It is the main aim of this book to develop at an accessible, moderate level an $L_2$ theory for elliptic differential operators of second order on bounded smooth domains in Euclidean n-space, including a priori estimates for boundary-value problems in terms of (fractional) Sobolev spaces on domains and on their boundaries, together with a related spectral theory. The presentation is preceded by an introduction to the classical theory for the Laplace-Poisson equation, and some chapters provide required ingredients such as the theory of distributions, Sobolev spaces and the spectral theory in Hilbert spaces. The book grew out of two-semester courses the authors have given several times over a period of ten years at the Friedrich Schiller University of Jena. It is addressed to graduate students and mathematicians who have a working knowledge of calculus, measure theory and the basic elements of functional analysis (as usually covered by undergraduate courses) and who are seeking an accessible introduction to some aspects of the theory of function spaces and its applications to elliptic equations. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Dorothee D. Haroske; Hans Triebel
Publisher: European Mathematical Society
Year: 2007
Language: English
Pages: 305
Preface......Page 6
Contents......Page 8
Introduction, basic definitions, and plan of the book......Page 12
Fundamental solutions and integral representations......Page 14
Green's functions......Page 17
Harmonic functions......Page 21
The Dirichlet problem......Page 28
The Poisson equation......Page 31
Notes......Page 34
The spaces D(Omega) and D'(Omega)......Page 36
Regular distributions, further examples......Page 38
Derivatives and multiplications with smooth functions......Page 42
Localisations, the spaces E'(Omega)......Page 44
The space S(R^n), the Fourier transform......Page 49
The space S'(R^n)......Page 54
The Fourier transform in S'(R^n)......Page 58
The Fourier transform in L_p(R^n)......Page 60
Notes......Page 64
The spaces Wk_p(Rn)......Page 67
The spaces Hs(Rn)......Page 70
Embeddings......Page 81
Extensions......Page 83
Traces......Page 90
Notes......Page 93
Basic definitions......Page 98
Extensions and intrinsic norms......Page 99
Odd and even extensions......Page 105
Periodic representations and compact embeddings......Page 108
Traces......Page 114
Notes......Page 123
Boundary value problems......Page 128
Outline of the programme, and some basic ideas......Page 131
A priori estimates......Page 133
Some properties of Sobolev spaces on R^n_+......Page 141
The Laplacian......Page 146
Homogeneous boundary value problems......Page 156
Inhomogeneous boundary value problems......Page 162
Smoothness theory......Page 165
The classical theory......Page 171
Green's functions and Sobolev embeddings......Page 175
Degenerate elliptic operators......Page 179
Notes......Page 181
Introduction and examples......Page 193
Spectral theory of self-adjoint operators......Page 198
Approximation numbers and entropy numbers: definition and basic properties......Page 201
Approximation numbers and entropy numbers: spectral assertions......Page 205
The negative spectrum......Page 211
Associated eigenelements......Page 216
Notes......Page 217
Introduction......Page 225
Compact embeddings: sequence spaces......Page 226
Compact embeddings: function spaces......Page 232
Spectral theory of elliptic operators: the self-adjoint case......Page 238
Spectral theory of elliptic operators: the regular case......Page 240
Spectral theory of elliptic operators: the degenerate case......Page 243
The negative spectrum......Page 245
Notes......Page 249
Basic notation and basic spaces......Page 256
Domains......Page 257
Integral formulae......Page 258
Surface area......Page 259
B Orthonormal bases of trigonometric functions......Page 261
Operators in Banach spaces......Page 263
Symmetric and self-adjoint operators in Hilbert spaces......Page 265
Semi-bounded and positive-definite operators in Hilbert spaces......Page 267
D Some integral inequalities......Page 270
Definitions, basic properties......Page 272
Special cases, equivalent norms......Page 275
Selected solutions......Page 280
Bibliography......Page 286
Author index......Page 294
List of figures......Page 296
Notation index......Page 298
Subject index......Page 302
