Hsiao g wendland w boundary integral equations

Applied Mathematical Sciences Volume 164 Editors S S Antman J E Marsden L Sirovich Advisors J K Hale P Holmes J Keener J Keller B J Matkowsky A Mielke C S Peskin K R Sreenivasan Applied Mathematical Sciences John: Partial Differential Equations, 4th ed Sirovich: Techniques of Asymptotic Analysis Hale: Theory of Functional Differential Equations, 2nd ed 32 Meis/Markowitz: Numerical Solution of Partial Differential Equations 33 Grenander: Regular Structures: Lectures in Pattern Theory, Vol III Percus: Combinatorial Methods 34 Kevorkian/Cole: Perturbation Methods in Applied Mathematics von Mises/Friedrichs: Fluid Dynamics 35 Carr: Applications of Centre Manifold Theory Freiberger/Grenander: A Short Course in Computational Probability and Statistics 36 Bengtsson/Ghil/Källén: Dynamic Meteorology: Data Assimilation Methods Pipkin: Lectures on Viscoelasticity Theory 37 Saperstone: Semidynamical Systems in Infinite Dimensional Spaces Giacaglia: Perturbation Methods in Non-linear Systems 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Topology 45 Glashoff/Gustafson: Linear Operations and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs 17 Collatz/Wetterling: Optimization Problems 18 Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I 46 Wilcox: Scattering Theory for Diffraction Gratings 19 Marsden/McCracken: Hopf Bifurcation and Its Applications 47 Hale et al.: Dynamics in Infinite Dimensions, 2nd ed 20 Driver: Ordinary and Delay Differential Equations 48 Murray: Asymptotic Analysis 21 Courant/Friedrichs: Supersonic Flow and Shock Waves 49 Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics 22 Rouche/Habets/Laloy: Stability Theory by Liapunov’s Direct Method 50 Wilcox: Sound Propagation in Stratified Fluids 23 Lamperti: Stochastic Processes: A Survey of the Mathematical Theory 24 Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol II 25 Davies: Integral Transforms and Their Applications, 3rd ed 51 Golubitsky/Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol I 52 Chipot: Variational Inequalities and Flow in Porous Media 53 Majda: Compressible Fluid Flow and System of Conservation Laws in Several Space Variables 54 Wasow: Linear Turning Point Theory 26 Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems 55 Yosida: Operational Calculus: A Theory of Hyperfunctions 27 de Boor: A Practical Guide to Splines, Revised Edition 56 Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory and Applications 28 Keilson: Markov Chain Models-Rarity and Exponentiality 57 Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations 29 de Veubeke: A Course in Elasticity 58 Dwoyer/Hussaini/Voigt (eds): Theoretical Approaches to Turbulence 30 Sniatycki: Geometric Quantization and Quantum Mechanics 31 Reid: Sturmian Theory for Ordinary Differential Equations 59 Sanders/Verhulst: Averaging Methods in Nonlinear Dynamical Systems (continued following index) G eorge C Hsia o W olfga ng L W endla nd Boundary Integral Eq uations 123 George C Hsiao Wolfgang L Wendland Department of Mathematical Sciences University of Delaware 528 Ewing Hall Newark, DE 19716-2553 USA hsiao@math.udel.edu Universitaă t Stuttgart Institut fuă r Angewandte Analysis und Numerische Simulation Pfaffenwaldring 57 70569 Stuttgart Germany wendland@mathematik.uni-stuttgart.de Editors: S S Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA ssa@math.umd.edu ISBN 978-3-540-15284-2 Applied Mathematical Sciences J E Marsden Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA marsden@cds.caltech.edu L Sirovich Laboratory of Applied Mathematics Department of Biomathematical Science Mount Sinai School of Medicine New York, NY 10029-6574 USA chico@camelot.mssm.edu e-ISBN 978-3-540-68545-6 ISSN 0066-5452 Library of Congress Control Number: 2008924867 Mathematics Subject Classification (2001): 47G10-30, 35J55, 45A05, 31A10, 73C02, 76D07 © 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, 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 Cover design: WMX Design, Heidelberg Printed on acid-free paper springer.com To our families for their love and understanding Preface 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 [139] and Schanz and Steinbach (eds.) [267]) Although we 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 [107] and Gauss [95, 96] 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 VIII Preface and the potential energies of their solution fields, allowing us to consider 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 0.1 gives a sketch of the topics contained in this book Chapters through 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 Chapter 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 Preface IX In Chapter 7, 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 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 with a presentation of the transformation formulae and invariance properties under the change of coordinates Chapter is devoted to the relation between the classical pseudodifferential operators and boundary integral operators For smooth boundaries, all of our examples in Chapters and 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 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, and 10, contain concrete examples of boundary integral equations in the framework of pseudodifferential operators on the boundary manifold In Chapter 9, 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 10, 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 These local coordinates may also serve for a more detailed analysis for Lipschitz domains X Preface Classical model problems (Chap and Chap 2) ψdOs in Ω ⊂ Rn (Chap 6) (Chap 5) (Chap 10) ❇ ❆❅ ✁✁ ❇❆❅ ✁ ❇❆ ❅ ✁ ❇❆ ❅ ✁ ❄ ❄ ❇ ❆ ✁❅ Generalized multilayer Classical ψdOs and IOs ❇ ❆ ✁ ❅ ❅ potentials on Γ = ∂Ω in Ω ⊂ Rn ❇ ✁❆ ❅ ❇ ❆   (Chap 7) (Chap 3) ❅ ✁❇ ❆   ✁❅   ❇ ❆ ✁   ❅ ❄ ❄ ✁   ❇ ❅❆ ❇ ❆ ✁  ❅ Sobolev spaces and trace ✁ BIEs and ψdOs on Γ ❇ ❅ ❆❆   ❇   theorems (Chap 4) (Chap and Chap 9) ❇   ❇   ❇ ❄ ❄   ❇   Fourier representation of ❇ Variational formulations of ❇ BIOs and ψdOs on Γ ⊂ R2   BVPs and BIEs 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 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 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 606 References 161 Kă onig, H.: An explicit formula for fundamental solutions of linear partial differential equations with constant coefficients Proc Amer Math Soc 120 (1994) 1315–1318 162 Kohr, M.: The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in IRn Math Nachr (2007) 163 Kohr, M and Pop, I.: Viscous Incompressible Flow WIT Press, Southampton 2004 164 Kohr, M and Wendland, W.L.: Variational boundary integral equations for the Stokes system Applicable Anal 85 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for Elliptic Systems Cambridge Univ Press, Cambridge 1995 324 Yu, De–hao: Natural Boundary Integral Method and its Applications Kluwer Acad Publ., Dordrecht 2002 325 Zeidler, E.: Nonlinear Functional Analysis and its Applications Vol II/A: Linear Monotone Operators Springer–Verlag, Berlin 1990 Index β , α m 96 S (Ω × Rn ), 303 c →, 167 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 s (Ω), 160 H0m (Ω), 167 s (Ω), 168 H00 s (Ω), 169 Hcomp s (Ω), 169 Hloc Km , 366 L2 (Γ ), 170 L2 (Ω), 159 L∞ (Ω), 160 Lp (Ω), 159 OP S m (Ω × Rn ), 305 P , 130 P , 130 [s], 174 Ψ hfq , 101, 353 Ψ hkq (Ω), 354 , 96 α”, 96 (ν) δΓ , 129 δy (x), 131 γ0 u, 171 γ0 , 171 N0 , 95 Lm (Ω), 314 n Lm c (Ω × R ), 319 T ∗ (Γ ), 417 GradΓ u, 117 HΓ−1 (Ω), 196 Q, 458 u ⊗ δΓ , 422 D(Ω), 97, 98 D := D (Rn ), 98 E(Ω), 98 E (Ω), 98 Hs (Γ ), 175 HE (Ω c ), 210 QΩ , 142 QΩ c , 142 S , 99 Hs (Γ ), 176 a priori estimate, 295 adjoint – equation, 260, 483 – operator, 226, 317 – system, 63 Agmon–Douglis–Nirenberg system, 328, 329, 341, 