Introduction: What Are Partial Differential Equations? 1
1. The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order 7
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 Principle 16
2. The Maximum Principle 33
2.1 The Maximum Principle of E.Hopf 33
2.2 The Maximum Principle of Alexandrov and Bakelman 39
2.3 Maximum Principles for Nonlinear Differential Equations 44
3. Existence TechniquesⅠ: Methods Based on the Maximum Principle 53
3.1 Difference Methods: Discretization of Differential Equations 53
3.2 The Perron Method 62
3.3 The Alternating Method of H.A.Schwarz 66
3.4 Boundary Regularity 71
4. Existence Techniques Ⅱ: Parabolic Methods.The Heat Equation 79
4.1 The Heat Equation:Definition and Maximum Principles 79
4.2 The Fundamental Solution of the Heat Equation.The Heat Equation and the Laplace Equation 91
4.3 The Initial Boundary Value Problem for the Heat Equation 98
4.4 Discrete Methods 114
5. Reaction-Diffusion Equations and Systems 119
5.1 Reaction-Diffusion Equations 119
5.2 Reaction-Diffusion Systems 126
5.3 The turing Mechanism 130
6. The Wave Equation and its Connections with the Laplace and Heat Equations 139
6.1 The One-Dimensional Wave Equation 139
6.2 The Mean Value Method: Solving the Wave Equation through the Darboux Equation 143
6.3 The Energy Inequality and the Relation with the Heat Equation 147
7. The Heat Equation, Semigroups, and Brownian Motion 153
7.1 Semigroups 153
7.2 Infinitesimal Generators of Semigroups 155
7.3 Brownian Motion 171
8. The Dirichlet Principle.Variational Methods for the Solu-tion of PDEs （Existence Techniques Ⅲ） 183
8.1 Dirichlet's Principle 183
8.2 The Sobolev Space W1,2 186
8.3 Weak Solutions of the Poisson Equation 196
8.4 Quadratic Variational Problems 198
8.5 Abstract Hilbert Space Formulation of the Variational Prob-lem.The Finite Element Method 201
8.6 Convex Variational Problems 209
9. Sobolev Spaces and L2 Regularity Theory 219
9.1 General Sobolev Spaces.Embedding Theorems of Sobolev,Morrey, and John-Nirenberg 219
9.2 L2-Regularity Theory:Interior Regularity of Weak Solutions of the Poisson Equation 234
9.3 Boundary Regularity and Regularity Results for Solutions of General Linear Elliptic Equations 241
9.4 Extensions of Sobolev Functions and Natural Boundary Con-ditions 249
9.5 Eigenvalues of Elliptic Operators 255
10. Strong Solutions 271
10.1 The Regularity Theory for Strong Solutions 271
10.2 A Survey of the Lp-Regularity Theory and Applications to Solutions of Semilinear Elliptic Equations 276
11. The Regularity Theory of Schauder and the Continuity Method（Existence Techniques Ⅳ） 283
11.1 Cα-Regularity Theory for the Poisson Equation 283
11.2 The Schauder Estimates 293
11.3 Existence Techniques Ⅳ: The Continuity Method 299
12. The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash 305
12.1 The Moser-Harnack Inequality 305
12.2 Properties of Solutions of Elliptic Equations 317
12.3 Regularity of Minimizers of Variational Problems 321
Appendix.Banach and Hilbert Spaces.The Lp-Spaces 339
References 347
Index of Notation 349
Index 353