifferentialEquations
32. Meis/Markowitz: Numerical Solution of PartialDifferential Equations
33. Grenander: Regular Structures: Lectures in PatternTheory, Vol. III
34. Kevorkian/Cole: Perturbation Methods in AppliedMathematics
35. Carr: Applications of Centre Manifold Theory
36. Bengtsson/Ghil/Källén: Dynamic Meteorology:Data Assimilation Methods
37. Saperstone: Semidynamical Systems in InfiniteDimensional Spaces
38. Lichtenberg/Lieberman: Regular and ChaoticDynamics, 2nd ed.
39. Piccini/Stampacchia/Vidossich: OrdinaryDifferential Equations inRn
40. Naylor/Sell: Linear Operator Theory inEngineering and Science
41. Sparrow: The Lorenz Equations: Bifurcations,Chaos, and Strange Attractors
42. Guckenheimer/Holmes: Nonlinear Oscillations,Dynamical Systems, and Bifurcations of VectorFields
43. Ockendon/Taylor: Inviscid Fluid Flows
44. Pazy: Semigroups of Linear Operators andApplications to Partial Differential Equations
45. Glashoff/Gustafson: Linear Operations andApproximation: An Introduction to theTheoretical Analysis and Numerical Treatment ofSemi-Infinite Programs
46. Wilcox: Scattering Theory for Diffraction Gratings
47. Hale et al.: Dynamics in Infinite Dimensions,2nd ed.
48. Murray: Asymptotic Analysis
49. Ladyzhenskaya: The Boundary-Value Problems ofMathematical Physics
50. Wilcox: Sound Propagation in Stratified Fluids
51. Golubitsky/Schaeffer: Bifurcation and Groups inBifurcation Theory, Vol. I
52. Chipot: Variational Inequalities and Flow inPorous Media
53. Majda: Compressible Fluid Flow and System ofConservation Laws in Several Space Variables
54. Wasow: Linear Turning Point Theory
55. Yosida: Operational Calculus: A Theory ofHyperfunctions
56. Chang/Howes: Nonlinear Singular PerturbationPhenomena: Theory and Applications
57. Reinhardt: Analysis of Approximation Methods forDifferential and Integral Equations
58. Dwoyer/Hussaini/Voigt (eds): TheoreticalApproaches to Turbulence
59. Sanders/Verhulst: Averaging Methods in NonlinearDynamical Systems
(continued following index)
123
W olfga ng L . W endla nd
BoundaryIntegral Eq uations
G eorge C . Hsia o
Editors:
S. S. AntmanDepartment of MathematicsandInstitute for Physical Scienceand TechnologyUniversity of MarylandCollege Park, MD [email protected]
J. E. MarsdenControl and DynamicalSystems, 107-81California Instituteof TechnologyPasadena, CA [email protected]
L. SirovichLaboratory of AppliedMathematicsDepartment ofBiomathematical ScienceMount Sinai Schoolof MedicineNew York, NY [email protected]
George C. Hsiao
University of Delaware
Wolfgang L. WendlandDepartment of Mathematical Sciences
528 Ewing Hall Institut f r AngewandteAnalysis und Numerische Simulation
u¨
e-ISBN 978-3-540-68545-6
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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even inthe absence of a specific statement, that such names are exempt from the relevant protective laws and regulations andtherefore free for general use.
Mathematics Subject Classification (2001): 47G10-30, 35J55, 45A05, 31A10, 73C02, 76D07
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ISBN 978-3-540-15284-2
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Preface
This book is devoted to the mathematical foundation of boundary integralequations. The combination of finite element analysis on the boundary withthese equations has led to very efficient computational tools, the boundaryelement methods (see e.g., the authors [139] and Schanz and Steinbach (eds.)[267]). Although we do not deal with the boundary element discretizationsin this book, the material presented here gives the mathematical foundationof these methods. In order to avoid over generalization we have confinedourselves to the treatment of elliptic boundary value problems.
