From Convexity to Nonconvexity
This collection of papers is dedicated to the memory of Gaetano Fichera, a great mathematician and also a good friend to the editors. Regrettably it took an unusual amount of time to bring this collection out. This was primarily due to the fact that the m
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Nonconvex Optimization and Its Applications Volume 55 Managing Editor: Panos Pardalos University of Florida, U.S.A.
Advisory Board:
l.R. Birge Northwestern University, U.S.A. Ding-Zhu Du University of Minnesota, U.S.A. C. A. F10udas
Princeton University, U.S.A.
l. Mockus Lithuanian Academy of Sciences, Lithuania H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A. G. Stavroulakis University of Ioannina, Greece
The titles published in this series are listed at the end of this volume.
From Convexity to N onconvexity Edited by
R.P. Gilbert University of Delaware
P.D. Panagiotopoulos and
P.M. Pardalos University of Florida
KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13:978-1-4613-7979-9 001: 10.10071978-1-4613-0287-2
e-ISBN -13: 978-1-4613-0287-2
Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved © 2001 Kluwer Academic Publishers Sof tcover reprint of the hardcover Ist edition 200 I No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents
Preface
xi
1
Frictional contact problems
1
Lars-Erik Andersson, Anders Klarbring
1.1 1.2 1.3 1.4 1.5 1.6 1.7
Introd uction Elementary example of non-uniqueness and non-existence Classical formulation of the quasistatic frictional contact problem The static problem Steady sliding problem Existence results for quasistatic friction problems Conclusion
1 2 4 5 8 8
10
References
11
2 Solutions for quasilinear hemivariational inequalities
15
Siegfried Carl
2.1 2.2 2.3 2.4
Introduction Notations, hypotheses and the main result Auxiliary results Proof of the main result 2.4.1 Static problem 2.4.2 Example 2.4.3 Concluding remarks
15 16
20 24 25 25 26
References
27
3 A Survey on Nonsmooth Critical Point Theory
29
M area Degiovanni
3.1 3.2 3.3 3.4 3.5 3.6
Introduction Critical point theory in metric spaces Subdifferential calculus Functionals of the calculus of variations Functionals with quadratic dependence on the gradient Area-type functionals
29 31 33 35 36 38 v
vi
FROM CONVEXITY TO NON CONVEXITY
References 4 Exhaustive families of approximations revisited V.F.Demyanov A.M.Rubinov 4.1 Directional derivatives and generalizations 4.2 Exhaustive families of upper and lower approximations
References
5
Optimal Zdzislaw 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
39 43 43 45
49
shape design Denkowski
51
Introd uction Preliminaries State relations for physical systems A
