Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems

384 Pages·2003·10.406 MB·English

by  D. Motreanu, V. Rǎdulescu (auth.)

Variational and Non-variational Methods in Nonlinear Analysis and
Boundary Value Problems
Nonc onvex Optimization and Its Applications
Volume 67
Managing Editor:
Panos Pardalos
Advisory Board:
J.R. Birge
Northwestern University, U.S.A.
Ding-Zhu Du
University qf Minnesota, U.S.A.
C. A. Floudas
Princeton University, U.S.A.
1. Mockus
Lithuanian Academy of Sciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.S.A.
G. Stavroulakis
Technical University Braunschweig, Germany
The titles published in this series are listed at the end of" this volume.
Variational and
Non-variational Methods in
Nonlinear Analysis and
Boundary Value Problems
by
D. Motreanu
Department (~l Mathematics,
University (~l Perpignan,
Perpignan, France
and
V. Radulescu
Department ol Mathematics,
University ol Craiova,
Craiova, Romania
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C. J.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-5248-6 ISBN 978-1-4757-6921-0 (eBook)
DOI 10.1007/978-1-4757-6921-0
Printed on acid-free paper
All Rights Reserved
© 2003 Springer Science+ Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2003
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from 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 of the work.
Contents
Preface IX
Introduction XI
1. ELEMENTS OF NONSMOOTH ANALYSIS 1
1 Generalized Gradients of Locally Lipschitz Functionals 1
2 Palais-Smale Condition and Coerciveness for a Class of
N onsmooth Functionals 12
3 Nonsmooth Analysis in the Sense of Degiovanni 19
2. CRITICAL POINTS FOR NONSMOOTH FUNCTIONALS 31
1 Critical Point Theory for Locally Lipschitz Functionals 31
2 Critical Point Theory for Convex Perturbations of Locally
Lipschitz Functionals 46
3 A Critical Point Theory in Metric Spaces 55
3. VARIATIONAL METHODS 67
1 Critical Point Theory for Convex Perturbations of Locally
Lipschitz Functionals in the Limit Case 67
2 Examples 83
4. MULTIVALUED ELLIPTIC PROBLEMS IN VARIATIONAL
FORM 99
1 Multiplicity for Locally Lipschitz Periodic Functionals 99
2 The Multivalued Forced-pendulum Problem 106
3 Hemivariational Inequalities Associated to Multivalued
Problems with Strong Resonance 111
4 A Parallel Nonsmooth Critical Point Theory. Approach
mn
to Stationary Schrodingcr Type Equations in 122
v
Vl VARIATIONAL AND NON-VARIATIONAL METHODS
5. BOUNDARY VALUE PROBLEMS IN NON-VARIATIONAL
FORM 139
1 The General Setting and Assumptions 139
2 Extremal Solutions of Quasilinear Parabolic Inclusion (5.1)
144
3 Proof of the Existence Result in Proposition 5.1 and an
Example 161
6. VARIATIONAL, HEMIVARIATIONAL AND VARIATIONAL-
HEMIVARIATIONAL INEQUALITIES 169
1 Hartman-Stampacchia Type Results for Hemivariational
Inequalities 170
2 Variational-Hemivariational Inequality Problems with Lack
of Convexity 178
3 Double Eigenvalue Hemivariational Inequalities with Non-
locally Lipschitz Energy Functional 191
4 Applications 202
7. EIGENVALUE PROBLEMS WITH SYMMETRIES 211
1 Orbits of Critical Points 211
2 Multiple Eigensolutions for Symmetric Functionals 219
3 Periodic Solutions of Hemivariational Inequalities. Multiple
Eigensolutions 225
4 Multiple Solutions for a Double Eigenvalue Hemivariational
Inequality with Constraints 233
8. NON-SYMMETRIC PERTURBATIONS OF SYMMETRIC
EIGENVALUE PROBLEMS 245
1 Non-symmetric Perturbations of Eigenvalue Problems for
Periodic Hemivariational Inequalities with Constraints 246
2 Perturbations of Double Eigenvalue Problems for General
Hemivariational Inequalities with Constraints 258
9. LOCATION OF SOLUTIONS FOR GENERAL NONSMOOTH
PROBLEMS 2~
1 Existence of Solutions by Minimax Methods for Variational-
Hemivariational Inequalities 273
2 Location ofEigensolutions to Variational-Hemivariational
Inequalities 288
3 Location of Solutions to Nonlinear Dirichlet Problems 299
Contents Vll
10. NONSMOOTH EVOLUTION PROBLEMS 307
1 First Order Evolution Variational Inequalities 307
2 Second Order Evolution Variational Equations 319
3 Stability Properties for Evolution Variational Inequalities 325
11. INEQUALITY PROBLEMS IN BV AND GEOMETRIC
APPLICATIONS 349
1 The General Framework 350
2 Area Type Functionals 360
3 A Result of Clark Type 367
4 An Inequality Problem with Superlinear Potential 370
Preface
This book reflects a significant part of authors' research activity dur
ing the last ten years. The present monograph is constructed on the
results obtained by the authors through their direct cooperation or due
to the authors separately or in cooperation with other mathematicians.
All these results fit in a unitary scheme giving the structure of this work.
The book is mainly addressed to researchers and scholars in Pure and
Applied Mathematics, Mechanics, Physics and Engineering.
We are greatly indebted to Viorica Venera Motreanu for the careful
reading of the manuscript and helpful comments on important issues.
We are also grateful to our Editors of Kluwer Academic Publishers
for their professional assistance.
Our deepest thanks go to our numerous scientific collaborators and
friends, whose work was so important for us.
D. Motreanu and V. Radulescu
IX
Introduction
The present monograph is based on original results obtained by the
authors in the last decade. This book provides a comprehensive expo
sition of some modern topics in nonlinear analysis with applications to
the study of several classes of boundary value problems. Our framework
includes multivalued elliptic problems with discontinuities, variational
inequalities, hemivariational inequalities and evolution problems. The
treatment relies on variational methods, monotonicity principles, topo
logical arguments and optimization techniques.
Excepting Sections 1 and 3 in Chapter 1 and Sections 1 and 3 in
Chapter 2, the material is new in comparison with any other book,
representing research topics where the authors contributed. The outline
of our work is the following.
Chapter 1 contains basic facts on nonsmooth analysis in the frame
work of Clarke's and Degiovanni's theories in Sections 1 and 3, while
Section 2 concerns the relationship between the Palais-Smale condition
and coerciveness.
Chapter 2 is devoted to modern nonsmooth critical point theories
(Chang, Degiovanni, Goeleven-Motreanu-Panagiotopoulos, Szulkin).
Chapter 3 deals with general results ensuring the existence of critical
points of nondifferentiable functionals.
Chapter 4 presents an abstract multiplicity theorem of Ljusternik
Schnirelman type and a comparison treatment of a class of stationary
Schrodinger equations with lack of compactness.
Chapter 5 studies extremal solutions for initial boundary value prob
lems of parabolic type involving Clarke's gradient.
Chapter 6 focuses on existence results for various classes of boundary
value problems expressed by variational, hemivariational or variational
hernivariational inequalities.
Xl