The central idea of eliminating the field equations in the domain and re-ducing boundary value problems to equivalent equations only on the bound-ary requires the knowledge of corresponding fundamental solutions, and thisidea has a long history dating back to the work of Green [107] and Gauss[95, 96]. Today the resulting boundary integral equations still serve as amajor tool for the analysis and construction of solutions to boundary valueproblems.
As is well known, the reduction to equivalent boundary integral equationsis by no means unique, and there are primarily two procedures for this re-duction, the ‘direct’ and ‘indirect’ approaches. The direct procedure is basedon Green’s representation formula for solutions of the boundary value prob-lem, whereas the indirect approach rests on an appropriate layer ansatz. Inour presentation we concentrate on the direct approach although the corre-sponding analysis and basic properties of the boundary integral operatorsremain the same for the indirect approaches. Roughly speaking, one obtainstwo kinds of boundary integral equations with both procedures, those of thefirst kind and those of the second kind.
The basic mathematical properties that guarantee existence of solutionsto the boundary integral equations and also stability and convergence analy-sis of corresponding numerical procedures hinge on Garding inequalities forthe boundary integral operators on appropriate function spaces. In addition,contraction properties allow the application of Carl Neumann’s classical suc-cessive iteration procedure to a class of boundary integral equations of thesecond kind. It turns out that these basic features are intimately relatedto the variational forms of the underlying elliptic boundary value problems
VIII Preface
and the potential energies of their solution fields, allowing us to considerthe boundary integral equations in the form of variational problems on theboundary manifold of the domain.
On the other hand, the Newton potentials as the inverses of the ellipticpartial differential operators are particular pseudodifferential operators onthe domain or in the Euclidean space. The boundary potentials (or Poissonoperators) are just Newton potentials of distributions with support on theboundary manifold and the boundary integral operators are their tracesthere. Therefore, it is rather natural to consider the boundary integral op-erators as pseudodifferential operators on the boundary manifold. Indeed,most of the boundary integral operators in applications can be recast as suchpseudodifferential operators provided that the boundary manifold is smoothenough.
With the application of boundary element methods in mind, where strongellipticity is the basic concept for stability, convergence and error analysis ofcorresponding discretization methods for the boundary integral equations,we are most interested in establishing strong ellipticity in terms of Garding’sinequality for the variational formulation as well as strong ellipticity of thepseudodifferential operators generated by the boundary integral equations.The combination of both, namely the variational properties of the ellipticboundary value and transmission problems as well as the strongly ellipticpseudodifferential operators provides us with an efficient means to analyze alarge class of elliptic boundary value problems.
This book contains 10 chapters and an appendix. For the reader’s benefit,Figure 0.1 gives a sketch of the topics contained in this book. Chapters 1through 4 present various examples and background information relevant tothe premises of this book.
In Chapter 5, we discuss the variational formulation of boundary inte-gral equations and their connection to the variational solution of associatedboundary value or transmission problems. In particular, continuity and co-erciveness properties of a rather large class of boundary integral equationsare obtained, including those discussed in the first and second chapters. InChapter 4, we collect basic properties of Sobolev spaces in the domain andtheir traces on the boundary, which are needed for the variational formula-tions in Chapter 5.
Chapter 6 presents an introduction to the basic theory of classicalpseudodifferential operators. In particular, we present the construction ofa parametrix for elliptic pseudodifferential operators in subdomains of IRn.Moreover, we give an iterative procedure to find Levi functions of arbitraryorder for general elliptic systems of partial differential equations. If the fun-damental solution exists then Levi’s method based on Levi functions allowsits construction via an appropriate integral equation.
Preface IX
In Chapter 7, we show that every pseudodifferential operator is anHadamard’s finite part integral operator with integrable or nonintegrablekernel plus possibly a differential operator of the same order as that of thepseudodifferential operator in case of nonnegative integer order. In addition,we formulate the necessary and sufficient Tricomi conditions for the inte-gral operator kernels to define pseudodifferential operators in the domain byusing the asymptotic expansions of the symbols and those of pseudohomoge-neous kernels. We close Chapter 7 with a presentation of the transformationformulae and invariance properties under the change of coordinates.
Chapter 8 is devoted to the relation between the classical pseudodifferen-tial operators and boundary integral operators. For smooth boundaries, all ofour examples in Chapters 1 and 2 of boundary integral operators belong tothe class of classical pseudodifferential operators on compact manifolds hav-ing symbols of the rational type. If the corresponding class of pseudodifferen-tial operators in the form of Newton potentials is applied to tensor productdistributions with support on the boundary manifold, then they generate, in anatural way, boundary integral operators which again are classical pseudodif-ferential operators on the boundary manifold. Moreover, for these operatorsassociated with regular elliptic boundary value problems, it turns out thatthe corresponding Hadamard’s finite part integral operators are invariant un-der the change of coordinates, as considered in Chapter 3. This approach alsoprovides the jump relations of the potentials. We obtain these properties byusing only the Schwartz kernels of the boundary integral operators. However,these are covered by Boutet de Monvel’s work in the 1960’s on regular ellipticproblems involving the transmission properties.
The last two chapters, 9 and 10, contain concrete examples of bound-ary integral equations in the framework of pseudodifferential operators onthe boundary manifold. In Chapter 9, we provide explicit calculations of thesymbols corresponding to typical boundary integral operators on closed sur-faces in IR3. If the fundamental solution is not available then the boundaryvalue problem can still be reduced to a coupled system of domain and bound-ary integral equations. As an illustration we show that these coupled systemscan be considered as some particular Green operators of the Boutet de Mon-vel algebra. In Chapter 10, the special features of Fourier series expansionsof boundary integral operators on closed curves are exploited.
We conclude the book with a short Appendix on differential operators inlocal coordinates with minimal differentiability. Here, we avoid the explicituse of the normal vector field as employed in Hadamard’s coordinates inChapter 3. These local coordinates may also serve for a more detailed analysisfor Lipschitz domains.
X Preface
Variational formulations ofBVPs and BIEs
(Chap. 5)
Sobolev spaces and trace
theorems (Chap. 4)
Generalized multilayerpotentials on Γ = ∂Ω
(Chap. 3)
Classical model problems
(Chap. 1 and Chap. 2)
Fourier representation of
BIOs and ψdOs on Γ ⊂ R2
(Chap. 10)
BIEs and ψdOs on Γ
(Chap. 8 and Chap. 9)
Classical ψdOs and IOsin Ω ⊂ R
n
(Chap. 7)
ψdOs in Ω ⊂ Rn
(Chap. 6)
Abbreviations:
Ω ⊂ IRn – A given domain with compact boundary ΓBVPs – Boundary value problemsBIEs – Boundary integral equationsψdOs – Pseudodifferential operatorsIOs – Integral operatorsBIOs – Boundary integral operators
Fig. 0.1. A schematic sketch of the topics and their relations
Our original plan was to finish this book project about 10 years ago.However, many new ideas and developments in boundary integral equationmethods appeared during these years which we have attempted to incorpo-rate. Nevertheless, we regret to say that the present book is by no meanscomplete. For instance, we only slightly touch on the boundary integral op-erator methods involving Lipschitz boundaries which have recently becomemore important in engineering applications. We do hope that we have madea small step forward to bridge the gap between the theory of boundary inte-gral equation methods and their applications. We further hope that this bookwill lead to better understanding of the underlying mathematical structure ofthese methods and will serve as a mathematical foundation of the boundaryelement methods.
