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1074 Edward W. Stredulinsky

Weighted Inequalities and Degenerate Elliptic Partial Differential Equations

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Author Edward W.Stredulinsky Department of Mathematics, University of Wisconsin Madison, Wisconsin 53706, USA

AMS Subject Classification (1980): 35J 70 ISBN 3-540-13370-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13370-4 Springer-Verlag New York Heidelberg Berlin Tokyo This work IS subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to 'Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

TABLE OF CONTENTS

Introduction . . . • . . . . . • • . . . . . • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter O.

Notation •.•.•...•...•.•••..•.....•...•..••.•••.•.•.•

3

Chapter 1.

Preliminary Analysis .••........•.••••.••.•••••......

3

1.1.0

Calculus in Measure Spaces ••••.•.•.••••.•.•••••••••.

4

1. 2.0

Weighted Hardy Inequalities .•....••••••••••.•••.....

12

1. 3.0

Equivalence of Capacities ..•.•.••••••••••••••••••.•.

25

Chapter 2.

Basic Results ..•.•.•.•.•.•..•••.••.•.•••..•..••••.••

32

2. (.0

Weighted Sobolev Inequalities •...•.•..•.••••....••••

33

2.2.0

Properties of Sobolev Spaces, Capacities and Sobolev Inequalities for Applications to Differential Equations .•••••••...•.••••..••••......•

2.3.0

Chapter 3.

45

Higher Integrability from Reverse Holder Inequalities ..•.•...•••..•....•....•••.••••..•......

89

Applications .•.••...•.••.•...•..•.•....•••••...•..••

96

3.1.0

A Harnack Inequality and Continuity of Weak

3.2.0

Modulus of Continuity Estimates for Weighted

3.3.0

Higher Integrability for the Gradient of

Solutions for Degenerate Elliptic Equations

.

97

Sobolev Functions and V.M.O. Functions ..•••••..•.•.•

126

Solutions of Elliptic Systems and Applications to Continuity of Solutions ••....••.••..•••.•••••.•..

134

Bibliography •••••••••••••••.•.••••••.•.....••..•••••••••••..••••

140

Index .•.•...•....•••..•.......•...•.•••..........•.....•..•.....

143

INTRODUCTION The main purpose of these notes is the analysis of various weighted spaces and weighted inequalities which are relevant to the study of degenerate partial differential equations.

The usefulness of

these results is demonstrated in the later part of the text where they are used to establish continuity for weak solutions of degenerate elliptic equations. The most important inequalities dealt with are certain weighted Sobolov and Poincare inequalities for which the admissible weights are characterized.

Weighted reverse Holder inequalities and

inequalities for the