Elliptic and Parabolic
PROBLEMS Rolduc and Gaeta 2001
Ed/tors
Josef Bemelmans | Bernard Brighi | Alain _ Michel Chipot | Francis Conrad | Itai Shafrir Vanda Valente | Giorgio Vergara-Caffarelli
Proceedings
of
the
European
Conference
/ 2 i ^ Proceedings of the
^ t h European Conference
Elliptic and Parabolic
'ROBLEM~
RoLduc and Gaeta 2001
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i-Mi
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Proceedings of t h e x
M
European Conference
Elliptic and Parabolic Rolduc and Gaeta 2001 Rolduc, Netherlands • 18-22 June 2001 Gaeta, Italy • 24-28 September 2001
Editors
Josef Bemelmans RWTH Aachen, Germany
Bernard Brighi Alain Brillard Universite de Haute Alsace, France
Michel Chipot Universitdt Zurich, Switzerland
Francis Conrad Universite Henri Poincare-Nancy 1, France
Itai Shafrir Technion-IIT, Israel
Vanda Valente IAC, CNR, Italy
Giorgio Vergara-Caffarelli Universita di Roma "La Sapienza", Italy
V f e World Scientific wB
New Jersey • London • Singapore • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Proceedings of the Fourth European Conference on ELLIPTIC AND PARABOLIC PROBLEMS — ROLDUC AND GAETA 2001 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-238-045-0
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Preface
This book is a collection of texts from some conferences that were given in Rolduc in June 2001 and Gaeta in September 2001, during the Fourth European Conference on Elliptic and Parabolic Problems. The subjects adressed in this volume are dealing with Calculus of Variations, Free Boundary Problems, Homogenization, Modeling, Numerical Analysis and various applications in physics, mechanics and engineering. We would like to thank all the participants to these meetings for their help in making it successful. Special thanks go to the contributors of this volume, and in particular to E. Maitre. These meetings have been made possible by grants from the IAC in Rome and from the Universities of Mulhouse, Nancy, Roma "La Sapienza", Zurich, the RWTH of Aachen and the Technion of Haifa. We express our deep appreciation to these institutions. Finally, we thank World Scientific for helping us to publish these proceedings.
J. Bemelmans, B. Brighi, A. Brillard, M. Chipot, F. Conrad, I. Shafrir, V. Valente, G. Vergara-Cafarelli
V
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VII
CONTENTS Preface
v
1. Rolduc Models for shape memory alloys described by subdifferentials of indicator functions T. Aiki and N. Kenmochi
1
Semilinear elliptic boundary value problems with critical Sobolev exponent C. Bandle
11
Diffraction problems for quasilinear elliptic and parabolic systems . . . G. Boyadjiev and N. Kutev Extremality and compactness results for elliptic hemivariational inequalities S. Carl On an operator equation for Stokes boundary value problems T. D. Chandra and J. de Graaf Local stability under changes of boundary conditions at a far away location M. Chipot and A. Rougirel On some variational problems with an infinite number of wells A. Elfanni Boltzmann equation for quantum particles and Fokker Planck approximation M. Escobedo Boundedness of global solutions of nonlinear parabolic equations . . . . M. Fila Transmission problems for degenerate parabolic equations T. Fukao
21
30 40
52 66
75 88 103
Existence of solutions of a segregation model arising in population dynamics G. Galiano, M. L. Garzon and A. Jungel
113
The nonstationary Stokes and Navier-Stokes equations in aperture domains T. Hishida
126