Preface XI
In closing, we would also like to mention some other relevant books relatedto boundary integral methods such as the classical books on potential the-ory by Kellogg [155] and Gunter [113], the mathematical books on boundaryintegral equations by Hackbusch [116], Jaswon and Symm [148], Kupradze[175, 176, 177], Schatz, Thomee and Wendland [268], Mikhlin [211, 212, 213],Nedelec [231, 234], Colton and Kress [47, 48], Mikhlin, Morozov and Paukshto[214], Mikhlin and Prossdorf [215], Dautray and Lions [60], Chen and Zhou[40], Gatica and Hsiao [93], Kress [172], McLean [203], Yu, De–hao [324],Steinbach [290], Freeden and Michel [83], Kohr and Pop [163], Sauter andSchwab [266], as well as the Encyclopedia articles by Maz’ya [202], Prossdorf[253], Agranovich [4] and the authors [141]. For engineering books on bound-ary integral equations, we suggest the books by Brebbia [23], Crouch andStarfield [57], Brebbia, Telles and Wrobel [24], Manolis and Beskos [197],Balas, Sladek and Sladek [11], Pozrikidis [252], Power and Wrobel [251], Bon-net [18], Gaul, Kogel and Wagner [94].
Acknowledgements:
We are very grateful for the continuous support and encouragement by ourstudents, colleagues, and friends. During the course of preparing this book wehave benefitted from countless discussions with so many excellent individualsincluding Martin Costabel, Gabriel Gatica, Olaf Steinbach, Ernst Stephanand the late Siegfried Prossdorf; to name a few. Moreover, we are indebtedto our reviewers of the first draft of the book manuscript for their criticalreviews and helpful suggestions which helped us to improve our presentation.We would like to extend our thanks to Greg Silber, Clemens Forster andGulnihal Meral for careful and critical proof reading of the manuscript.
We take this opportunity to acknowledge our gratitude to our universi-ties, the University of Delaware at Newark, DE. U.S.A. and the Universityof Stuttgart in Germany; the Alexander von Humboldt Foundation, the Ful-bright Foundation, and the German Research Foundation DFG for repeatedsupport within the Priority Research Program on Boundary Element Meth-ods and within the Collaborative Research Center on Multifield Problems,SFB 404 at the University of Stuttgart, the MURI program of AFOSR atthe University of Delaware, and the Necas Center in Prague. We express inparticular our gratitude to the Oberwolfach Research Institute in Germanywhich supported us three times through the Research in Pairs Program, wherewe enjoyed the excellent research environment and atmosphere. It is also apleasure to acknowledge the generous attitude, the unfailing courtesy, andthe ready cooperation of the publisher.
Last, but by no means least, we are gratefully indebted to Gisela Wend-land for her highly skilled hands in the LATEX typing and preparation of thismanuscript.
Newark, Delaware George C. HsiaoStuttgart, Germany, 2008 Wolfgang L. Wendland
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Green Representation Formula . . . . . . . . . . . . . . . . . . . . . . . 11.2 Boundary Potentials and Calderon’s Projector . . . . . . . . . . . . . 31.3 Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.1 The Exterior Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 131.4.2 The Exterior Neumann Problem . . . . . . . . . . . . . . . . . . . 15
1.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Low Frequency Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 The Lame System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1 The Interior Displacement Problem . . . . . . . . . . . . . . . . . 472.2.2 The Interior Traction Problem . . . . . . . . . . . . . . . . . . . . . 552.2.3 Some Exterior Fundamental Problems . . . . . . . . . . . . . . 562.2.4 The Incompressible Material . . . . . . . . . . . . . . . . . . . . . . . 61
2.3 The Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.3.1 Hydrodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 652.3.2 The Stokes Boundary Value Problems . . . . . . . . . . . . . . . 662.3.3 The Incompressible Material — Revisited . . . . . . . . . . . 75
2.4 The Biharmonic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.4.1 Calderon’s Projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.4.2 Boundary Value Problems and Boundary
Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3. Representation Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.1 Classical Function Spaces and Distributions . . . . . . . . . . . . . . . . 953.2 Hadamard’s Finite Part Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 101
XVI Table of Contents
3.3 Local Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.4 Short Excursion to Elementary Differential Geometry . . . . . . . 111
