Variational and Hemivariational Inequalities Theory, Methods and Applications: Volume I: Unilateral Analysis and Unilateral Mechanics

417 Pages·2003·17.024 MB·English

by  D. Goeleven, D. Motreanu, Y. Dumont, M. Rochdi (auth.)

Variational and Hemivariational Inequalities.
Theory, Methods and Applications
Volume I
Nonconvex Optimization and Its Applications
Volume 69
Managing Editor:
Panos Pardalos
University ofF lorida, U.SA.
Advisory Board:
J. R. Birge
University ofM ichigan, U.S.A.
Ding-ZhuDu
University ofM innesota, U.S.A.
c. A. Floudas
Princeton University, U.SA.
J. Mockus
Lithuanian Academy ofS ciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.SA.
G. Stavroulakis
Technical University Braunschweig, Germany
VARIATIONAL AND HEMIVARIATIONAL
INEQUALITIES
Theory, Methods and Applications
Volume I: UNILATERAL ANALYSIS AND
UNILATERAL MECHANICS
D.GOELEVEN
IREMIA, University of La Reunion,
FRANCE
D.MOTREANU
University of Perpignan.
FRANCE
Y. DUMONT
IREMIA, University of La Reunion,
FRANCE
M. ROCHDI
IREMIA, University of La Reunion,
FRANCE
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication
CIP info or:
Title: Variational and Hemivariational Inequalities: Theory, Methods and Applications
Volume I: Unilateral Analysis and Unilateral Mechanics
Author: Goeleven, Motreanu, Dumont, Rochdi
ISBN 978-1-4613-4646-3 ISBN 978-1-4419-8610-8 (eBook)
DOI 10.1007/978-1-4419-8610-8
Copyright © 2003 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2003
Softcover reprint of the hardcover 1s t edition 2003
All rights reserved. No part of this pUblication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means, electronic, mechanical, photo-copying,
microfilming, recording, or otherwise, without the prior written permission of the publisher, with
the exception of any material supplied specifically for the purpose of being entered and executed
on a computer system, for exclusive use by the purchaser ofthe work.
Permissions for books published in the USA: perm; 55; oDs@wkap com
Permissions for books published in Europe: [email protected]
Printed on acid-free paper.
This book is dedicated
to the IlleulOry of Prof.
P.D. Panagiotopoulos.
Contents
List of Figures
Xl
Acknowledgments xiii
l. UNILATERAL ANALYSIS 1
1.1 Basic Mathematical Tools 1
1.2 Unilateral Analysis in Lloc(X; JR) 34
1.3 Unilateral Analysis in ro(X; JR U +00) 66
1.4 Asymptotic Unilateral Analysis 85
2. UNILATERAL MECHANICS 111
2.1 Mathematical Formalism 112
2.2 Principle of Virtual Power 115
2.3 Principle of Virtual Work 118
2.4 Convex Superpotentials 118
2.5 Nonconvex Superpotentials 121
2.6 Monotone Unilateral Boundary Conditions 122
2.7 Monotone Interior Unilateral Conditions 129
2.8 Nonmonotone Unilateral Boundary Conditions 131
2.9 Nonmonotone Interior Unilateral Conditions 134
2.10 Mathematical Modelling of Unilateral Conditions
and Unilateral B.V.Ps in Functional Spaces 135
Vlll VARIATIONAL AND HEMIVARIATIONAL INEQUALITIES
2.11 Variational and Hemivariational Inequalities in Unilateral
Mechanics 163
3. FUNDAMENTAL EXISTENCE THEORY OF INEQUALITY
PROBLEMS 207
3.1 The Hartman-Stampacchia Theorem in lRn 208
3.2 The Lions-Stampacchia Theorem and Generalized
Projection Mappings 209
3.3 The Browder Theorem for Monotone and
Hemicontinuous Variational Inequalities 212
3.4 The Fichera's Approach for Semicoercive Variational
Inequalities 214
3.5 The Recession Approach for Noncoercive Variational
Inequalities 223
3.6 The Monotonicity Principle and the Method of Lower and
Upper Solutions for Variational Inequalities 230
3.7 The Generalized Hille-Yosida Theorem and the Semi-Group
Approach for Evolution Variational Inequalities 239
3.8 The Brezis Approach for Evolution Variational
Inequalities 251
3.9 The Maximal Monotone Approach 258
3.10 The Generalized Hartman-Stampacchia Theorem for
Variational-Hemivariational Inequalities 263
3.11 Coercive Variational-Hemivariational Inequalities 267
3.12 Noncoercive Variational-Hemivariational Inequalities 271
3.13 A fixed Point Approach for a Class of Evolution
Variational Inequalities 277
4. MINIMAX METHODS FOR INEQUALITY PROBLEMS 281
4.1 The General Setting 282
Contents IX
4.2 A Deformation Result 296
4.3 Minimax Principles for Functionals of Type (H) 305
4.4 Multiplicity Theorems for Even Functionals of Type (H) 313
4.5 Examples and Applications 323
5. TOPOLOGICAL METHODS FOR INEQUALITY PROBLEMS335
5.1 Topological Tools 335
5.2 Fixed Point Formulations 341
5.3 An Alternative Theorem 345
5.4 Existence of Global Continua 348
5.5 A Topological Approach For Noncoercive Evolution
Variational Inequalities 350
5.6 The Asymptotic Relaxation Principle 354
5.7 A Degree Theoretic Approach for
Variational-Hemivariational Inequalities 361
Appendices 401
A- List of Notations 401
List of Figures
1.1.1 Convex set - non Convex set 2
1.1.2 Geometrical interpretation of the definition of convexity 3
1.1.3 l.s.c. functional - not l.s.c. functional 5
1.1.4 non convex cone - convex cone 6
1.1.5 Tangent and normal cones to a set 10
1.1.6 Tangent and normal cones to a convex set 12
1.1.7 Separation of convex sets 16
1.2.1 Subdifferentiation of integral functionals 54
1.3.1 The graphs of (1.3.15) and (1.3.16) 79
1.4.1 Recession cones 89
2.6.1 Unilateral contact boundary conditions 126
2.6.2 The friction boundary condition 128
2.8.1 Nonmonotone unilateral boundary conditions 132
2.8.2 Nonmonotone Winkler law 135
2.11.1 Adhesive contact problem 175
2.11.2 Notation and geometry of the laminated plate 182
2.11.3 On the buckling of a von Karman plate in adhesive
contact 192
2.11.4 Loading-unloading hysteresis loops 197
2.11.5 Control signal 203