Partial Differential Equations

Chapter 2 The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order

Chapter 3 The Maximum Principle

Chapter 4 Existence Techniques I: Methods Based on the Maximum Principle

Chapter 5 Existence Techniques II: Parabolic Methods. The Heat Equation

Chapter 6 The Wave Equation and Its Connections with the Laplace and Heat Equations

Chapter 7 The Heat Equation, Semigroups, and Brownian Motion

Chapter 8 The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III)

Chapter 9 Sobolev Spaces and L 2 Regularity Theory

Chapter 10 Strong Solutions

Chapter 11 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV)

Chapter 12 The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash