Nonlinear Differential Equations of Monotone Types in Banach Spaces
This book is concerned with basic results on Cauchy problems associated with nonlinear monotone operators in Banach spaces with applications to partial differential equations of evolutive type. This is a monograph about the most significant results obtain
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Viorel Barbu
Nonlinear Differential Equations of Monotone Types in Banach Spaces
Viorel Barbu Fac. Mathematics Al. I. Cuza University Blvd. Carol I 11 700506 Iasi Romania [email protected]
ISSN 1439-7382 ISBN 978-1-4419-5541-8 e-ISBN 978-1-4419-5542-5 DOI 10.1007/978-1-4419-5542-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009943993 M athematics Subject Classification (2010): 34G20, 34G25, 35A16 © Springer Science+Business Media, LLC 2010 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 Springer is part of Springer Science+Business Media (www.springer.com)
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1
Fundamental Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Geometry of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Convex Functions and Subdifferentials . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Sobolev Spaces and Linear Elliptic Boundary Value Problems . . . . . 10 1.4 Infinite-Dimensional Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 21 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2
Maximal Monotone Operators in Banach Spaces . . . . . . . . . . . . . . . . . . 2.1 Minty–Browder Theory of Maximal Monotone Operators . . . . . . . . . 2.2 Maximal Monotone Subpotential Operators . . . . . . . . . . . . . . . . . . . . . 2.3 Elliptic Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Nonlinear Elliptic Problems of Divergence Type . . . . . . . . . . . . . . . . . Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Accretive Nonlinear Operators in Banach Spaces . . . . . . . . . . . . . . . . . . 97 3.1 Definition and General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.2 Nonlinear Elliptic Boundary Value Problem in L p . . . . . . .
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