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\begin{center}
\begin{Large}
{\bf Abstracts - Main speakers}\bigskip\bigskip\\
E.~Acerbi,\\
University of Parma\smallskip\\
From $p$-laplacian to $p(x)$-laplacian
\end{Large}
\end{center}
We discuss various regularity results for equations or systems related to
the 
$p(x)$-laplacian operator $-\mbox{div}\;|Du|^{p(x)-2}Du$, including
elliptic or parabolic systems of the types arising from the theory of
electrorheological fluids, and equations of the type
$$-\mbox{div}\;|Du|^{p(x)-2}Du=-\mbox{div}\;|f|^{p(x)-2}f\;,$$
where $|f|^{q(x)}\in L^{1}$ for some function $q>p$, thereby proving
higher integrability of $Du$.
This is joint work with G.~Mingione.
\na
\begin{center}
\begin{Large}
Giovanni Alberti,\\
University of Pisa,\smallskip\\
Ginzburg-Landau functionals and the Plateau problem.\\
A variational approach.
\end{Large}
\end{center}
I will describe a $\Gamma$-convergence result for functionals of
Ginzburg-Landau type obtained in collaboration with S. Baldo and G. Orlandi.
The main consequence is the following: the Jacobian determinants of minimizers
with suitable Dirichlet conditions converge to minimal surfaces $M$ with
prescribed boundary, while the corresponding energy densities converge to the
area of $M$.
\na
\begin{center}
\begin{Large}
K.~Bhattacharya,\\
California Institute of Technology\smallskip\\
Free boundary problems in heterogeneous bodies
\end{Large}
\end{center}
We discuss the homogenization of some free boundary
problems that are motivated by the propagation of phase boundaries in
heterogeneous bodies.  This is based on joint work with Bogdan Craciun.
\na
\begin{center}
\begin{Large}
Andrea Braides,\\
University of Rome ``Tor Vergata'',\medskip\\
Multiple scale effects in the passage from discrete
to continuum variational problems
\end{Large}
\end{center}
The study of variational limits of discrete energies
allows to produce multi-scale phenomena from seemingly
inoffensive pair potentials. In this way, we may recover
continuum energies characteristic of theories of fracture,
softening, nonlinear elasticity, phase transitions,
non-local damage, etc., as first or second-order limit effects of
simple lattice theories as the lattice spacing tends to
zero. Conversely, the introduction of a new `microscopic
dimension' leads to original interesting phenomena, that often
require a continuum formulation different from the `classical'
ones.
\na
\begin{center}
\begin{Large}
Giuseppe Buttazzo,\\
University of Pisa\smallskip\\
Optimal networks in mass transportation problems
\end{Large}
\end{center}
Mass transportation problems deal with a metric space
$(X,d)$ and two probability measures $f^+$ and $f^-$ on
$X$. The cost functional $$F(d)=\inf\Big\{\int_{X\times
X}d(x,y)\,d\gamma(x,y)\ : \ \gamma\hbox{ has marginals
$f^+$ and $f^-$}\Big\}$$ can then be defined, and its value
depends on the distance $d$ we considered on $X$.\\
We give a model for the description of an urban
transportation network and we consider the related
optimization problem which consists in finding the design
of the network which has the best transportation
performances.
\na\newpage
\begin{center}
\begin{Large}
B. Dacorogna,\\
University of Lausanne\smallskip\\
Viscosity and almost everywhere solutions of first order
partial differential equations
\end{Large}
\end{center}
Consider the
differential inclusion $Du(x)$ belongs to $E$. We exhibit an
explicit solution that we call fundamental. It turns out to
be also a viscosity solution, when properly defining this
notion. Finally we consider a Dirichlet problem associated
to the differential inclusion and we give an iterative
procedure for finding a solution.
\smallskip\\
\begin{center}
\begin{Large}
G.~Dal Maso,\\
SISSA, Trieste\smallskip\\
Existence results for variational models in fracture mechanics
\end{Large}
\end{center}
The talk will present some recent existence results for the variational model of
crack growth in brittle materials proposed by Francfort and Marigo. These
results, obtained in collaboration with Francfort and Toader, deal with the
$n$-dimensional case, with a quasiconvex bulk energy and with time dependent
prescribed boundary deformations and time dependent applied loads.\smallskip\\
\begin{center}
\begin{Large}
Georg Dolzmann,\\
University of Maryland \smallskip\\
Variational problems in the analysis of solid-solid phase transformations
\end{Large}
\end{center}
Materials undergoing solid-solid phase transformations often have               
surprising                                                                      
mechanical properties as, for example, the shape memory effect.                 
In this talk, we study a remarkable consequence of the abundance of             
geometric                                                                       
patterns formed by the martensitic phases in a shape memory material.           
In fact, the material displays theoretically a fluid-like behavior in the       
sense that the set of all approximately energy free                             
affine deformations has an open interior with                                   
respect to the constraint of incompressibility. We also address regularity      
questions for generalized minimizers of non-convex variational problems         
describing these materials.\newpage
\begin{center}
\begin{Large}
Gero Friesecke,\\
Warwick\smallskip\\
Minimum energy configurations of large atomistic systems
\end{Large}
\end{center}
We discuss some examples of variational problems for large
particle systems arising in physics, including
\begin{itemize}
\item
a classical mechanics model for electrons in a large atoms
\item
a two-dimensional non-smooth interatomic force model introduced
by Radin to explain crystallization
\item a two-dimensional mass-spring model which can be proved to exhibit
crystallization (joint work with F.Theil, using a recent continuum
rigidity result obtained jointly with R.D.James and S.M\"uller).
