Partial Differential Equations

Preface to the First Edition.- Preface to the Second Edition.- Introduction.- The Laplace equation as the prototype of an elliptic partial differential equation of 2nd 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 reqularity theorem of de Giorgi and Nash * Appendix: Banach and Hilbert spaces. The Lp-spaces.- References.- Index of Notation.- Index.-