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.