3.4.1 Second Order Differential Operatorsin Divergence Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.5 Distributional Derivatives and Abstract Green’sSecond Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.6 The Green Representation Formula . . . . . . . . . . . . . . . . . . . . . . . 1303.7 Green’s Representation Formulae in Local Coordinates . . . . . . 1353.8 Multilayer Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.9 Direct Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . 145
3.9.1 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . 1453.9.2 Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4. Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.1 The Spaces Hs(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.2 The Trace Spaces Hs(Γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.2.1 Trace Spaces for Periodic Functions on a SmoothCurve in IR2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.2.2 Trace Spaces on Curved Polygons in IR2 . . . . . . . . . . . . . 1854.3 The Trace Spaces on an Open Surface . . . . . . . . . . . . . . . . . . . . . 1894.4 Weighted Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5. Variational Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.1 Partial Differential Equations of Second Order . . . . . . . . . . . . . 195
5.1.1 Interior Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1995.1.2 Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045.1.3 Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.2 Abstract Existence Theorems for Variational Problems . . . . . . 2185.2.1 The Lax–Milgram Theorem . . . . . . . . . . . . . . . . . . . . . . . . 219
5.3 The Fredholm–Nikolski Theorems . . . . . . . . . . . . . . . . . . . . . . . . 2265.3.1 Fredholm’s Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.3.2 The Riesz–Schauder and the Nikolski Theorems . . . . . . 2355.3.3 Fredholm’s Alternative for Sesquilinear Forms . . . . . . . . 2405.3.4 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
5.4 Garding’s Inequality for Boundary Value Problems . . . . . . . . . 2435.4.1 Garding’s Inequality for Second Order Strongly
Elliptic Equations in Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435.4.2 The Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2505.4.3 Garding’s Inequality for Exterior Second Order
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2545.4.4 Garding’s Inequality for Second Order Transmission
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2595.5 Existence of Solutions to Boundary Value Problems . . . . . . . . . 259
5.5.1 Interior Boundary Value Problems . . . . . . . . . . . . . . . . . . 260
Table of Contents XVII
5.5.2 Exterior Boundary Value Problems . . . . . . . . . . . . . . . . . 2645.5.3 Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.6 Solution of Integral Equations via Boundary Value Problems . 2655.6.1 The Generalized Representation Formula for Second
Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.6.2 Continuity of Some Boundary Integral Operators . . . . . 2675.6.3 Continuity Based on Finite Regions . . . . . . . . . . . . . . . . . 2705.6.4 Continuity of Hydrodynamic Potentials . . . . . . . . . . . . . 2725.6.5 The Equivalence Between Boundary Value Problems
and Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2745.6.6 Variational Formulation of Direct Boundary Integral
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775.6.7 Positivity and Contraction of Boundary Integral
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2875.6.8 The Solvability of Direct Boundary Integral Equations 2915.6.9 Positivity of the Boundary Integral Operators
of the Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2925.7 Partial Differential Equations of Higher Order . . . . . . . . . . . . . . 2935.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.8.1 Assumptions on Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995.8.2 Higher Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . 2995.8.3 Mixed Boundary Conditions and Crack Problem . . . . . 300
6. Introduction to Pseudodifferential Operators . . . . . . . . . . . . . 3036.1 Basic Theory of Pseudodifferential Operators . . . . . . . . . . . . . . 3036.2 Elliptic Pseudodifferential Operators on Ω ⊂ IRn . . . . . . . . . . . 326
6.2.1 Systems of Pseudodifferential Operators . . . . . . . . . . . . . 3286.2.2 Parametrix and Fundamental Solution . . . . . . . . . . . . . . 3316.2.3 Levi Functions for Scalar Elliptic Equations . . . . . . . . . 3346.2.4 Levi Functions for Elliptic Systems . . . . . . . . . . . . . . . . . 3416.2.5 Strong Ellipticity and Garding’s Inequality . . . . . . . . . . 343
6.3 Review on Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . 3466.3.1 Local Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . 3476.3.2 Fundamental Solutions in IRn for Operators
with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 3486.3.3 Existing Fundamental Solutions in Applications . . . . . . 352