351 Airy stress function, 79 almost incompressible materials, 75 amplitude function, 314 angle of rotation, 81 associated pressure, 64 asymptotic – expansion – – of equations, 31 – – of kernels, 383, 503, 576 – – of symbols, 309, 316, 317 – representation, 399 atlas A, 414, 415, 418, 421, 467 Banach space, 97 Banach’s fixed point principle, 224 Beltrami operator, 119, 597 bending moment, 81 Bessel potential operator, 343, 563 Betti–Somigliana formula, 45 614 Index biharmonic equation, 79, 85, 294, 582, 590 bijective transformation, 403 bilinear form, 80, 251, 565 boundary – conditions, 146 – – normal, 146 – integral – – equations of the first kind, 92 – – operator, 140 – operators, 421, 448 – – generated by domain pseudodifferential operators, 421 – – mapping properties, 418 – sesquilinear forms, 283 – surface, 108 – traction, 247 Boutet de Monvel – algebra, 494, 495 – operator, 494 Calder´ on projector, 9, 14, 67, 85, 141, 270, 582, 587 canonical – extension, 370, 473 – representation of Γ , 554 Carl Neumann’s method, 287, 290 Cauchy – data, 3, 65, 81, 149 – principal value integral, 47, 105 – singular integral equation, 564, 569 change of coordinates, 320, 321, 394, 398 characteristic determinant, 329 Christoffel symbols, 112, 115 circulation, 17 classical Sobolev spaces, 167 closed range, 227 coercive, 295 combined Dirichlet–Neumann problem, 203, 215 commutative diagram, 321 commutator, 537 compact – imbedding, 167 – operator, 219 – support, 96 compatibility – conditions, 66, 71, 81, 86, 142, 186, 214, 359, 372 – relations, 415 complementary – boundary operator, 200, 201 – tangential boundary operators, 146 complete symbol, 316, 317, 323, 366, 502, 520, 581 – of the elastic double layer operator, 580 composition of operators, 317 computation of stress and strain, 543 conormal derivative, 196 continuous – extension, 453 – functional, 98, 100 – operator, 305, 458 contour integral, 449 contraction, 287, 289 contravariant coordinates, 417 convergence – in C0∞ , 97 – in S , 99 convergent iterations, 291 cotangent bundle, 417 coupled system of integral equations, 496 curved polygon, 185 deflection, 81 derivatives of boundary potentials, 541, 547 diffeomorphism, 111, 399, 410, 414, 446, 593 direct formulation, 11, 66 Dirichlet – problem, 27, 66, 81 – to Neumann map, 288 displacement problem, 47 distribution, 97, 305 – with compact support, 98 – homogeneous, 370 domain integral equations, 342 double layer potential, 4, 509, 513, 576, 577 – biharmonic, 81 – hydrodynamic, 65 dual – norm, 164 – operator, 98 duality, 164, 175, 185 dynamic viscosity, 62 eigensolutions, 261, 482 elasticity tensor, 246 elliptic, 326, 327, 329, 341, 499, 563 – formally positive, 245, 248 – in the sense Index – – of Agmon–Douglis–Nirenberg, 568 – – of Petrovski, 350 – on Γ , 419 – regular, 147 energy – space, 258, 278, 289 enforced constraints, 216 equivalence – between boundary value problems and integral equations, 274 – classes of Cauchy sequences, 160 – norms, 168 – theorem, 151 essential boundary condition, 200 essentially bounded, 159 exceptional or irregular frequencies, 28 explicit formulae for the symbol, 558 extension – conditions, 445, 468 – operator, 162 exterior – boundary value problem, 152, 264 – combined Dirichlet–Neumann problem, 214 – Dirichlet problem, 14, 28, 86, 207 – displacement problem, 56 – energy space, 216 – Green’s formula, 198 – Neumann problem, 212, 213 – problems, 65 – representation formula, 138 – traction problem, 60 finite – atlas, 109 – energy, 211 – part integral, 355, 397, 405, 473 first Green’s formula, 196, 487 fluid density, 62 Fourier – integral operators, 314 – series – – expansion, 550, 578, 580 – – norm, 183 – – representation, 555, 557, 575, 577, 581 – – representation of the simple layer potential operator V , 550 – transform, 99, 165, 185, 304 Frechet space, 169 Fredholm – alternative, 226, 240, 482 – index, 242, 563, 564, 567, 568, 572 – integral equation, 337, 342 615 – – of the first kind, 11, 15, 86 – – of the second kind, 12, 15 – operator, 242, 563 – theorem, 226 free space Green’s operator, 356 Frenet’s formulae, 554 Friedrichs inequality, 169 fundamental – assumption, 146, 149 – matrix, 505 – solution, 2, 45, 63, 81, 131, 206, 331, 346 – velocity tensor, 64 Gă unter derivative, 116, 117, 536 Gaussian bracket, 101, 174 – curvature, 114 general – exterior – – Dirichlet problem, 211 – – Neumann–Robin problem, 213 – Green representation formula, 130 – interior – – Dirichlet problem, 200 – – Neumann–Robin problem, 201, 203 generalized – first Green’s formula, 294 – Newton potential, 499, 500 – plane stress, 45 – Plemelj–Sochozki jump relations, 474 – polynomials, 211 – representation formula, 266 – second Green’s formula, 261 generalized representation formula, 495 global parametric representation, 181 graph norm, 251 Green’s – formula, 80 – identity, 128 – operator, 331, 494 growth conditions at infinity, 65 G˚ arding’s – inequality, 218, 243, 245, 248, 254, 256, 259, 278, 283, 286, 292, 343, 420, 482, 567 theorem, 244 Hă older continuity, 49 – modulus, 97 – norm, 97 – spaces, 616 Index Hadamard – coordinates, 111, 113 – finite part integral, 103, 104 – – operator, 354, 366 Helmholtz – equation, 354, 502, 574 – operator, 350 Hilbert – space, 160, 170, 174, 210 – – formulation, 209 – transform, 564, 567 homogeneous, 372 – degree, 310 – distribution, 369 – equation, 260 – function, 353 – polynomial, 369 – principal symbol, 319, 326 – symbol, 389, 575 – – expansion, 505 hydrodynamic boundary – integral operator, 292 – traction, 66 hypersingular – boundary – – integral equation, 13 – – potential, 535 – elastic operator, 580 – integral equation of the first kind, 16 – operator, 49, 68, 555, 578 – surface potential, 525 kernel – function, 354 – of the double layer potential, 554 Kieser’s theorem, 524 kinematic viscosity, 62 Korn’s inequality, 247, 248, 251 Kutta–Joukowski condition, 16 incompressible viscous fluid, 62 induced mapping, 415 inner product, 160, 166, 174, 211 insertion problem, 93 interior – boundary value problems, 260 – Dirichlet problem, 86, 200, 590 – displacement problem, 75 – Neumann problem, 201, 253, 591 invariance property, 405, 406 invariant parity conditions, 411 inverse – Fourier transform, 100, 165 – trace theorem, 180 isomorphism, 100, 165 iterated Laplacian, 349 mapping properties of potentials, 268, 282, 497 Maue formula, 578 mean curvature, 114 mixed boundary conditions, 91 modified Calder´ on projector, 150 multi–index, 95 multiple layer potentials, 139, 142 multiplication by ϕ, 169 multiply connected domains, 572 jump relation, 136, 145, 269, 297, 454, 513, 517, 530 – for the derivatives of boundary potentials, 536 Lam´e – constants, 45 – system, 505, 578 Laplace–Beltrami operator, 122 Laplacian, 1, 122 large–time behavior, 31 Lax–Milgram theorem, 195, 219, 223 Levi function, 334, 336, 341, 496, 503 Lipschitz – boundary, 110 – continuous, 97 – norm, 97 local – chart, 414, 421, 467 – fundamental solutions, 346 – operator, 414 – pseudodifferential operator, 416 – spaces, 169 locally – convex topological vector space, 97 Hă older continuous, 97 LopatinskiShapiro condition, 147 Lyapounov boundaries, 19, 110 natural – boundary condition, 201 – trace space, 171 necessary compatibility condition, 141 Neumann – boundary condition, 81 – problem, 66 – series, 335 Newton potential, 45, 500, 502, 505, 506 Index Nikolski’s Theorem, 237 norm, 176, 177 oddly elliptic, 565, 567, 570 one scalar second order equation, 243 order of a symbol, 304 orthogonality conditions, 28, 71, 76, 261, 482 orthonormal matrix, 593 oscillatory integrals, 307 Paley–Wiener–Schwartz theorem, 100, 305, 357 parametric 1–periodic representation, 549 parametrix, 327, 330, 332, 356, 419 parity condition, 389, 390, 399, 404–406, 445, 447 Parseval’s formula, 100 Parseval–Plancherel