Table Of ContentThe IMA Volumes
in Mathematics
and Its Applications
Volume 3
Series Editors
George R. Sell Hans Weinberger
Institute for Mathematics and Its Applications
IMA
The Institute for Mathematics and Its Applications was established by a grant
from the National Science Foundation to the University of Minnesota in 1982. The
IMA seeks tp encourage the development and study of fresh mathematical concepts
and questions of concern to the other sciences by bringipg together mathematicians and
scientists from diverse fields in an atmosphere that will stimulate discussion and col
laboration.
The IMA Volumes are intended to involve the broader scientific community in this
process.
Hans Weinberger, Director
George R. Sell, Associate Director
IMA Programs
1982-1983 Statistical and Continuum Approaches to Phase Transition
1983-1984 Mathematical Models for the Economics of Decentralized Resource Allocation
1984-1985 Continuum Physics and Partial Differential Equations
1985-1986 Stochastic Differential Equations ~nd Their Applications
1986--1987 Scientific Computation
1987-1988 Applied Cpmbinatorics
1988-1989 Nonlinear Waves
Springer Lecture Notes from the IMA
The Mathematics and Physics of Disordered Media
Editors: Barry Hughes and Barry Ninliam
(Lecture Notes in Mathematics, Volume 1035, 1983)
Orienting Polymers
Editor: J. L. Ericksen
(Lecture Notes in Mathematics, Volume 1063, 1984)
New Perspectives in Thermodynamics
Editor: James Serrin
(Springer-Verlag, 1986)
Models of Economic Dynamics
Editor: Hugo Sonnenschein
(Lecture Notes in Economics, Volume 264, 1986)
Metastability
and
Incompletely Posed Problems
Edited by
Stuart S. Antman, J.L. Ericksen,
David Kinderlehrer, and Ingo Milller
With 46 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Stuart S. Antrnan
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
J.L. Ericksen
School of Mathematics and Department of Aerospace Engineering and Mechanics, University of
Minnesota, Minneapolis, MN 55455, U.S.A.
David Kinderlehrer
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Ingo Miiller
FB9-Hermann Fottinger Institut, Technical University, Berlin, F.R.G.
AMS Classification: 35-06, 35R25, 35B35
Library of Congress Cataloging-in-Publication Data
Metastability and incompletely posed problems.
(The IMA volumes in mathematics and its
applications; v. 3)
Proceedings of a workshop held during the IMA
1984-85 program on continuum physics and partial
differential equations.
I. Differential equations, Partial-Improperly
posed problems-Congresses. 2. Field theory
(Physics)-Congresses. 3. Stability-Congresses.
\. Antman, S. S. (Stuart S.) II. University of
Minnesota. Institute for Mathematics and its
Applications. III. Title.
QA377.M47 1986 515.3'53 86-28033
© 1987 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1987
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010,
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
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The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the
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Permission to photocopy for internal or personal use, or the internal or personal use of specific
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directly to Springer-Verlag New York, 175 Fifth Avenue, New York, New York 10010, U.S.A.
96462-211987 $0.00 + .20
9 8 7 6 5 4 3 2 I
ISBN-13: 978-1-4613-8706-0 e-ISBN-13: 978-1-4613-8704-6
DOl: 10.1007/978-1-4613-8704-6
The IMA Volumes in Mathematics and
Its Applications
Current Volumes:
Volume 1: Homogenization and Effective Moduli of Materials and Media
Editors: 1.L. Ericksen, David Kinderlehrer, Robert Kohn, and 1.-L. Lions
Volume 2: Oscillation Theory, Computation, and Methods of Compensated
Compactness
Editors: Constantine Dafermos, 1.L. Ericksen, David Kinderlehrer, and
Marshall Slemrod
Volume 3: Metastability and Incompletely Posed Problems
Editors: Stuart S. Antman, 1.L. Ericksen, David Kinderlehrer, and Ingo
Muller
Forthcoming Volumes:
1984-1985: Continuum Physics and Partial Differential Equations
Theory and Applications of Liquid Crystals
Amorphous Polymers and Non-Newtonian Fluids
Dynamical Problems in Continuum Physics
1985-1986: Stochastic Differential Equations and Their Applications
Random Media
Percolation Theory and Ergodic Theory of Infinite Particle Systems
Hydrodynamic Behavior and Interacting Particle Systems and Applications
Stochastic Differential Systems, Stochastic Control Theory and Applications
CONTENTS
Foreword ..............................................................•.. i x
Preface ...............................•................. ,................ xi
Dissipative Mechanisms .................................................. .
