Strongly Elliptic Systems and Boundary Integral Equations

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Cambridge University Press, Jan 28, 2000 - Mathematics - 357 pages
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Partial differential equations provide mathematical models of many important problems in the physical sciences and engineering. This book treats one class of such equations, concentrating on methods involvingthe use of surface potentials. It provides the first detailed exposition of the mathematical theory of boundary integral equations of the first kind on non-smooth domains. Included are chapters on three specific examples: the Laplace equation, the Helmholtz equation and the equations of linear elasticity.The book is designed to provide an ideal preparation for studying the modern research literature on boundary element methods.
 

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Contents

Introduction
1
Exercises
15
Abstract Linear Equations
17
The Kernel and Image
18
Duality
20
Compactness
27
Fredholm Operators
32
Hilbert Spaces
38
The Third Green Identity
200
Jump Relations and Mapping Properties
202
Duality Relations
211
Exercises
215
Boundary Integral Equations
217
Integral Representations
219
The Dirichlet Problem
226
The Neumann Problem
229

Coercivity
42
Elementary Spectral Theory
45
Exercises
52
Sobolev Spaces
57
Convolution
58
Differentiation
61
Schwartz Distributions
64
Fourier Transforms
69
Sobolev Spaces First Definition
73
Sobolev Spaces Second Definition
75
Equivalence of the Norms
79
Localisation and Changes of Coordinates
83
Density and Imbedding Theorems
85
Lipschitz Domains
89
Sobolev Spaces on the Boundary
96
The Trace Operator
100
VectorValued Functions
106
Exercises
107
Strongly Elliptic Systems
113
Strongly Elliptic Operators
118
Boundary Value Problems
128
Regularity of Solutions
133
The Transmission Property
141
Estimates for the SteklovPoincare Operator
145
Exercises
156
Homogeneous Distributions
158
FinitePart Integrals
159
Extension from ℝⁿ 0 to ℝⁿ
166
Fourier Transforms
169
Change of Variables
174
FinitePart Integrals on Surfaces
181
Exercises
187
Surface Potentials
191
Parametrices
192
Fundamental Solutions
197
Mixed Boundary Conditions
231
Exterior Problems
234
Regularity Theory
239
Exercises
241
The Laplace Equation
246
Fundamental Solutions
247
Spherical Harmonics
250
Behaviour at Infinity
258
Solvability for the Dirichlet Problem
260
Solvability for the Neumann Problem
266
Exercises
268
The Helmholtz Equation
276
Separation of Variables
277
The Sommerfeld Radiation Condition
280
Uniqueness and Existence of Solutions
286
A Boundary Integral Identity
289
Exercises
293
Linear Elasticity
296
Korns Inequality
297
Fundamental Solutions
299
Uniqueness Results
301
Exercises
305
Extension Operators for Sobolev Spaces
309
Exercises
315
Interpolation Spaces
317
The KMethod
318
The JMethod
321
Interpolation of Sobolev Spaces
329
Exercises
333
Further Properties of Spherical Harmonics
334
Exercises
338
References
341
Index
347
Index of Notation
353
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