formula, 165 partial differential equation, 130 partie finie integral, 355, 405 partition of unity, 170, 417 periodic – functions, 181 – pseudodifferential operators, 558 plane strain, 45 Poincar´e inequality, 168 Poisson – equation, – operator, 413, 423, 441, 448, 494 – ratio, 45, 80 polar coordinates, 399 polyhomogeneous, 310 positively homogeneous – function, 304, 380 – principal symbol, 369 positivity, 287 pressure operator, 581 principal – fundamental solutions, 348 – symbol, 316, 320, 416, 417, 574 – – of the acoustic double layer boundary integral operator, 519 – – of the hypersingular boundary integral operator of linear elasticity, 533 – – of the operators Kk and Kk , 518 – – of the simple layer boundary integral operator, 521 product with a distribution, 98 projection theorem, 220 properly supported, 310, 312, 313 617 pseudodifferential operator, 305, 494 – classical, 310, 319, 355, 380 – general representation, 355 – integro–differential operator, 306 – mapping properties, 316, 418 – standard, 304 – transposed, 317 pseudohomogeneous – distribution, 372 – expansion, 354, 558, 577, 586, 587 – function, 101, 359, 372, 375, 381, 389 – kernel, 354, 394 – – expansion, 575 – – under the change of coordinates, 394 pullback, 320, 415 pushforward, 320, 415 radiation condition, 56, 134, 142, 204, 208 rank condition, 248 rapidly decreasing, 99 rational function, 446 regular – diffeomorphism, 112 – elliptic boundary value problem, 147 regularizer, 315 Rellich Lemma, 251 representation – formula, 80, 82, 93, 131, 135, 138, 265 – of pseudodifferential operators, 383 Riemannian metric, 112, 398, 594 Riesz representation theorem, 221 Riesz–Schauder Theorem, 236 right inverse, 180 rigid motion, 253 saddle point problem, 251 scalar differential equation, 119, 348, 510 scalar product, 174, 177 Schwartz – kernel, 305, 335, 356, 381, 383, 420 – space S, 99 second – fundamental form, 114 – Green’s formula, 63, 130, 261 – order system, 148, 195 sectional trace, 466, 478 sesquilinear form, 196, 199, 223, 247, 259 shear force, 81 618 Index shift–theorem, 483 simple layer – hydrodynamic potential, 65 – potential, 3, 81, 508, 514, 550, 574, 578 singular Green’s operator, 495 singular perturbation, 33 skew–symmetric bilinear form, 202 Slobodetskii norm, 161 smoothing – operator, 315, 415 – property, 415 Sobolev – imbedding theorems, 167 – spaces of negative order, 163 Sommerfeld – matrix, 246 – radiation conditions, 26 special parity conditions, 472 Steklov–Poincar´e operator, 288 Stokes system, 61, 62, 329 stream function, 79 stress – operator, 63 – tensor, 63 strong – ellipticity, 219, 512, 556, 564, 569 – Lipschitz domain, 110 strongly elliptic, 326, 343, 419, 480, 513, 515, 565, 570 – second order systems, 510 – system of pseudodifferential operators, 482 supplementary transmission problem, 270 support of a distribution, 98 surface – gradient, 117 – integral, 170 – potential in the half space, 467 symbol, 313, 316, 446, 582 – class, 303 – expansion, 587 – matrix, 330, 524 – of rational type, 446, 447, 475, 478 – of the hypersingular integral operator, 527 symmetric part, 210 system – of pseudodifferential equations, 568, 591 – of periodic integral equations, 572 tangent – bundle, 417 – space, 417 tangential differential operator, 116, 455, 471 tempered distribution, 99, 205 tensor product, 135 thin plate, 80, 81 trace – of Newton potential, 502 – on Γ , 170, 171 – operator, 171, 177, 478, 494 – sectional, 466, 478 – spaces, 169, 171, 178, 180, 181 – – on an open surface, 189 – theorem, 177 traction problem, 55 traditional transformation formula, 394 transformed kernel, 394 transmission – condition, 216, 413, 424, 433, 444, 445, 447, 494 – problems, 215, 264 transposed differential operator, 130 Tricomi conditions, 357, 381, 383, 412, 445, 468 tubular neighbourhood, 421 two–dimensional – Laplacian, 550 – potential flow, 16 uniform cone property, 161 uniformly strongly elliptic, 130, 343 unique – continuation property, 205 – solution, 262, 264 variational formulation, 195, 200, 204, 208, 214, 282, 480 very strongly elliptic, 245 volume potential, 45, 331, 476 vorticity, 80 wave number, 25 weakly singular, 4, 68 weighted Sobolev spaces, 191 Applied Mathematical Sciences (continued from page ii) 60 Ghil/Childress: Topics in Geophysical Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics 61 Sattinger/Weaver: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics 62 LaSalle: The Stability and Control of Discrete Processes 63 Grasman: Asymptotic Methods of Relaxation Oscillations and Applications 64 Hsu: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems 65 Rand/Armbruster: Perturbation Methods, Bifurcation Theory and Computer Algebra 66 Hlavácek/Haslinger/Necasl/Lovísek: Solution of Variational Inequalities in Mechanics 67 Cercignani: The Boltzmann Equation and Its Application 68 Temam: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed 69 Golubitsky/Stewart/Schaeffer: Singularities and Groups in Bifurcation Theory, Vol II 70 Constantin/Foias/Nicolaenko/Temam: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations 71 Catlin: Estimation, Control, and The Discrete Kalman Filter 72 Lochak/Meunier: Multiphase Averaging for Classical Systems 73 Wiggins: Global Bifurcations and Chaos 74 Mawhin/Willem: Critical Point Theory and Hamiltonian Systems 75 Abraham/Marsden/Ratiu: Manifolds, Tensor Analysis, and Applications, 2nd ed 76 Lagerstrom: Matched Asymptotic Expansions: Ideas and Techniques 77 Aldous: Probability Approximations via the Poisson Clumping Heuristic 78 Dacorogna: Direct Methods in the Calculus of Variations 79 Hernández-Lerma: Adaptive Markov Processes 80 Lawden: Elliptic Functions and Applications 81 Bluman/Kumei: Symmetries and Differential Equations 82 Kress: Linear Integral Equations, 2nd ed 83 Bebernes/Eberly: Mathematical Problems from Combustion Theory 84 Joseph: Fluid Dynamics of Viscoelastic Fluids 85 Yang: Wave Packets and Their Bifurcations in Geophysical Fluid Dynamics 86 Dendrinos/Sonis: Chaos and Socio-Spatial Dynamics 87 Weder: Spectral and Scattering Theory for wave Propagation in Perturbed Stratified Media 88 Bogaevski/Povzner: Algebraic Methods in Nonlinear Perturbation Theory 89 O’Malley: Singular Perturbation Methods for Ordinary Differential Equations 90 Meyer/Hall: Introduction to Hamiltonian Dynamical Systems and the N-body Problem 91 Straughan: The Energy Method, Stability, and Nonlinear Convection 92 Naber: The Geometry of Minkowski Spacetime 93 Colton/Kress: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed 94 Hoppensteadt: Analysis and Simulation of Chaotic Systems 95 Hackbusch: Iterative Solution of Large Sparse Systems of Equations 96 Marchioro/Pulvirenti: Mathematical Theory of Incompressible Nonviscous Fluids 97 Lasota/Mackey: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd ed 98 de Boor/Höllig/Riemenschneider: Box Splines 99 Hale/Lunel: Introduction to Functional Differential Equations 100 Sirovich (ed): Trends and Perspectives in Applied Mathematics 101 Nusse/Yorke: Dynamics: Numerical Explorations, 2nd ed 102 Chossat/Iooss: The Couette-Taylor Problem 103 Chorin: Vorticity and Turbulence 104 Farkas: Periodic Motions 105 Wiggins: Normally Hyperbolic Invariant Manifolds in Dynamical Systems 106 Cercignani/Ilner/Pulvirenti: The Mathematical Theory of Dilute Gases 107 Antman: Nonlinear Problems of Elasticity, 2nd ed 108 Zeidler: Applied Functional Analysis: Applications to Mathematical Physics 109 Zeidler: Applied Functional Analysis: Main Principles and Their Applications 110 Diekman/van Gils/Verduyn Lunel/Walther: Delay Equations: Functional-, Complex-, and Nonlinear Analysis 111 Visintin: Differential Models of Hysteresis 112 Kuznetsov: Elements of Applied Bifurcation Theory, 2nd ed 113 Hislop/Sigal: Introduction to Spectral