This book is intended for students who wish to get an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. 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. This book can be utilized for a one-year course on partial differential equations.

Klappentext

Modern and systematic treatment of main approaches; Several additions have been made to the German edition, most notably coverage of eigenvalues and expansions; Emphasis on methods relevant for both linear and nonlinear equations; Contains chapter summaries, detailed illustrations and numerous exercises

Inhalt

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 Head equation The wave equation and its connections with the Laplace and heat equation The heat equation, semigroups, and Brownian motion The Dirichlet principle. Variational methods for the solution of PDE (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 Banach and Hilbert spaces. The Lp-spaces * Bibliography