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This IMA Volume in Mathematics and its Applications AMORPHOUS
POLYMERS AND NON-NEWTONIAN FLUIDS is in part the proceedings of a
workshop which was an integral part of the 1984-85 IMA program on
CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS We are
grateful to the Scientific Committee: Haim Brezis Constantine
Dafermos Jerry Eri cksen David Kinderlehrer for planning and
implementing an exciting and stimulating year-long program. We espe
cially thank the Program Organizers, Jerry Ericksen, David
Kinderlehrer, Stephen Prager and Matthew Tirrell for organizing a
workshop which brought together scientists and mathematicians in a
variety of areas for a fruitful exchange of ideas. George R. Sell
Hans Weinberger Preface The diversity of experimental phenomena and
the range of applications of liquid crystals present timely and
challenging questions for experimentalists, mechanists, and
mathematicians. The scope of this workshop was to bring together
research workers and practitioners in these areas from
laboratories, industry, and universities to explore common issues.
The contents of this volume vary from descriptions of experimental
phenomena, of which our understanding is insufficient, to questions
of a mathematical nature and of efficient computation."
This IMA Volume in Mathematics and its Applications AMORPHOUS
POLYMERS AND NON-NEWTONIAN FLUIDS is in part the proceedings of a
workshop which was an integral part of the 1984-85 IMA program on
CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS We are
grateful to the Scientific Committee: Haim Brezis Constantine
Dafermos Jerry Ericksen David Kinderlehrer for planning and
implementing an exciting and stimulating year-long program. We espe
cially thank the Program Organizers, Jerry Ericksen, David
Kinderlehrer, Stephen Prager and Matthew Tirrell for organizing a
workshop which brought together scientists and mathematicians in a
variety of areas for a fruitful exchange of ideas. George R. Sell
Hans Weinberger Preface Experiences with amorphous polymers have
supplied much of the motivation for developing novel kinds of
molecular theory, to try to deal with the more significant features
of systems involving very large molecules with many degrees
offreedom. Similarly, the observations of many unusual macroscopic
phenomena has stimulated efforts to develop linear and nonlinear
theories of viscoelasticity to describe them. In either event, we
are confronted not with a well-established, specific set of
equations, but with a variety of equations, conforming to a loose
pattern and suggested by general kinds of reasoning. One challenge
is to devise techniques for finding equations capable of delivering
definite and reliable predictions. Related to this is the issue of
discovering ways to better grasp the nature of solutions ofthose
equations showing some promise.
This IMA Volume in Mathematics and its Applications Homogenization
and Effective Moduli of Materials and Media represents the
proceedings of a workshop which was an integral part of the 19R4-R5
IMA program on CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL
EQUATIONS. We are grateful to the Scientific Committee: J . L.
Ericksen D. Kinderlehrer H. Brezis C. Dafermos for their dedication
and hard work in rleveloping an imaginative, stimulating, and
productive year-long program. George R. Sell Hans Weinherger
PREFACE The papers in this volume were presented at a workshop on
homogenization of differential equations and the determination of
effective moduli of materials and media, primarily in the context
of continuum theory. These areas are closely linked to a variety of
phenomena, such as the elastic and dielectric responses of
composites, and the effective properties of shales and soils. For
instance, the ability to predict the effective stiffness response
of a composite across a broad range of frequencies allows its
performance under given circumstances to be assessed by means of
nondestructive testing. A fundamental mathematical tool is
homogenization, the study of partial differential equations with
rapidly varying coefficients or boundary conditions. The recent
alliance of homogenization with optimal design has stimulated the
development of both fields. The presentations at the workshop
emphasized recent advances and open questions.
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.
The work of Hans Lewy (1904--1988) has had a profound influence in
the direction of applied mathematics and partial differential
equations, in particular, from the late 1920s. Two of the
particulars are well known. The Courant--Friedrichs--Lewy condition
(1928), or CFL condition, was devised to obtain existence and
approximation results. This condition, relating the time and
spatial discretizations for finite difference schemes, is now
universally employed in the simulation of solutions of equations
describing propagation phenomena. Lewy's example of a linear
equation with no solution (1957), with its attendant consequence
that most equations have no solution, was not merely an unexpected
fact, but changed the viewpoint of the entire field. Lewy made
pivotal contributions in many other areas, for example, the
regularity theory of elliptic equations and systems, the Monge--
AmpSre Equation, the Minkowski Problem, the asymptotic analysis of
boundary value problems, and several complex variables. He was
among the first to study variational inequalities. In much of his
work, his underlying philosophy was that simple tools of function
theory could help one understand the essential concepts embedded in
an issue, although at a cost in generality. This approach was
extremely successful. In this two-volume work, most all of Lewy's
papers are presented, in chronological order. They are preceded by
several short essays about Lewy himself, prepared by Helen Lewy,
Constance Reid, and David Kinderlehrer, and commentaries on his
work by Erhard Heinz, Peter Lax, Jean Leray, Richard MacCamy,
Fran?ois Treves, and Louis Nirenberg. Additionally, there are
Lewy's own remarks on the occasion of his honorary degree from the
University of Bonn.
