Modern Methods in the Calculus of Variations: Lp Spaces
This is the first of two books on methods and techniques in the calculus of variations. Contemporary arguments are used throughout the text to streamline and present in a unified way classical results, and to provide novel contributions at the forefront o
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Irene Fonseca Giovanni Leoni
Modern Methods in the Calculus p of Variations: L Spaces
Irene Fonseca
Giovanni Leoni
Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA [email protected]
Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA [email protected]
ISBN: 978-0-387-35784-3
e-ISBN: 978-0-387-69006-3
Library of Congress Control Number: 2007931775 Mathematics Subject Classification (2000): 49-00, 49-01, 49-02, 49J45, 28-01, 28-02, 28B20, 52A © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY, 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com
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Preface
In recent years there has been renewed interest in the calculus of variations, motivated in part by ongoing research in materials science and other disciplines. Often, the study of certain material instabilities such as phase transitions, formation of defects, the onset of microstructures in ordered materials, fracture and damage, leads to the search for equilibria through a minimization problem of the type min {I (v) : v ∈ V} , where the class V of admissible functions v is a subset of some Banach space V . This is the essence of the calculus of variations: the identification of necessary and sufficient conditions on the functional I that guarantee the existence of minimizers. These rest on certain growth, coercivity, and convexity conditions, which often fail to be satisfied in the context of interesting applications, thus requiring the relaxation of the energy. New ideas were needed, and the introduction of innovative techniques has resulted in remarkable developments in the subject over the past twenty years, somewhat scattered in articles, preprints, books, or available only through oral communication, thus making it difficult to educate young researchers in this area. This is the first of two books in the calculus of variations and measure theory in which many results, some now classical and others at the forefront of research in the subject, are gathered in a unified, consistent way. A main concern has been to use contemporary arguments throughout the text to revisit and streamline well-known aspects of the theory, while providing novel contributions. The core of this book is the analysis of necessary and sufficient conditions for sequential
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