Convex Variational Problems Linear, Nearly Linear and Anisotropic Gr
The author emphasizes a non-uniform ellipticity condition as the main approach to regularity theory for solutions of convex variational problems with different types of non-standard growth conditions. This volume first focuses on elliptic variat
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Michael Bildhauer
Convex Variational Problems Linear, Nearly Linear and Anisotropic Growth Conditions
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Author Michael Bildhauer Department of Mathematics Saarland University P.O. Box 151150 66041 Saarbr¨ucken Germany e-mail: [email protected]
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Mathematics Subject Classification (2000): 49-02, 49N60, 49N15, 35-02, 35J20, 35J50 ISSN 0075-8434 ISBN 3-540-40298-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10935590
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Preface
In recent years, two (at first glance) quite different fields of mathematical interest have attracted my attention. • •
Elliptic variational problems with linear growth conditions. Here the notion of a “solution” is not obvious and, in fact, the point of view has to be changed several times in order to get some deeper insight. The study of the smoothness properties of solutions to convex anisotropic variational problems with superlinear growth.
It took some time to realize that, in spite of the fundamental differences and with the help of some suitable theorems on the existence and uniqueness of solutions in the case of linear growth conditions, a non-uniform ellipticity condition serves as the main tool towards a unified view of the regularity theory for both kinds of problems. This is roughly speaking the background of my habilitations thesis at the Saarland University which is the basis for this presentation. Of course there is a long list of people who have contributed to this monograph in one or the other way and I express my thanks to each of them. Without trying to list them all, I really want to mention: Prof.
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