Preface. - Introduction: What are Partial Differential Equations?. - 1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order. - 2 The Maximum Principle. - 3 Existence Techniques I: Methods Based on the Maximum Principle. - 4 Existence Techniques II: Parabolic Methods. The Heat Equation. - 5 Reaction-Diffusion Equations and Systems. - 6 Hyperbolic Equations. - 7 The Heat Equation Semigroups and Brownian Motion. - 8 Relationships between Different Partial Differential Equations. - 9 The Dirichlet Principle. Variational Methods for the Solutions of PDEs (Existence Techniques III). - 10 Sobolev Spaces and L^2 Regularity theory. - 11 Strong solutions. - 12 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV). - 13The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash. - Appendix: Banach and Hilbert spaces. The L^p-Spaces. - References. - Index of Notation. - Index. Language: English