Boundary Integral Equations (eBook)

George C. Hsiao, Wolfgang L. Wendland (Autoren)

eBook Download: PDF

2008 | 2008
XIX, 620 Seiten
Springer Berlin (Verlag)
978-3-540-68545-6 (ISBN)

Infos

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.

Inhalt

Preface 7
Table of Contents 13
1. Introduction 18
1.1 The Green Representation Formula 18
1.2 Boundary Potentials and Calder´on’s Projector 20
1.3 Boundary Integral Equations 27
1.4 Exterior Problems 30
1.5 Remarks 36
2. Boundary Integral Equations 41
2.1 The Helmholtz Equation 41
2.2 The Lam´e System 61
2.3 The Stokes Equations 78
2.4 The Biharmonic Equation 95
2.5 Remarks 107
3. Representation Formulae, Local Coordinates and Direct Boundary Integral Equations 111
3.1 Classical Function Spaces and Distributions 111
3.2 Hadamard’s Finite Part Integrals 117
3.3 Local Coordinates 124
3.4 Short Excursion to Elementary Differential Geometry 127
3.5 Distributional Derivatives and Abstract Green’s Second Formula 142
3.6 The Green Representation Formula 146
3.7 Green’s Representation Formulae in Local Coordinates 151
3.8 Multilayer Potentials 155
3.9 Direct Boundary Integral Equations 161
3.10 Remarks 173
4. Sobolev Spaces 175
4.1 The Spaces Hs(O) 175
4.2 The Trace Spaces Hs(G) 185
4.3 The Trace Spaces on an Open Surface 205
4.4 The Weighted Sobolev Spaces Hm(Oc .)and Hm(IRn .)
5.1 Partial Differential Equations of Second Order 211
5.2 Abstract Existence Theorems for Variational Problems 234
5.3 The Fredholm–Nikolski Theorems 242
5.4 G°arding’s Inequality for Boundary Value Problems 259
5.5 Existence of Solutions to Strongly Elliptic Boundary Value Problems 275
5.6 Solutions of Certain Boundary Integral Equations and Associated Boundary Value Problems 281
5.7 Partial Differential Equations of Higher Order 309
5.8 Remarks 315
6. Introduction to Pseudodifferential Operators 319
6.1 Basic Theory of Pseudodifferential Operators 319
6.2 Elliptic Pseudodifferential Operators on O IRn 342
6.3 Review on Fundamental Solutions 362
7. Pseudodifferential Operators as Integral Operators 369
7.1 Pseudohomogeneous Kernels 369
7.2 Coordinate Changes and Pseudohomogeneous Kernels 410
8. Pseudodifferential and Boundary Integral Operators 429
8.1 Pseudodifferential Operators on Boundary Manifolds 430
8.2 Boundary Operators Generated by Domain Pseudodifferential Operators 437
8.3 Surface Potentials on the Plane IRn 1 439
8.4 Pseudodifferential Operators with Symbols of Rational Type 462
8.5 Surface Potentials on the Boundary Manifold G 483
8.6 Volume Potentials 492
8.7 Strong Ellipticity and Fredholm Properties 495
8.8 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations 501
8.9 Remarks 507
9. Integral Equations on G IR3 Recast as Pseudodifferential Equations 509
9.1 Newton Potential Operators for Elliptic Partial Differential Equations and Systems 515
9.2 Surface Potentials for Second Order Equations 523
9.3 Invariance of Boundary Pseudodifferential Operators 540
9.4 Derivatives of Boundary Potentials 551
9.5 Remarks 563
10. Boundary Integral Equations on Curves in IR2 564
10.1 Representation of the basic operators for the 2D– Laplacian in terms of Fourier series 565
10.2 The Fourier Series Representation of Periodic Operators A Lmc(G) 571
10.3 Ellipticity Conditions for Periodic Operators on G 577
10.4 Fourier Series Representation of some Particular Operators 589
10.5 Remarks 606
A. Differential Operators in Local Coordinates with Minimal Differentiability 608
References 614
Index 628

Produkt-Details

• Erscheint lt. Verlag: 7.5.2008
• Reihe/Serie: Applied Mathematical Sciences
• Reihe/Serie: Applied Mathematical Sciences
• Zusatzinfo: XIX, 620 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Boundary integral and pseudodifferential operators • differential equation • first kind and second kind integral equations • fluid mechanics • Fredholm alternative • fundamental solutions • Garding' s inequality • Green' s formulae • integral equation • Mechanics • Operator • Partial differential equations • pseudo-differential operators • scattering • Sobolev spaces • strong ellipticity • variational methods • weak solutions
• ISBN-10: 3-540-68545-6 / 3540685456
• ISBN-13: 978-3-540-68545-6 / 9783540685456