目錄
Introduction: What Are Partial Differential Equations?
1.The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order
1.1 Harmonic functions. Representation Formula for the Solution of the Dirichlet Problem on the Ball (Existence Techniques 0)
1.2 Mean Value Properties of Harmonic Functions. Subharmonic Functions. The Maximum Principle
2.The Maximum Principle
2.1 The Maximum Principle of E.Hopf
2.2 The Maximum Principle of Alexandrov and Bakelman
2.3 Maximum Principles for Nonlinear Differential Equations
3.Existence Techniques Ⅰ: Methods Based on the Maximum Principle
3.1 Difference Methods: Discretization of Differential Equations
3.2 The Perron Method
3.3 The Alternating Method of H.A.Schwarz
3.4 Boundary Regularity
4.Existence Techniques Ⅱ: Parabolic Methods. The Heat Equation
4.1 The Heat Equation: efinition and Maximum Principles
4.2 The Fundamental Solution of the Heat Equation. The Heat Equation and the Laplace Equation
4.3 The Initial Boundary Value Problem for the Heat Equation
4.4 Discrete Methods
5.Reaction-Diffusion Equations and Systems
5.1 Reaction-Diffusion Equations
5.2 Reaction-Diffusion Systems
5.3 The Turing Mechanism
6.The Wave Equation and its Connections with the Laplace and Heat Equations
6.1 The One-Dimensional Wave Equation
6.2 The Mean Value Method: Solving the Wave Equation Through the Darboux Equation
6.3 The Energy Inequality and the Relation with the Heat Equation
7.The Heat Equation, Semigroups, and Brownian Motion
7.1 Semigroups
7.2 Infinitesimal Generators of Semigroups
7.3 Brownian Motion
8.The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques Ⅲ)
8.1 Dirichlet's Principle
8.2 The Sobolev Space W1,2
8.3 Weak Solutions of the Poisson Equation
8.4 Quadratic Variational Problems
8.5 Abstract Hilbert Space Formulation of the Variational Problem. The Finite Element Method
8.6 Convex Variational Problems
9.Sobolev Spaces and L2 Regularity Theory
9.1 General Sobolev Spaces. Embedding Theorems of Sobolev, Morrey, and John-nirenberg
9.2 L2-Regularity Theory: Interior Regularity of Weak Solutions of the Poisson Equation
9.3 Boundary Regularity and Regularity Results for Solutions of General Linear Elliptic Equations
9.4 Extensions of Sobolev Functions and Natural Boundary Conditions
9.5 Eigenvalues of Elliptic Operators
10.Strong Solutions
10.1 The Regularity Theory for Strong Solutions
10.2 A Survey of the Lp-Regularity Theory and Applications to Solutions of Semilinear Elliptic Equations
11.The Regularity Theory of Schauder and the Continuity Method (Existence Techniques Ⅳ)
11.1 Cα-Regularity Theory for the Poisson Equation
11.2 The Schauder Estimates
11.3 Existence Techniques Ⅳ: The Continuity Method
12.The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash
12.1 The Moser-Harnack Inequality
12.2 Properties of Solutions of Elliptic Equations
12.3 Regularity of Minimizers of Variational Problems
Appendix.Banach and Hilbert Spaces. The Lp-Spaces
References
Index of Notation
Index
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