7. Pseudodifferential Operators as Integral Operators . . . . . . . 3537.1 Pseudohomogeneous Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
7.1.1 Integral Operators as Pseudodifferential Operatorsof Negative Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
7.1.2 Non–Negative Order Pseudodifferential Operatorsas Hadamard Finite Part Integral Operators . . . . . . . . . 380
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7.1.3 Parity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3897.1.4 A Summary of the Relations between Kernels
and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3927.2 Coordinate Changes and Pseudohomogeneous Kernels . . . . . . . 394
7.2.1 The Transformation of General Hadamard Finite PartIntegral Operators under Change of Coordinates . . . . . 397
7.2.2 The Class of Invariant Hadamard Finite Part IntegralOperators under Change of Coordinates . . . . . . . . . . . . . 404
8. Pseudodifferential and Boundary Integral Operators . . . . . . 4138.1 Pseudodifferential Operators on Boundary Manifolds . . . . . . . . 414
8.1.1 Ellipticity on Boundary Manifolds . . . . . . . . . . . . . . . . . . 4188.1.2 Schwartz Kernels on Boundary Manifolds . . . . . . . . . . . . 420
8.2 Boundary Operators Generated by DomainPseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
8.3 Surface Potentials on the Plane IRn−1 . . . . . . . . . . . . . . . . . . . . . 4238.4 Pseudodifferential Operators with Symbols of Rational Type . 4468.5 Surface Potentials on the Boundary Manifold Γ . . . . . . . . . . . . 4678.6 Volume Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4768.7 Strong Ellipticity and Fredholm Properties . . . . . . . . . . . . . . . . 4798.8 Strong Ellipticity of Boundary Value Problems
and Associated Boundary Integral Equations . . . . . . . . . . . . . . . 4858.8.1 The Boundary Value and Transmission Problems . . . . . 4858.8.2 The Associated Boundary Integral Equations
of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4888.8.3 The Transmission Problem and Garding’s inequality . . 489
8.9 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
9. Integral Equations on Γ ⊂ IR3 Recastas Pseudodifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4939.1 Newton Potential Operators for Elliptic Partial Differential
Equations and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4999.1.1 Generalized Newton Potentials for the Helmholtz
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5029.1.2 The Newton Potential for the Lame System. . . . . . . . . . 5059.1.3 The Newton Potential for the Stokes System . . . . . . . . . 506
9.2 Surface Potentials for Second Order Equations . . . . . . . . . . . . . 5079.2.1 Strongly Elliptic Differential Equations . . . . . . . . . . . . . . 5109.2.2 Surface Potentials for the Helmholtz Equation . . . . . . . 5149.2.3 Surface Potentials for the Lame System . . . . . . . . . . . . . 5199.2.4 Surface Potentials for the Stokes System . . . . . . . . . . . . 524
9.3 Invariance of Boundary Pseudodifferential Operators . . . . . . . . 5249.3.1 The Hypersingular Boundary Integral Operators
for the Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . 525
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9.3.2 The Hypersingular Operator for the Lame System . . . . 5319.3.3 The Hypersingular Operator for the Stokes System . . . 535
9.4 Derivatives of Boundary Potentials . . . . . . . . . . . . . . . . . . . . . . . 5359.4.1 Derivatives of the Solution to the Helmholtz Equation 5419.4.2 Computation of Stress and Strain on the Boundary
for the Lame System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5439.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
10. Boundary Integral Equations on Curves in IR2 . . . . . . . . . . . . 54910.1 Fourier Series Representation of the Basic Operators . . . . . . . . 55010.2 The Fourier Series Representation of Periodic Operators
A ∈ Lmc(Γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
10.3 Ellipticity Conditions for Periodic Operators on Γ . . . . . . . . . . 56210.3.1 Scalar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56310.3.2 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56810.3.3 Multiply Connected Domains . . . . . . . . . . . . . . . . . . . . . . 572
10.4 Fourier Series Representation of some Particular Operators . . 57410.4.1 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 57410.4.2 The Lame System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57810.4.3 The Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58110.4.4 The Biharmonic Equation . . . . . . . . . . . . . . . . . . . . . . . . . 582
10.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
A. Differential Operators in Local Coordinateswith Minimal Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