Stuart S. Antman and Reza Malek-Madani
Does Rank-One Convexity Imply Quasiconvexity? 17
J.M. Ball
Metastable Harmonic Maps................................................. 33
Haim Brezis
Bifurcation of Constrained Problems in Thermoelasticity .................. 43
M.C. Calderer
The Compressible Reynolds Lubrication Equation 61
Michel Chi pot and Mitchell Luskin
Twinning of Crystals I 77
Jerry L. Ericksen
Quasiconvexity and Partial Regularity in the Calculus of Variations ...... 95
L. C. Evans
Introduction to Pattern Selection in Dendritic Solidification .•...•...... 103
Nigi I Goldenfeld
Some Results and Conjectures in the Gradient Th~ory of Phase Transitions. 135
Morton E. Gurtin
The Stabil ity and Metastability of Quartz ................................ 147
R.D. James
Continuation Theorems for Schrodinger Operators .......................... 177
Ca r I os E. Ken i g
Twinning of Crystals II .................................................. 185
David Kinderlehrer
Simulation of Pseudo-Elastic Behaviour in a System of Rubber Ballons ..... 213
Wolfgang Kitsche. Ingo Muller and Peter Strehlow
Asymptotic Problems in Distributed Systems ............................... 241
J.-L. Lions
Stability of Nonlinear Waves .............................................. 259
Tai-Ping Liu
The Nash-Moser Technique For an Inverse Problem in Potential Theory Related
to Geodesy ............................................................... 269
C. Maderna. C. Pagani. and S. Salsa
Variational Stabi lity and Relaxed Dirichlet Problems ..................... 277
Umberto Mosco
A Contribution to the Description of Natural States for Elastic Crystalline
So lids ................................................................... 295
Mario Pi tteri
Nonlocal Problems in Electromagnetism 311
Robert C. Rogers
Hyperbolic Aspects in the Theory of the Porous ~ledium Equation ........... 325
Juan Vazquez
Green's Formulas for Linearized Problems with Live Loads ................. 343
Giorgio Vergara-Caffarelli
Some Aspects of Adiabatic Shear Bands.................................... 353
T.W. Wright
Information About Other Volumes in this Program.......................... 373
FOREWORD
This IMA Volume in Mathematics and its Applications
Metastability and Incompletely Posed Problems
represents the proceedings of a workshop which was an integral part of the
19R4-R5 IMA program on CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EOIIATIONS.
We are grateful to the Scientific Committee:
,I.L. Eri cksen
D. Kinderlehrer
H. Rrezis
C. Dafermos
for their dedication and hard work in developing an imaginative, stimulating, and
productive year-long program.
George R. Sell
Hans Weinberger
Preface
Most equilibrium events in nature do not realize configurations of minimum
energy. They are only metastable. Available knowledge of constitutive relations
and environmental interactions may be limiterl. As a result, many configurations
may he compatible with the rlata. Such questions are incompletely poserl. The
papers in this volume address a wide variety of these issues as they are perceived
by the material scientist and the mathematician. They represent a portion of the
significant activity which has been underway in recent years, from the experimental
arena and physical theory to the analysis of differential equations and computation.
One may ask what circumstances tend to promote stahility or how does metastahility
manifest itself in nature. Traditionally, knowledge of a material or process is
achieved by observation and analysis of small departures from an equilihrium con
figuration. With i'ncreasing frequency, however, phenomena encountered display not
only larger but more dramatic changes, such as occur in phase transitions. Ry
grappling with specific problems, the authors provide a little experience from
which we might aspire to abstract viable methods for the analysis and prediction
of metastable behavior. There is a speculative character to much of this work,
partly because of problems stated hut not resolved.
These issues were among the principal themes of the 1984-19R5 I.M.A. Program,
Continuum Physics and Partial Oifferential Equations. Additional contrihutions to
this subject may be found in the other volumes of this series listerl at the end of
thi s book.
The workshop hrought together researchers in a numher of areas of engineering,
mathematics, and physics. The conference committee greatly appreciates the concerted
efforts of the speakers and discussants to make their presentations intelligihle
to a mixed audience. Included in this volume are papers connected to the topic of
the workshop which were presented at other times during the year.
