Nonsmooth Variational Problems and Their Inequalities Comparison Pri
This monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as is multivalued elliptic problems, variational inequalities, hemivariational inequalit
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Siegfried Carl Vy Khoi Le Dumitru Motreanu
Nonsmooth Variational Problems and Their Inequalities Comparison Principles and Applications
Siegfried Carl Institut für Mathematik Martin-Luther-Universität Halle-Wittenberg D-06099 Halle Germany [email protected]
Vy Khoi Le Department of Mathematics and Statistics University of Missouri-Rolla Rolla, MO 65409 U.S.A [email protected]
Dumitru Motreanu Département de Mathématiques Université de Perpignan 66860 Perpignan France [email protected]
Mathematics Subject Classifications (2000): (Primary) 35B05, 35J20, 35J85, 35K85, 35R70, 47J20, 47J35, 49J52, 49J53; (Secondary) 35J60, 35K55, 35R05, 35R45, 49J40, 58E35 Library of Congress Control Number: 2006933727 ISBN-13: 978-0-387-30653-7
e-ISBN-13: 978-0-387-46252-3
Printed on acid-free paper. © 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, U.S.A.), 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. 9 8 7 6 5 4 3 2 1 springer.com
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Preface
Nonsmooth variational problems have their origin in the study of nondifferentiable energy functionals, and they arise as necessary conditions of critical points of such functionals. In this way, variational inequalities are related with convex energy or potential functionals, whereas the new class of hemivariational inequalities arise in the study of nonconvex potential functionals that are, in general, merely locally Lipschitz. The foundation of variational inequalities is from Fichera, Lions, and Stampacchia, and it dates back to the 1960s. Hemivariational inequalities were first introduced by Panagiotopoulos about two decades ago and are closely related with the development of the new concept of Clarke’s generalized gradient. By using this new type of inequalities, Panagiotopoulos was able to solve various open questions in mechanics and engineering. This book focuses on nonsmooth variational problems not necessarily related with some potential or energy functional, which arise, e.g., in the study of boundary value problems with nonsmooth data and/or nonsmooth constraints such as multivalued elliptic problems with multifunctions of Clarke’s subgradient type, variational inequalities, hemivariational inequalities, and their corresponding evolutionary counterparts. The main purpose is to provide a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution me
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