Linear and semilinear partial differential equations: an introduction
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
De Gruyter
2013
|
| Schriftenreihe: | de Gruyter Textbook
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XIV, 280 S. 240 mm x 170 mm |
| ISBN: | 311026904X 9783110269048 |
Internformat
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Datensatz im Suchindex
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| adam_text |
Titel: Linear and semilinear partial differential equations
Autor: Precup, Radu
Jahr: 2013
Contents
Preface vii
Notation ix
I Classical Theory
1 Preliminaries 3
1.1 Basic Differential Operators . 3
1.2 Linear and Quasilinear Partial Differential Equations. 5
1.3 Solutions of Some Particular Equations . 8
1.4 Boundary Välue Problems. 10
1.4.1 Boundary Value Problems for Poisson's Equation. 10
1.4.2 Boundary Value Problems for the Heat Equation. 11
1.4.3 Boundary Value Problems for the Wave Equation . 12
2 Partial Differential Equations and Mathematical Modeling 13
2.1 Conservation Laws: Continuity Equations . 13
2.2 Reaction-Diffusion Systems . 16
2.3 The One-Dimensional Wave Equation . 17
2.4 Other Equations in Mathematical Physics. 18
3 Elliptic Boundary Value Problems 21
3.1 Green's Formulas. 21
3.2 The Fundamental Solution of Laplace's Equation. 22
3.3 Mean Value Theorems for Harmonic Functions. 25
3.4 The Maximum Principie . 26
3.5 Uniqueness and Continuous Dependence on Data for the Dirichlet
Problem. 29
3.6 Green's Function ofthe Dirichlet Problem. 30
3.7 Poisson's Formula. 31
3.8 Dirichlet's Principie. 34
xii Contents
3.9 The Generalized Solution of the Dirichlet Problem . 37
3.10 Abstract Fourier Series . 42
3.11 The Eigenvalues and Eigenfunctions of the Dirichlet Problem . 45
3.12 The Case of Elliptic Equations in Divergence Form. 50
3.13 The Generalized Solution ofthe Neumann Problem. 51
3.14 Complements. 55
3.14.1 Harnack's Inequality. 55
3.14.2 Hopfs Maximum Principie. 57
3.14.3 The Newtonian Potential. 59
3.14.4 Perron's Method . 62
3.14.5 Layer Potentials. 68
3.14.6 Fredholm's Method of Integral Equations . 70
3.15 Problems . 71
4 Mixed Problems for Evolution Equations 87
4.1 The Maximum Principie for the Heat Equation . 87
4.2 Vector-Valued Functions. 90
4.3 The Cauchy-Dirichlet Problem for the Heat Equation. 91
4.4 The Cauchy-Dirichlet Problem for the Wave Equation . 99
4.5 Problems . 102
5 The Cauchy Problem for Evolution Equations 109
5.1 The Fourier Transform . 109
5.1.1 The Fourier Transform onL^R"). 109
5.1.2 Fourier Transform and Convolution . 110
5.1.3 The Fourier Transform on the Schwartz Space S (R") . 112
5.2 The Cauchy Problem for the Heat Equation . 116
5.3 The Cauchy Problem for the Wave Equation . 119
5.4 Nonhomogeneous Equations: Duhamel's Principie . 123
5.5 Problems . 125
II Modern Theory
6 Distributions 131
6.1 The Fundamental Spaces of the Theory of Distributions . 131
6.2 Distributions: Examples; Operations with Distributions. 133
Contents xiii
6.2.1 Regular Distributions . 133
6.2.2 The Dirac Distribution . 134
6.2.3 Differentiation. 134
6.2.4 Multiplication by a Smooth Function. 136
6.2.5 Composition with a Smooth Function. 137
6.2.6 Convolution. 137
6.2.7 Distributions of Compact Support. 139
6.2.8 Weyl's Lemma. 142
6.3 The Fourier Transform of Tempered Distributions. 142
6.3.1 The Fourier Transform on S' (BP) . 143
6.3.2 The Fourier Transform on L2 (R"). 144
6.3.3 Convolution in $' . 144
6.4 Problems . 145
7 Sobolev Spaces 149
7.1 The Sobolev Spaces Hm (ß). 149
7.2 The Extension Operator. 152
7.3 The Sobolev Spaces H™ (ß). 156
7.4 Sobolev's Continuous Embedding Theorem. 159
7.5 Rellich-Kondrachov's Compact Embedding Theorem. 163
7.6 The Embedding of Hm (ß) into C (ß). 165
7.7 The Sobolev Space H~m (ß). 167
7.8 Fourier Series in H~l (ß) . 172
7.9 Generalized Solutions ofthe Cauchy Problems . 175
8 The Variational Theory of Elliptic Boundary Value Problems 180
8.1 The Variational Method for the Dirichlet Problem. 180
8.2 The Variational Method for the Neumann Problem . 184
8.3 Maximum Principies for Weak Solutions . 186
8.4 Regularity of Weak Solutions . 191
8.5 Regularity of Eigenfunctions. 198
8.6 Problems . 201
m Semilinear Equations
9 Semilinear Elliptic Problems 208
9.1 The Nemytskii Superposition Operator. 208
xiv Contents
9.2 Application of Banach's Fixed Point Theorem. 211
9.3 Application of Schauder's Fixed Point Theorem. 213
9.4 Application ofthe Leray-Schauder Fixed Point Theorem . 215
9.5 The Monotone Iterative Method . 218
9.6 The Critical Point Method. 220
9.7 Problems . 225
10 The Semilinear Heat Equation 227
10.1 The Nonhomogeneous Heat Equation in H~l (ß). 227
10.2 Regularity Results. 233
10.3 Application of Banach's Fixed Point Theorem. 238
10.4 Application of Schauder's Fixed Point Theorem. 241
10.5 Application ofthe Leray-Schauder Fixed Point Theorem . 245
11 The Semilinear Wave Equation 248
11.1 The Nonhomogeneous Wave Equation in H~* (ß). 248
11.2 Application of Banach's Fixed Point Theorem. 252
11.3 Application of the Leray-Schauder Fixed Point Theorem. 257
12 Semilinear Schrödinger Equations 262
12.1 The Nonhomogeneous Schrödinger Equation. 262
12.2 Properties of the Schrödinger Solution Operator. 266
12.3 Applications of Banach's Fixed Point Theorem . 268
12.4 Applications of Schauder's Fixed Point Theorem. 272
Bibliography 275
Index 278 |
| any_adam_object | 1 |
| author | Precup, Radu 1955- |
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| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
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| title | Linear and semilinear partial differential equations an introduction |
| title_auth | Linear and semilinear partial differential equations an introduction |
| title_exact_search | Linear and semilinear partial differential equations an introduction |
| title_full | Linear and semilinear partial differential equations an introduction Radu Precup |
| title_fullStr | Linear and semilinear partial differential equations an introduction Radu Precup |
| title_full_unstemmed | Linear and semilinear partial differential equations an introduction Radu Precup |
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| title_sort | linear and semilinear partial differential equations an introduction |
| title_sub | an introduction |
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