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. 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. Connections between elliptic, parabolic, and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups. This book can be utilized for a one-year course on partial differential equations.
For the new edition the author has added a new chapter on reaction-diffusion equations and systems. There is also new material on Neumann boundary value problems, Poincar? inequalities, expansions, as well as a new proof of the Hölder regularity of solutions of the Poisson equation.
Jürgen Jost is Co-Director of the Max Planck Institute for Mathematics in the Sciences and Professor of Mathematics at the University of Leipzig. He is the author of a number of Springer books, including Dynamical Systems (2005), Postmodern Analysis (3rd ed. 2005, also translated into Japanese), Compact Riemann Surfaces (3rd ed. 2006) and Riemannian Geometry and Geometric Analysis (4th ed., 2005). The present book is an expanded translation of the original German version, Partielle Differentialgleichungen (1998). TOC: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.
