Second Order Parabolic Differential Equations

The following sections are included:

• PREFACE

• PREFACE TO REVISED EDITION

• CONTENTS

The following sections are included:

• Outline of this book

• Further remarks

• Notation

The following sections are included:

• Introduction

• The weak maximum principle

• The strong maximum principle

• A priori estimates

• Notes

• Exercises

The following sections are included:

• Introduction

• The theory of weak derivatives

• The method of continuity

• Problems in small balls

• Global existence and the Perron process

• Notes

• Exercises

The following sections are included:

• Introduction

• Hölder continuity

• Campanato spaces

• Interior estimates

• Estimates near a flat boundary

• Regularized distance

• Intermediate Schauder estimates

• Curved boundaries and nonzero boundary data

• Two special mixed problems

• Notes

• Exercises

The following sections are included:

• Introduction

• Uniqueness of solutions

• The Cauchy-Dirichlet problem with bounded coefficients

• The Cauchy-Dirichlet problem with unbounded coefficients

• The oblique derivative problem

• Notes

• Exercises

The following sections are included:

• Introduction

• Notation and basic results

• Differentiability of weak solutions

• Sobolev inequalities

• Poincaré's inequality

• Global boundedness

• Local estimates

• Consequences of the local estimates

• Boundary estimates

• More Sobolev-type inequalities

• Conormal problems

• A special mixed problem

• Solvability in Hölder spaces

• The parabolic DeGiorgi classes

• Notes

• Exercises

The following sections are included:

• Introduction

• Maximum principles

• Basic results from harmonic analysis

• L^p estimates for constant coefficient divergence structure equations

• Interior L^p estimates for solutions of nondivergence form constant coefficient equations

• An interpolation inequality

• Interior L^p estimates

• Boundary and global estimates

• estimates for the oblique derivative problem

• The local maximum principle

• The weak Harnack inequality

• Boundary estimates

• Notes

• Exercises

The following sections are included:

• Introduction

• The Schauder fixed point theorem

• Applications of the Schauder theorem

• A theorem of Caristi and its applications

• Notes

• Exercises

The following sections are included:

• Introduction

• Comparison principles

• Maximum estimates

• Comparison principles for divergence form operators

• The maximum principle for divergence form operators

• Notes

• Exercises

The following sections are included:

• Introduction

• The boundary gradient estimate in general domains

• Convex-increasing domains

• The spatial distance function

• Curvature conditions

• Nonexistence results

• The case of one space dimension

• Continuity estimates

• Notes

• Exercises

The following sections are included:

• Introduction

• Global gradient bounds for general equations

• Examples
  • Uniformly parabolic equations
  • Prescribed mean curvature equations
  • A false mean curvature equation

• Local gradient bounds

• The Sobolev theorem of Michael and Simon

• Estimates for equations in divergence form

• The case of one space dimension

• A gradient bound for an intermediate situation

• Notes

• Exercises

The following sections are included:

• Introduction

• Interior estimates for equations in divergence form

• Equations in one space dimension

• Interior estimates for equations in general form

• Boundary estimates

• Improved results for nondivergence equations

• Selected existence results
  • Uniformly parabolic equations
  • Mean curvature equations
  • False mean curvature equations
  • Equations in convex-increasing domains
  • Continuous boundary values
  • One-dimensional problems

• Notes

• Exercises

The following sections are included:

• Introduction

• Maximum estimates

• Gradient estimates for the conormal problem
  • The prescribed mean curvature equation
  • Uniformly parabolic equations

• Gradient bounds for uniformly parabolic problems in general form

• The Hölder gradient estimate for the conormal problem

• Nonlinear boundary conditions with linear equations

• The Hölder gradient estimate for quasilinear equations

• Existence theorems

• Notes

• Exercises

The following sections are included:

• Introduction
  • Monge-Ampère type operators
  • Pucci extremal operators
  • Parabolic Bellman equations

• Comparison and maximum principles

• Simple uniformly parabolic equations

• Higher regularity of solutions

• The Cauchy-Dirichlet problem

• Boundary second derivative estimates

• The oblique derivative problem

• The case of one space dimension

• Notes

• Exercises

The following sections are included:

• Introduction

• General results for Hessian equations

• Estimates on solutions

• Existence of solutions

• Properties of symmetric polynomials

• The parabolic analog of the Monge-Ampère equation

• Notes

• Exercises

The following sections are included:

• Bibliography

• Index