Boundary Integral Equations

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Springer Science & Business Media, May 7, 2008 - Mathematics - 620 pages
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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.
 

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Contents

58 Remarks
299
583 Mixed Boundary Conditions and Crack Problem
300
6 Introduction to Pseudodifferential Operators
303
62 Elliptic Pseudodifferential Operators on 𝛺 C IR
326
621 Systems of Pseudodifferential Operators
328
622 Parametrix and Fundamental Solution
331
623 Levi Functions for Scalar Elliptic Equations
334
624 Levi Functions for Elliptic Systems
341

Boundary Integral Equations
25
211 Low Frequency Behaviour
31
22 The Lamé System
45
221 The Interior Displacement Problem
47
222 The Interior Traction Problem
55
223 Some Exterior Fundamental Problems
56
224 The Incompressible Material
61
23 The Stokes Equations
62
231 Hydrodynamic Potentials
65
232 The Stokes Boundary Value Problems
66
233 The Incompressible Material Revisited
75
24 The Biharmonic Equation
79
241 Calderóns Projector
83
242 Boundary Value Problems and Boundary Integral Equations
85
25 Remarks
91
3 Representation Formulae
95
32 Hadamards Finite Part Integrals
101
33 Local Coordinates
108
34 Short Excursion to Elementary Differential Geometry
111
341 Second Order Differential Operators in Divergence Form
119
35 Distributional Derivatives and Abstract Greens Second Formula
126
36 The Green Representation Formula
130
37 Greens Representation Formulae in Local Coordinates
135
38 Multilayer Potentials
139
39 Direct Boundary Integral Equations
145
392 Transmission Problems
155
310 Remarks
157
4 Sobolev Spaces
159
42 The Trace Spaces HГ
169
421 Trace Spaces for Periodic Functions on a Smooth Curve in IR2
181
422 Trace Spaces on Curved Polygons in IR2
185
43 The Trace Spaces on an Open Surface
189
44 Weighted Sobolev Spaces
191
5 Variational Formulations
195
511 Interior Problems
199
512 Exterior Problems
204
513 Transmission Problems
215
52 Abstract Existence Theorems for Variational Problems
218
521 The LaxMilgram Theorem
219
53 The FredholmNikolski Theorems
226
532 The RieszSchauder and the Nikolski Theorems
235
533 Fredholms Alternative for Sesquilinear Forms
240
534 Fredholm Operators
241
54 Gardings Inequality for Boundary Value Problems
243
542 The Stokes System
250
543 Gardings Inequality for Exterior Second Order Problems
254
544 Gardings Inequality for Second Order Transmission Problems
259
551 Interior Boundary Value Problems
260
552 Exterior Boundary Value Problems
264
56 Solutions of Integral Equations via Boundary Value Problems
265
562 Continuity of Some Boundary Integral Operators
267
563 Continuity Based on Finite Regions
270
564 Continuity of Hydrodynamic Potentials
272
565 The Equivalence Between Boundary Value Problems and Integral Equations
274
566 Variational Formulation of Direct Boundary Integral Equations
277
567 Positivity and Contraction of Boundary Integral Operators
287
568 The Solvability of Direct Boundary Integral Equations
291
569 Positivity of the Boundary Integral Operators of the Stokes System
292
57 Partial Differential Equations of Higher Order
293
625 Strong Ellipticity and Gardings Inequality
343
63 Review on Fundamental Solutions
346
631 Local Fundamental Solutions
347
632 Fundamental Solutions in IR for Operators with Constant Coefficients
348
633 Existing Fundamental Solutions in Applications
352
7 Pseudodifferential Operators as Integral Operators
353
711 Integral Operators as Pseudodifferential Operators of Negative Order
356
712 NonNegative Order Pseudodifferential Operators as Hadamard Finite Part Integral Operators
380
713 Parity Conditions
389
714 A Summary of the Relations between Kernels and Symbols
392
72 Coordinate Changes and Pseudohomogeneous Kernels
394
721 The Transformation of General Hadamard Finite Part Integral Operators under Change of Coordinates
397
722 The Class of Invariant Hadamard Finite Part Integral Operators under Change of Coordinates
404
8 Pseudodifferential and Boundary Integral Operators
413
81 Pseudodifferential Operators on Boundary Manifolds
414
811 Ellipticity on Boundary Manifolds
418
812 Schwartz Kernels on Boundary Manifolds
420
82 Boundary Operators Generated by Domain Pseudodifferential Operators
421
83 Surface Potentials on the Plane IR
423
84 Pseudodifferential Operators with Symbols of Rational Type
446
85 Surface Potentials on the Boundary Manifold Г
467
86 Volume Potentials
476
87 Strong Ellipticity and Fredholm Properties
479
88 Strong Ellipticity of Boundary Value Problems and Associated Boundary Integral Equations
485
882 The Associated Boundary Integral Equations of the First Kind
488
883 The Transmission Problem and Gardings inequality
489
89 Remarks
491
9 Integral Equations on Γ IR3 Recast as Pseudodifferential Equations
493
91 Newton Potential Operators for Elliptic Partial Differential Equations and Systems
499
911 Generalized Newton Potentials for the Helmholtz Equation
502
912 The Newton Potential for the Lamé System
505
913 The Newton Potential for the Stokes System
506
92 Surface Potentials for Second Order Equations
507
921 Strongly Elliptic Differential Equations
510
922 Surface Potentials for the Helmholtz Equation
514
923 Surface Potentials for the Lamé System
519
924 Surface Potentials for the Stokes System
524
931 The Hypersingular Boundary Integral Operators for the Helmholtz Equation
525
932 The Hypersingular Operator for the Lamé System
531
933 The Hypersingular Operator for the Stokes System
535
941 Derivatives of the Solution to the Helmholtz Equation
541
942 Computation of Stress and Strain on the Boundary for the Lamé System
543
95 Remarks
547
10 Boundary Integral Equations on Curves in IR˛
548
101 Fourier Series Representation of the Basic Operators
550
102 The Fourier Series Representation of Periodic Operators
556
103 Ellipticity Conditions for Periodic Operators on Г
562
1031 Scalar Equations
563
1032 Systems of Equations
568
1033 Multiply Connected Domains
572
104 Fourier Series Representation of some Particular Operators
574
1042 The Lamé System
578
1043 The Stokes System
581
1044 The Biharmonic Equation
582
105 Remarks
591
A Differential Operators in Local Coordinates with Minimal Differentiability
593
References
599
Index
613
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