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偏微分方程 第2版 英文PDF|Epub|txt|kindle电子书版本网盘下载
- （德）约斯特著 著
- 出版社： 世界图书出版公司北京公司
- ISBN：7510032967
- 出版时间：2011
- 标注页数：359页
- 文件大小：9MB
- 文件页数：372页
- 主题词：
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图书目录
Introduction: What Are Partial Differential Equations?1
1. The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order7
1.1 Harmonic Functions.Representation Formula for the Solution of the Dirichlet Problem on the Ball （Existence Techniques 0）7
1.2 Mean Value Properties of Harmonic Functions.Subharmonic Functions.The Maximum Principle16
2. The Maximum Principle33
2.1 The Maximum Principle of E.Hopf33
2.2 The Maximum Principle of Alexandrov and Bakelman39
2.3 Maximum Principles for Nonlinear Differential Equations44
3. Existence TechniquesⅠ: Methods Based on the Maximum Principle53
3.1 Difference Methods: Discretization of Differential Equations53
3.2 The Perron Method62
3.3 The Alternating Method of H.A.Schwarz66
3.4 Boundary Regularity71
4. Existence Techniques Ⅱ: Parabolic Methods.The Heat Equation79
4.1 The Heat Equation:Definition and Maximum Principles79
4.2 The Fundamental Solution of the Heat Equation.The Heat Equation and the Laplace Equation91
4.3 The Initial Boundary Value Problem for the Heat Equation98
4.4 Discrete Methods114
5. Reaction-Diffusion Equations and Systems119
5.1 Reaction-Diffusion Equations119
5.2 Reaction-Diffusion Systems126
5.3 The turing Mechanism130
6. The Wave Equation and its Connections with the Laplace and Heat Equations139
6.1 The One-Dimensional Wave Equation139
6.2 The Mean Value Method: Solving the Wave Equation through the Darboux Equation143
6.3 The Energy Inequality and the Relation with the Heat Equation147
7. The Heat Equation, Semigroups, and Brownian Motion153
7.1 Semigroups153
7.2 Infinitesimal Generators of Semigroups155
7.3 Brownian Motion171
8. The Dirichlet Principle.Variational Methods for the Solu-tion of PDEs （Existence Techniques Ⅲ）183
8.1 Dirichlet's Principle183
8.2 The Sobolev Space W1,2186
8.3 Weak Solutions of the Poisson Equation196
8.4 Quadratic Variational Problems198
8.5 Abstract Hilbert Space Formulation of the Variational Prob-lem.The Finite Element Method201
8.6 Convex Variational Problems209
9. Sobolev Spaces and L2 Regularity Theory219
9.1 General Sobolev Spaces.Embedding Theorems of Sobolev,Morrey, and John-Nirenberg219
9.2 L2-Regularity Theory:Interior Regularity of Weak Solutions of the Poisson Equation234
9.3 Boundary Regularity and Regularity Results for Solutions of General Linear Elliptic Equations241
9.4 Extensions of Sobolev Functions and Natural Boundary Con-ditions249
9.5 Eigenvalues of Elliptic Operators255
10. Strong Solutions271
10.1 The Regularity Theory for Strong Solutions271
10.2 A Survey of the Lp-Regularity Theory and Applications to Solutions of Semilinear Elliptic Equations276
11. The Regularity Theory of Schauder and the Continuity Method（Existence Techniques Ⅳ）283
11.1 Cα-Regularity Theory for the Poisson Equation283
11.2 The Schauder Estimates293
11.3 Existence Techniques Ⅳ: The Continuity Method299
12. The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash305
12.1 The Moser-Harnack Inequality305
12.2 Properties of Solutions of Elliptic Equations317
12.3 Regularity of Minimizers of Variational Problems321
Appendix.Banach and Hilbert Spaces.The Lp-Spaces339
References347
Index of Notation349
Index353