This IMA Volume in Mathematics and its Applications MICROSTRUCTURE
AND PHASE TRANSITION is based on the proceedings of a workshop
which was an integral part of the 1990-91 IMA program on "Phase
Transitions and Free Boundaries." We thank R. Fosdick, M.E. Gurtin,
W.-M. Ni and L.A. Peletier for organizing the year-long program
and, especially, D. Kinderlehrer, R. James, M. Luskin and J.
Ericksen for organizing the meeting and editing these proceedings.
We also take this opportunity to thank those agencies whose
financial support made the workshop possible: the Army Research
Office, and the National Science Foun dation. A vner Friedman
Willard Miller. Jr. PREFACE Much of our traditional knowledge of
materials and processes is achievf'd by observa tion and analysis
of small departures from equilibrium. Many materials, especially
modern alloys, ceramics, and their composites, experience not only
larger but more dramatic changes, such as the occurrence of phase
transitions and t.he creation of defect structures, when viewed at
the microscopic scale. How is this observed, how can it be
interpreted, and how does it influence macroscopic behavior? These
are the principle concerns of this volume, which constitutes the
proceedings of an IMA workshop dedicated to these issues.
A traditional way to honor distinguished scientists is to combine
collections of papers solicited from friendly colleagues into
dedicatory volumes. To honor our friend and colleague Mort Gurtin
on the occasion of his sixty-fifth birthday, we followed a surer
path to produce a work of intrinsic and lasting scientific value:
We collected pa pers that we deemed seminal in the field of
evolving phase interfaces in solids, a field to which Mort Gurtin
himself has made fundamental contributions. Our failure for lack of
space to include in this volume every paper of major significance
is mitigated by the ma gisterial introduction prepared by Eliot
Fried, which assesses the contributions of nu merous works. We hope
that this collection will prove useful and stimulating to both
researchers and students in this exciting field. August 1998 JohnM.
Ball David Kinderlehrer Paulo Podio-Guidugli Marshall Slemrod
Contents Introduction: Fifty Years of Research on Evolving Phase
Interfaces By Eliot Fried. 0
************************************************ 0 ***** 1 I.
Papers on Materials Science Surface Tension as a Motivation for
Sintering By C. Herring 33 Two-Dimensional Motion of Idealized
Grain Boundaries By W. W. Mullins 0 *********** 0
******************* 70 Morphological. Stability of a Particle
Growing by Diffusion or Heat Flow By w. w. Mullins and R. F.
Sekerka 75 Energy Relations and the Energy-Momentum Tensor in
Continuum Mechanics By J. D. Eshelby 82 The Interactions of
Composition and Stress in Crystalline Solids By F. e. Larche and 1.
W. Cahn 120 II.
The work of Hans Lewy (1904-1988) has had a profound influence in
the direc tion of applied mathematics and partial differential
equations, in particular, from the late 1920s. We are all familiar
with two of the particulars. The Courant-Friedrichs Lewy condition
(1928), or CFL condition, was devised to obtain existence and ap
proximation results. This condition, relating the time and spatial
discretizations for finite difference schemes, is now universally
employed in the simulation of solutions of equations describing
propagation phenomena. His example of a linear equation with no
solution (1957), with its attendant consequence that most equations
have no solution, was not merely an unexpected fact, but changed
the viewpoint of the entire field. Lewy made pivotal contributions
in many other areas, for example, the regu larity theory of
elliptic equations and systems, the Monge-Ampere Equation, the
Minkowski Problem, the asymptotic analysis of boundary value
problems, and sev eral complex variables. He was among the first to
study variational inequalities. In much of his work, his underlying
philosophy was that simple tools of function theory could help us
understand the essential concepts embedded in an issue, although at
a cost in generality. This was extremely successful."
The work of Hans Lewy (1904--1988) has had a profound influence in
the direction of applied mathematics and partial differential
equations, in particular, from the late 1920s. Two of the
particulars are well known. The Courant--Friedrichs--Lewy condition
(1928), or CFL condition, was devised to obtain existence and
approximation results. This condition, relating the time and
spatial discretizations for finite difference schemes, is now
universally employed in the simulation of solutions of equations
describing propagation phenomena. Lewy's example of a linear
equation with no solution (1957), with its attendant consequence
that most equations have no solution, was not merely an unexpected
fact, but changed the viewpoint of the entire field. Lewy made
pivotal contributions in many other areas, for example, the
regularity theory of elliptic equations and systems, the
Monge--AmpA]re Equation, the Minkowski Problem, the asymptotic
analysis of boundary value problems, and several complex variables.