viii Global attractors for multivalued flows associated with subdifferentials N. Kenmochi and N. Yamazaki Quasiconvex extreme points of convex sets M. Kruzik The Richards equation for the modeling of a nuclear waste repository 0. Lafitte and C. Le Potier Trace identities and universal estimates for eigenvalues of linear pencils M. Levitin and L. Parnovski Universal estimates for the blow-up rate in a semilinear heat equation J. Matos and P. Souplet Asymptotic behavior of curves evolving by forced curvature flows . . . H. Ninomiya Existence of non-steady flows of an incompressible, viscous drop of fluid in a frame rotating with finite angular velocity M. Padula and V. A. Solonnikov On a convection-diffusion equation with partial diffusivity A. Pascucci Quasiconvexity and optimal design P. Pedregal A regularity criterion for the angular velocity component in the case of axi-symmetric Navier-Stokes equations M. Pokorny A comparison principle for the p-Laplacian A. Poliakovsky and I. Shafrir Supercritical variational problems with unique critical points W. Reichel Structure of the solution flow for steady-state problems of one-dimensional Fremond models of shape memory alloys K. Shirakawa Lagrangian coordinates in free boundary problems for multidimensional parabolic equations S. I. Shrnarev The Brezis-Nirenberg problem on H n . Existence and uniqueness of solutions S. Stapelkamp
135 145
152
160
165 175
180 204 215
233 243 253
263
274
283
IX
Solvability of general elliptic problems in Holder spaces V. Volpert and A. Volpert
291
2. G a e t a Nonlinear diffusion in irregular domains U. G. Abdulla Analysis of radiative transfer equation coupled with nonlinear heat conduction equation F. Asllanaj, G. Jeandel, J. R. Roche and D. Schmitt Viscosity Lyapunov functions for almost sure stability of degenerate diffusions M. Bardi and A. Cesaroni Mass transport through charged membranes D. Bothe and J. Priiss The Hopf solution of Hamilton-Jacobi equations /. C. Dolcetta Conservation of critical groups in a weaker topology for functionals associated with quasilinear elliptic equations J. N. Corvellec and H. Douik Approximating exterior flows by flows on truncated exterior domains: piecewise polygonal artificial boundaries P. Deuring On the Stefan problem with surface tension J. Escher, J. Priiss and G. Simonett Epidemic models with compartmental diffusion W. E. Fitzgibbon, M. Langlais and J. J. Morgan Algebraic multigrid for selected PDE systems T. Fullenbach and K. Stuben Numerical computation of electromagnetic guided waves in a general perturbed stratified medium D. G. Pedreira A non-stationary model for catalytic converters with cylindrical geometry J.-D. Hoernel Exact controllability of piezoelectric shells B. Miara Optimal control approach for the fluid-structure interaction problems C. M. Murea
302
311
322 332 343
352
364 377 389 399
411
424 434
442
X
Hyperbolic propagation of singularities in a parabolic system of shell theory J. Sanchez-Hubert Singular perturbations going out of the energy space. Layers in elliptic and parabolic cases E. Sanchez-Palencia Regularity and uniqueness results for a phase change problem in binary alloys J.-F. Scheid and G. Schimperna Bifurcation in population dynamics K. Umezu
451
461
475 485
Models for shape memory alloys described by subdifferentials of indicator functions* Toyohiko Aiki1 and Nobuyuki Kenmochi2 Department of Mathematics, Faculty of Education Gifu University Gifu 501-1193, Japan Department of Mathematics, Faculty of Education Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan Email :
[email protected] ;
[email protected] 1
2
Abstract We consider a system of nonlinear partial differential equations, which is one mathematical model describing the dynamics of shape memory alloys. We prove the existence and uniqueness of a solution of this one-dimensional model, using the theory of nonlinear evolution equations governed by subdifferential operators of time-dependent convex functions on Hilbert spaces, and applying some classical results dealing with /^-estimates for solutions of parabolic equations.
1
Introduction
T h e shape memory alloy problem under consideration is to find a triplet of functions, t h e temperature field 6 := 8(t, x), t h e stress a :— a(t, x) and t h e displacement u := u(t, x) on Q(T) := (0,T) x (0,1), 0 < T < oo, satisfying utt + ^uxxxx
- /iuxxt - ax = 0
0t - K-0XX = auxt + jx\uxt\2
in Q(T),
(1)
in Q(T),
o-t - v(Txx + dl(8, e; a) 3 cuxt
(2)
in Q(T),
u{t,0)=u(t,l)=uxx{t,0)=uxx{t,l)
=0
(3) for0
