Lieberman G.M. Second Order Parabolic Partial Differential Equations

World Scientific, 1996 (2005 reprint). - 439 pages.This book is an introduction to the general theory of second order parabolic differential equations, which model many important, time-dependent physical systems. It studies the existence, uniqueness, and regularity of solutions to a variety of problems with Dirichlet boundary conditions and general linear and nonlinear boundary conditions by means of a priori estimates. The first seven chapters give a description of the linear theory and are suitable for a graduate course on partial differential equations. The last eight chapters cover the nonlinear theory for smooth solutions. They include much of the author's research and are aimed at researchers in the field. A unique feature is the emphasis on time-varying domains.Maximum Principles
Introduction to the Theory of Weak Solutions
Hölder Estimates
Existence, Uniqueness, and Regularity of Solutions
Further Theory of Weak Solutions
Strong Solutions
Fixed Point Theorems and Their Applications
Comparison and Maximum Principles
Boundary Gradient Estimates
Global and Local Gradient Bounds
Hölder Gradient Estimates and Existence Theorems
The Oblique Derivative Problem for Quasilinear Parabolic Equations
Fully Nonlinear Equations I. Introduction
Fully Nonlinear Equations II. Hessian Equations