• PDF / 4,729,253 Bytes
• 126 Pages / 461 x 684 pts Page_size
• 35 Downloads / 207 Views

DOWNLOAD

REPORT

---

922 Bernard Dacorogna

Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals

Springer-Verlag Berlin Heidelberg New York 1982

Author

Bernard Dacorogna Departsrnent de Mathematiques Ecole Polytechnique Federale de Lausanne 61, Avenue de Cour, 1007 Lausanne, Switzerland

AMS Subject Classifications (1980): 46-XX ISBN 3-540-11488-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11488-2 Springer-Verlag New York Heidelberg Berlin 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 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

PREFACE

These notes are the result of a graduate course given at Brown during the first quarter of 1981. They should be considered as an introduction to the subject. They are not intended to be a complete presentati.on of all the resuIts in this area. The results presented here are not all new and obviously a large part of the first and second chapter owes much to various works of

F. Murat and L. Tartar on compensated compactness. I would like to thank, particularly,

C.M. Dafermos for his constant encou-

ragement and his many, always helpful, suggestions. Without hi.s help these notes would never have been written. I want also to thank W. Hrusa for carefully reading and correcting the manuscript. Finally my thanks go to Kate MacDougall for the very nice typing of these notes.

B. Dacorogna Providence, R.1.

July, 1981

This research has been supported in part by the National Science Foundation under Con t r a c t je N'Sft-Bng . CME80-23824.

WEAK CONTINUITY AND WEAK LOWER SEMICONTINUITY OF NON-LINEAR FUNCTIONALS by B. Dacorogna

ABSTRACT

These notes deal with the behavior of nonlinear functionals with respect to weak convergence.

In the first chapter- we investigate

several necessary and sufficient conditions in order that a nonlinear function is weakly continuous or weakly lower semicontinuous.

In

Chapter II we give some applications of the results of Chapter I to partial differential equations and to nonlinear elasticity.

In the

last chapter we deal with dual and relaxed variational problems.

TABLE OF CONTENTS Page Introduction. • • •• . • • . •• • •• • •• • • • • • • • . . . . •. • • • . ••• • • • • . • • • • • . • . • • • • • Chapter I.

1

Compensated Compactness

§l.

Preliminary Result (Case without Assumptions on the Derivatives). . • • • •• •• • • • • • • • . . . • • • • •• • • • • • • • • • • • . • . • • . . • • ••

7

§2.

Case with Assumptions on the Derivatives...................

11

§3.

Legendre-Hadamard Condition and other Necessary Conditions.

19

§4.

The Quadratic Case