This book develops the important approaches to the existence and regularity of partial differential equations, in particular of elliptic (and parabolic) type, particularly those aspects and methods that are relevant not only for linear, but also for nonlinear equations.

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.

Modern and systematic treatment of main approaches Several additions have been made to the 2nd edition, including a new chapter on Reacition-diffusion equations Emphasis on methods relevant for both linear and nonlinear equations Contains chapter summaries, detailed illustrations and numerous exercises

Autorentext

Dr. Jürgen Jost studierte Mathematik, Physik, Volkswirtschaftslehre und Philosophie an der Universität Bonn. Er promovierte in der Mathematik und wurde nach verschiedenen internationalen Forschungsaufenthalten als Professor für Mathematik an die Ruhr-Universität Bochum und 1996 als Direktor an das neu zu gründende Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig berufen. Er ist Autor von mehr als 20 Forschungsmonographien und Lehrbüchern und von über 400 wissenschaftlichen Fachpublikationen.

Inhalt
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.