Introduction: What Are Partial Differential Equations?. - The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order. - The Maximum Principle. - Existence Techniques I: Methods Based on the Maximum Principle. - Existence Techniques II: Parabolic Methods. The Heat Equation. - Reaction-Diffusion Equations and Systems. - The Wave Equation and its Connections with the Laplace and Heat Equations. - The Heat Equation Semigroups and Brownian Motion. - The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III). - Sobolev Spaces and L2 Regularity Theory. - Strong Solutions. - The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV). - The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash. Language: English