Theory 114 Kevorkian/Cole: Multiple Scale and Singular Perturbation Methods 115 Taylor: Partial Differential Equations I, Basic Theory 116 Taylor: Partial Differential Equations II, Qualitative Studies of Linear Equations (continued on next page) Applied Mathematical Sciences (continued from previous page) 117 Taylor: Partial Differential Equations III, Nonlinear Equations 142 Schmid/Henningson: Stability and Transition in Shear Flows 118 Godlewski/Raviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws 143 Sell/You: Dynamics of Evolutionary Equations 119 Wu: Theory and Applications of Partial Functional Differential Equations 120 Kirsch: An Introduction to the Mathematical Theory of Inverse Problems 121 Brokate/Sprekels: Hysteresis and Phase Transitions 144 Nédélec: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems 145 Newton: The N-Vortex Problem: Analytical Techniques 146 Allaire: Shape Optimization by the Homogenization Method 122 Gliklikh: Global Analysis in Mathematical Physics: Geometric and Stochastic Methods 147 Aubert/Kornprobst: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations 123 Khoi Le/Schmitt: Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems 148 Peyret: Spectral Methods for Incompressible Viscous Flow 124 Polak: Optimization: Algorithms and Consistent Approximations 149 Ikeda/Murota: Imperfect Bifurcation in Structures and Materials 125 Arnold/Khesin: Topological Methods in Hydrodynamics 150 Skorokhod/Hoppensteadt/Salehi: Random Perturbation Methods with Applications in Science and Engineering 126 Hoppensteadt/Izhikevich: Weakly Connected Neural Networks 151 Bensoussan/Frehse: Regularity Results for Nonlinear Elliptic Systems and Applications 127 Isakov: Inverse Problems for Partial Differential Equations, 2nd ed 152 Holden/Risebro: Front Tracking for Hyperbolic Conservation Laws 128 Li/Wiggins: Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations 153 Osher/Fedkiw: Level Set Methods and Dynamic Implicit Surfaces 129 Müller: Analysis of Spherical Symmetries in Euclidean Spaces 154 Bluman/Anco: Symmetries and Integration Methods for Differential Equations 130 Feintuch: Robust Control Theory in Hilbert Space 155 Chalmond: Modeling and Inverse Problems in Image Analysis 131 Ericksen: Introduction to the Thermodynamics of Solids, Revised Edition 156 Kielhöfer: Bifurcation Theory: An Introduction with Applications to PDEs 132 Ihlenburg: Finite Element Analysis of Acoustic Scattering 157 Kaczynski/Mischaikow/Mrozek: Computational Homology 133 Vorovich: Nonlinear Theory of Shallow Shells 158 Oertel: Prandtl’s Essentials of Fluid Mechanics, 10th Revised Edition 134 Vein/Dale: Determinants and Their Applications in Mathematical Physics 135 Drew/Passman: Theory of Multicomponent Fluids 159 Ern/Guermond: Theory and Practice of Finite Elements 136 Cioranescu/Saint Jean Paulin: Homogenization of Reticulated Structures 160 Kaipio/Somersalo: Statistical and Computational Inverse Problems 137 Gurtin: Configurational Forces as Basic Concepts of Continuum Physics 138 Haller: Chaos Near Resonance 139 Sulem/Sulem: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse 140 Cherkaev: Variational Methods for Structural Optimization 141 Naber: Topology, Geometry, and Gauge Fields: Interactions 161 Ting: Viscous Vortical Flows II 162 Ammari/Kang: Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory 163 Bernado/Budd/Champneys/Kowalczyk: Piecewisesmooth Dynamical Systems: Theory and Applications 164 Hsiao/Wendland: Boundary Integral Equations 165 Straughan: Stability and Wave Motion in Porous Media ... Differential Equations 59 Sanders/Verhulst: Averaging Methods in Nonlinear Dynamical Systems (continued following index) G eorge C Hsia o W olfga ng L W endla nd Boundary Integral Eq uations 123 George... 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... – – – – – 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