He was among the first to study variational inequalities. In much
of his work, his underlying philosophy was that simple tools of
function theory could help one understand the essential concepts
embedded in an issue, although at a cost in generality. This
approach was extremely successful. In this two-volume work, most
all of Lewy's papers are presented, in chronological order. They
are preceded by several short essays about Lewy himself, prepared
by Helen Lewy, Constance Reid, and David Kinderlehrer, and
commentaries on his work by Erhard Heinz, Peter Lax, Jean Leray,
Richard MacCamy, FranAois Treves, and Louis Nirenberg.
Additionally, there are Lewy's own remarks on the occasion of his
honorarydegree from the University of Bonn.
The work of Hans Lewy (1904--1988) has had a profound influence in
the direction of applied mathematics and partial differential
equations, in particular, from the late 1920s. Two of the
particulars are well known. The Courant--Friedrichs--Lewy condition
(1928), or CFL condition, was devised to obtain existence and
approximation results. This condition, relating the time and
spatial discretizations for finite difference schemes, is now
universally employed in the simulation of solutions of equations
describing propagation phenomena. Lewy's example of a linear
equation with no solution (1957), with its attendant consequence
that most equations have no solution, was not merely an unexpected
fact, but changed the viewpoint of the entire field. Lewy made
pivotal contributions in many other areas, for example, the
regularity theory of elliptic equations and systems, the Monge--
AmpSre Equation, the Minkowski Problem, the asymptotic analysis of
boundary value problems, and several complex variables. He was
among the first to study variational inequalities. In much of his
work, his underlying philosophy was that simple tools of function
theory could help one understand the essential concepts embedded in
an issue, although at a cost in generality. This approach was
extremely successful. In this two-volume work, most all of Lewy's
papers are presented, in chronological order. They are preceded by
several short essays about Lewy himself, prepared by Helen Lewy,
Constance Reid, and David Kinderlehrer, and commentaries on his
work by Erhard Heinz, Peter Lax, Jean Leray, Richard MacCamy,
Fran?ois Treves, and Louis Nirenberg. Additionally, there are
Lewy's own remarks on the occasion of his honorary degree from the
University of Bonn.
This IMA Volume in Mathematics and its Applications AMORPHOUS
POLYMERS AND NON-NEWTONIAN FLUIDS is in part the proceedings of a
workshop which was an integral part of the 1984-85 IMA program on
CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS We are
grateful to the Scientific Committee: Haim Brezis Constantine
Dafermos Jerry Ericksen David Kinderlehrer for planning and
implementing an exciting and stimulating year-long program. We espe
cially thank the Program Organizers, Jerry Ericksen, David
Kinderlehrer, Stephen Prager and Matthew Tirrell for organizing a
workshop which brought together scientists and mathematicians in a
variety of areas for a fruitful exchange of ideas. George R. Sell
Hans Weinberger Preface Experiences with amorphous polymers have
supplied much of the motivation for developing novel kinds of
molecular theory, to try to deal with the more significant features
of systems involving very large molecules with many degrees
offreedom. Similarly, the observations of many unusual macroscopic
phenomena has stimulated efforts to develop linear and nonlinear
theories of viscoelasticity to describe them. In either event, we
are confronted not with a well-established, specific set of
equations, but with a variety of equations, conforming to a loose
pattern and suggested by general kinds of reasoning. One challenge
is to devise techniques for finding equations capable of delivering
definite and reliable predictions. Related to this is the issue of
discovering ways to better grasp the nature of solutions ofthose
equations showing some promise."
Articles in this volume are based on lectures presented at the Park
City summer school on ``Mathematics and Materials'' in July 2014.
The central theme is a description of material behavior that is
rooted in statistical mechanics. While many presentations of
mathematical problems in materials science begin with continuum
mechanics, this volume takes an alternate approach. All the
lectures present unique pedagogical introductions to the rich
variety of material behavior that emerges from the interplay of
geometry and statistical mechanics. The topics include the
order-disorder transition in many geometric models of materials
including nonlinear elasticity, sphere packings, granular
materials, liquid crystals, and the emerging field of synthetic
self-assembly. Several lectures touch on discrete geometry
(especially packing) and statistical mechanics. The problems
discussed in this book have an immediate mathematical appeal and
are of increasing importance in applications, but are not as widely
known as they should be to mathematicians interested in materials
science. The volume will be of interest to graduate students and
researchers in analysis and partial differential equations,
continuum mechanics, condensed matter physics, discrete geometry,
and mathematical physics.
This unabridged republication of the 1980 text, an established
classic in the field, is a resource for many important topics in
elliptic equations and systems and is the first modern treatment of
free boundary problems. Variational inequalities (equilibrium or
evolution problems typically with convex constraints) are carefully
explained in An Introduction to Variational Inequalities and Their
Applications. They are shown to be extremely useful across a wide
variety of subjects, ranging from linear programming to free
boundary problems in partial differential equations. Exciting new
areas like finance and phase transformations along with more
historical ones like contact problems have begun to rely on
variational inequalities, making this book a necessity once again.
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