This book offers an ideal introduction to the theory of partial differential equations. It focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. These are maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. This book also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. It also explores connections between elliptic, parabolic, and hyperbolic equations as well as the connection with Brownian motion and semigroups.

This second edition features a new chapter on reaction-diffusion equations and systems. There is also new material on Neumann boundary value problems, Poincaré inequalities, expansions as well as a new proof of the Hölder regularity of solutions of the Poisson equation.

Inhalt

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.-