This book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations.

This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader. New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

New edition extensively revised and updated Features a systematic discussion of the relations between different types of partial differential equations Presents new Harnack type techniques

Autorentext

Jürgen Jost is Codirector of the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, an Honorary Professor at the Department of Mathematics and Computer Sciences at Leipzig University, and an External Faculty Member of the Santa Fe Institute for the Sciences of Complexity, New Mexico, USA. He is the author of a number of further Springer textbooks including Postmodern Analysis (1997, 2002, 2005), Compact Riemann Surfaces (1997, 2002, 2006), Partial Differential Equations (2002, 2007, 2013), Differentialgeometrie und Minimalflächen (1994, 2007, 2014, with J. Eschenburg), Dynamical Systems (2005), Mathematical Concepts (2015), as well as several research monographs, such as Geometry and Physics (2009), and many publications in scientific journals.

Inhalt
Preface.- Introduction: What are Partial Differential Equations?.- 1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order.- 2 The Maximum Principle.- 3 Existence Techniques I: Methods Based on the Maximum Principle.- 4 Existence Techniques II: Parabolic Methods. The Heat Equation.- 5 Reaction-Diffusion Equations and Systems.- 6 Hyperbolic Equations.- 7 The Heat Equation, Semigroups, and Brownian Motion.- 8 Relationships between Different Partial Differential Equations.- 9 The Dirichlet Principle. Variational Methods for the Solutions of PDEs (Existence Techniques III).- 10 Sobolev Spaces and L^2 Regularity theory.- 11 Strong solutions.- 12 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV).- 13The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash.- Appendix: Banach and Hilbert spaces. The L^p -Spaces.- References.- Index of Notation.- Index.