Nonlinear Partial Differential Equations of Second Order

✍ Scribed by Guang Chang Dong

• Publisher: American Mathematical Society
• Year: 1991
• Tongue: English
• Leaves: 263
• Series: Translations of Mathematical Monographs, Vol. 95
• Category: Library

✦ Synopsis

This book addresses a class of equations central to many areas of mathematics and its applications. Although there is no routine way of solving nonlinear partial differential equations, effective approaches that apply to a wide variety of problems are available. This book addresses a general approach that consists of the following: Choose an appropriate function space, define a family of mappings, prove this family has a fixed point, and study various properties of the solution. The author emphasizes the derivation of various estimates, including a priori estimates. By focusing on a particular approach that has proven useful in solving a broad range of equations, this book makes a useful contribution to the literature.

✦ Table of Contents

Cover

S Title

Nonlinear Partial Differential Equations of Second Order

Copyright

 ©1991 by the American Mathematical Society

 ISBN 0-8218-4554-3

 QA377.T8613 1991 515'.353-dc20

 LCCN 91-27853

Contents

Introduction

Notes for the English Translation

 (1) The uniqueness and existence of solutions to nonlinear oblique derivative boundary value problems for fully nonlinear parabolic equations satisfying natural structure conditions

 (2) The existence, uniqueness, and regularity for viscosity solutions of thefirst boundary value problem for fully nonlinear parabolic equations satisfying anatural structure conditio

 (3) Fully nonlinear degenerate parabolic equations

CHAPTER I The First Boundary Value Problemfor Second-Order Quasilinear Parabolic Equations with Principal Part in Divergence Form

 §1. Uniform and Holder estimates for the solut

 §2. A uniform bound for D_x u

 §3. A Holder estimate for D_x u

 §4. Existence and uniqueness of the solution for the first boundary value problem

CHAPTER II A Periodic Boundary Value Problem for a Nonlinear Telegraph Equation

 §1. Solvability for higher-dimensional telegraph equations in the nonresonance case

 §2. A discussion on the resonance case

 §3. Regularity of a generalized solution

CHAPTER III The Initial Value Problemfor a Nonlinear Schrodinger Equation

 §1. Background materials

 §2. The initial value problem for the linear Schrodinger equation

 §3. The initial value problem for a nonlinear Schrodinger equation

CHAPTER IV Multi-Dimensional Subsonic Flows Around an Obstacle

 §1. Introduction

 §2. Background material for the linear problem

 §3. Solution to the auxiliary problem

 §4. Resolution of the problem of a flow passing an obstacle and elementary properties of the solution

 §5. Further properties of the solution

CHAPTER V The Initial-Boundary Value Problem for Degenerate Quasilinear Parabolic Equations

 §1. Formulation of the problem and a Holder estimate for the solution

 §2. Solvability for the first boundary value problem

 §3. Uniqueness of the solution

CHAPTER VI The Speed of Propagation of the Solution of a Degenerate Quasilinear Parabolic Equation

 §1. An estimate on the domain of dependence

 §2. A lower estimate for the solution

CHAPTER VII Aleksandrov and Bony Maximum Principles for Parabolic Equations

 §1. Introduction

 §2. Some properties of convex functions

 §3. Convex envelopes

 §4. Several Aleksandrov maximum principles

 §5. Bony maximum principles

CHAPTER VIII The Density Theorem and Its Applications

 §1. The statement of the density theorem

 §2. Several lemmas and the proof of the density theorem

 §3. The Harnack inequality for parabolic equations with measurable coefficients

 §4. A Holder estimate for the solution of a quasilinear parabolic equation

 §5. A Holder estimate for the solution for a quasilinear parabolic system

CHAPTER IX Fully Nonlinear Parabolic Equations

 §1. A uniform bound for a solution u and an interior estimate for D_x u

 §2. An interior estimate for the second derivatives

 §3. An interior Holder estimate for the second derivatives

 §4. A near boundary Holder estimate for u

 §5. Uniform and Holder estimates for Dx u near the boundary

 §6. Near boundary uniform and Holder estimates for the second derivatives

 §7. Uniqueness and existence of a solution for the first boundary value problem under the natural structure conditions

CHAPTER X Fully Nonlinear Parabolic Equations (Continued)

 §1. The density theorem for quasilinear parabolic equations with natural structure condition of the second kind

 §2. A Holder estimate for the solution and unique solvability for the first boundary value problem

 §3. Certain apriori estimates for the solutions of a fully nonlinear parabolic equation with natural structure condition of the second kind and the unique solvability of the first boundary value problem

Symbols

References

Epilogue

Back Cover