Elements of Nonlinear AnalysisSpringer Science & Business Media, 01.11.2000 - 256 Seiten The goal of this book is to present some modern aspects of nonlinear analysis. Some of the material introduced is classical, some more exotic. We have tried to emphasize simple cases and ideas more than complicated refinements. Also, as far as possible, we present proofs that are not classical or not available in the usual literature. Of course, only a small part of nonlinear analysis is covered. Our hope is that the reader - with the help of these notes - can rapidly access the many different aspects of the field. We start by introducing two physical issues: elasticity and diffusion. The pre sentation here is original and self contained, and helps to motivate all the rest of the book. Then we turn to some theoretical material in analysis that will be needed throughout (Chapter 2). The next six chapters are devoted to various aspects of elliptic problems. Starting with the basics of the linear theory, we introduce a first type of nonlinear problem that has today invaded the whole mathematical world: variational inequalities. In particular, in Chapter 6, we introduce a simple theory of regularity for nonlocal variational inequalities. We also attack the question of the existence, uniqueness and approximation of solutions of quasilinear and mono tone problems (see Chapters 5, 7, 8). The material needed to read these parts is contained in Chapter 2. The arguments are explained using the simplest possible examples. |
Inhalt
Some Physical Motivations | 1 |
12 A problem in biology | 9 |
13 Exercises | 14 |
A Short Background in Functional Analysis | 15 |
22 Integration on boundaries | 18 |
23 Introduction to Sobolev spaces | 21 |
24 Exercises | 37 |
Elliptic Linear Problems | 39 |
85 Approximation of nonlinear problems | 122 |
86 Exercises | 129 |
Minimizers | 131 |
92 The direct method | 133 |
93 Applications | 135 |
94 The Euler Equation | 142 |
95 Exercises | 143 |
Minimizing Sequences | 145 |
32 The LaxMilgram theorem and its applications | 41 |
33 Exercises | 48 |
Elliptic Variational Inequalities | 49 |
42 Some applications | 52 |
43 Exercises | 58 |
Nonlinear Elliptic Problems | 59 |
52 A monotonicity method | 62 |
53 A generalization of variational inequalities | 66 |
54 Some multivalued problems | 71 |
55 Exercises | 82 |
A Regularity Theory for Nonlocal Variational Inequalities | 85 |
62 Applications to second order variational inequalities | 91 |
63 Exercises | 94 |
Uniqueness and Nonuniqueness Issues | 95 |
72 Nonuniqueness issues | 99 |
73 Exercises | 102 |
Finite Element Methods for Elliptic Problems | 105 |
82 Some simple finite elements | 106 |
83 Interpolation error | 113 |
84 Convergence results | 116 |
102 Young measures | 147 |
103 Construction of the minimizing sequences | 150 |
104 A more elaborate issue | 154 |
105 Numerical analysis of oscillations | 166 |
106 Exercises | 182 |
Linear Parabolic Equations | 185 |
113 The resolution of parabolic problems | 191 |
114 Applications | 198 |
115 Exercises | 205 |
Nonlinear Parabolic Problems | 207 |
122 Nonlocal problems | 215 |
123 Exercises | 220 |
Asymptotic Analysis | 221 |
132 The case of several stationary points | 222 |
133 A nonlinear case | 223 |
134 Blowup | 241 |
135 Exercises | 248 |
251 | |
255 | |
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Häufige Begriffe und Wortgruppen
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Verweise auf dieses Buch
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