Main Data
Author: George Hsiao, Wolfgang L. Wendland
Title: Boundary Integral Equations
Publisher: Springer-Verlag
ISBN/ISSN: 9783540685456
Edition: 1
Price: CHF 123.20
Publication date: 01/01/2008
Category: Wirtschaft/Management
Language: English
Technical Data
Pages: 620
Kopierschutz: DRM
Geräte: PC/MAC/eReader/Tablet
Formate: PDF
Table of contents
This book is devoted to the mathematical foundation of boundary integral equations. The combination of ?nite element analysis on the boundary with these equations has led to very e?cient computational tools, the boundary element methods (see e.g., the authors [139] and Schanz and Steinbach (eds.) [267]). Although we do not deal with the boundary element discretizations in this book, the material presented here gives the mathematical foundation of these methods. In order to avoid over generalization we have con?ned ourselves to the treatment of elliptic boundary value problems. The central idea of eliminating the ?eld equations in the domain and - ducing boundary value problems to equivalent equations only on the bou- ary 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.
Table of contents
Table of Contents13
1. Introduction18
1.1 The Green Representation Formula18
1.2 Boundary Potentials and Calder´ons Projector20
1.3 Boundary Integral Equations27
1.4 Exterior Problems30
1.5 Remarks36
2. Boundary Integral Equations41
2.1 The Helmholtz Equation41
2.2 The Lam´e System61
2.3 The Stokes Equations78
2.4 The Biharmonic Equation95
2.5 Remarks107
3. Representation Formulae, Local Coordinates and Direct Boundary Integral Equations111
3.1 Classical Function Spaces and Distributions111
3.2 Hadamards Finite Part Integrals117
3.3 Local Coordinates124
3.4 Short Excursion to Elementary Differential Geometry127
3.5 Distributional Derivatives and Abstract Greens Second Formula142
3.6 The Green Representation Formula146
3.7 Greens Representation Formulae in Local Coordinates151
3.8 Multilayer Potentials155
3.9 Direct Boundary Integral Equations161
3.10 Remarks173
4. Sobolev Spaces175
4.1 The Spaces Hs(O)175
4.2 The Trace Spaces Hs(G)185
4.3 The Trace Spaces on an Open Surface205
4.4 The Weighted Sobolev Spaces Hm(Oc .)and Hm(IRn .)
5.1 Partial Differential Equations of Second Order211
5.2 Abstract Existence Theorems for Variational Problems234
5.3 The FredholmNikolski Theorems242
5.4 G°ardings Inequality for Boundary Value Problems259
5.5 Existence of Solutions to Strongly Elliptic Boundary Value Problems275
5.6 Solutions of Certain Boundary Integral Equations and Associated Boundary Value Problems281
5.7 Partial Differential Equations of Higher Order309
5.8 Remarks315
6. Introduction to Pseudodifferential Operators319
6.1 Basic Theory of Pseudodifferential Operators319
6.2 Elliptic Pseudodifferential Operators on O IRn342
6.3 Review on Fundamental Solutions362
7. Pseudodifferential Operators as Integral Operators369
7.1 Pseudohomogeneous Kernels369
7.2 Coordinate Changes and Pseudohomogeneous Kernels410
8. Pseudodifferential and Boundary Integral Operators429
8.1 Pseudodifferential Operators on Boundary Manifolds430
8.2 Boundary Operators Generated by Domain Pseudodifferential Operators437
8.3 Surface Potentials on the Plane IRn 1439
8.4 Pseudodifferential Operators with Symbols of Rational Type462
8.5 Surface Potentials on the Boundary Manifold G483
8.6 Volume Potentials492
8.7 Strong Ellipticity and Fredholm Properties495
8.8 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations501
8.9 Remarks507
9. Integral Equations on G IR3 Recast as Pseudodifferential Equations509
9.1 Newton Potential Operators for Elliptic Partial Differential Equations and Systems515
9.2 Surface Potentials for Second Order Equations523
9.3 Invariance of Boundary Pseudodifferential Operators540
9.4 Derivatives of Boundary Potentials551
9.5 Remarks563
10. Boundary Integral Equations on Curves in IR2564
10.1 Representation of the basic operators for the 2D Laplacian in terms of Fourier series565
10.2 The Fourier Series Representation of Periodic Operators A Lmc(G)571
10.3 Ellipticity Conditions for Periodic Operators on G577
10.4 Fourier Series Representation of some Particular Operators589
10.5 Remarks606
A. Differential Operators in Local Coordinates with Minimal Differentiability608