Strongly Elliptic Systems and Boundary Integral EquationsCambridge University Press, 28 gen 2000 - 357 pagine 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. |
Sommario
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 |
| 341 | |
| 347 | |
| 353 | |
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Strongly Elliptic Systems and Boundary Integral Equations William McLean Anteprima non disponibile - 2000 |
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ó assume B₁ B₁u Banach space boundary integral equation boundary value problem bounded linear operator closed subspace coefficients coercive Comp compact support conormal derivative constant converges Corollary cutoff function Deduce define denote dense differential operator Dirichlet problem dist(u distribution double-layer potential eigenvalue Exercise exists finite formula Fourier transform Fredholm alternative fundamental solution grad Helmholtz equation Hence Hilbert space homogeneous of degree implies inequality inner product integral equation K₁ kernel Lemma Lipschitz domain mapping property Neumann problem normed space orthogonal partition of unity positive and bounded prove result satisfies self-adjoint sequence sesquilinear form Show Sobolev spaces spherical harmonics strongly elliptic supp Suppose third Green identity u₁ unique vector W₁ X₁ zero ди