\end{itemize}
\begin{center}
\begin{Large}
Adriana Garroni,\\
University of Rome ``La Sapienza''\smallskip\\
Gamma-limit of a variational model for dislocations
\end{Large}
\end{center}
We will study the asymptotic behaviour in terms of
$\Gamma$-convergence of a variational formulation for a phase field
model for dislocations. The energy is given by two main terms: the
Peierls energy obtained by an infinitely many wells potential and a
long range interaction elastic energy modelled by a non local term of
$H^{1/2}$ type.  To model a forest hardening we add zero boundary
condition on a perforated domain.  We analyse the asymptotic behaviour
for different scales.  One of the main tools is the introduction of
suitable notion of non local capacity and the corresponding cell
problem formula.\na
\begin{center}
\begin{Large}
R.~James,\\
University of Minnesota, Minneapolis\smallskip\\
The possible existence of some interesting\\
hybrid materials nominally forbidden by basic physics
\end{Large}
\end{center}
These speculations begin with the observation by
various scientists, probing new phenomena
through the use of first principles studies,
that the simultaneous occurrence of certain properties
is unlikely.  A prototype example is the simultaneous
occurrence in a single material of ferromagnetism and
ferroelectricity.  These studies typically do not consider
the  possibility of a first order phase transformation
(hence the word ``nominally'' in the
title), but there is a lot of indirect evidence
that, if the lattice parameters are allowed to change
significantly, then one might have co-existence of such
properties.  This seems to extend to diverse sets of
nominally incompatible properties.  Thus, one could try the
following: seek a reversible first order phase
transformation, necessarily also involving a distortion,
from, say, ferroelectric to ferromagnetic phases. If it
were highly reversible, there would be the extremely
interesting additional possibility of controlling the
volume fraction of phases with fields or stress. The key
point is reversibility.\\
Even big first order phase changes can be highly
reversible (liquid water to ice, shape memory materials),
and we argue that in solid materials it is the nature
of the shape change that is critical.  We suggest,
based on a close examination of measured
hysteresis loops in martensitic systems, that an idea based
on ``good fitting of the phases'' governs reversibility.
The mathematical status of this idea is not completely
clear, as I explain, but its possible connection with
quasiconvexity raises mathematical questions of fundamental
interest.
\na
\begin{center}
\begin{Large}
Robert V. Kohn,\\
Courant Institute, New York\smallskip\\
A continuum approach to structure of twist grain boundaries
\end{Large}
\end{center}
The structure of a twist-grain boundary is a lattice of
screw dislocations separating a periodic array of regions with good
registry. This structure is usually explained using atomic-scale models. 
My talk presents recent joint work with Antti Pihlaja (Merrill Lynch) 
exploring a different, continuum approach. We view the pattern as arising
from minimization of an ``incoherence energy''
$\int W(H) \, dx_1 \, dx_2$ where $H$ is the 
incoherency tensor of Cermelli and Gurtin. The rows of $H$ are
curl-free, so this resembles a problem of 2D elasticity. In particular,
when $W$ is not quasiconvex, minimizing sequences develop spatial patterns.
Our main contributions are (a) identification of the patterns and
Young measures associated with the standard picture of a twist grain 
boundary; and (b) identification of a specific energy density $W$
for which this pattern is optimal. The identification of a physically
plausible $W$ -- representing, perhaps, the continuum limit
of an atomic-scale model -- remains an open problem.%
\na
\begin{center}
\begin{Large}
Jan Mal\'y,\\
Charles University, Prague\smallskip\\
Scalar minimizers with fractal singular sets
\end{Large}
\end{center}
Lack of regularity of local minimizers for convex functionals with
non-standard growth  conditions is considered. It is shown
that there exists a function $a \in C^{\alpha}(\Omega)$ such that  the functional
$$
\mathcal{F}:u \mapsto \int_{\Omega} (|Du|^p + a(x)|Du|^q)\, dx
$$
admits a local minimizer $u \in W^{1,p}(\Omega)$ whose set of non-Lebesque
points is a closed set $\Sigma$ with Hausdorff dimension arbitrarily close to
$N-p$. Here $\Omega\subset\mathbb R^N$ is an open set and $1< p<N< N+\alpha<q<+\infty$.
This is joint work with Irene Fonseca and Giuseppe Mingione.\na
\begin{center}
\begin{Large}
Paolo Marcellini,\\
University of Florence\smallskip\\
Implicit partial differential equations
\end{Large}
\end{center}
With the title of \textit{Implicit partial differential equations}, we refer
to differential equations or systems of \textbf{first} order (we consider
systems of second order too) of the form
\begin{eqnarray*}
\left\{
\begin{array}{c}
F_{1}(x,u(x),Du(x))=0 \\
F_{2}(x,u(x),Du(x))=0 \\
\cdots \\
F_{I}(x,u(x),Du(x))=0%
\end{array}%
\right. ,\mbox{\ \ \ \ \ \ a.e. }x\in \Omega ,
\end{eqnarray*}
where $\Omega \subset \mathbb{R}^{n}$ is an open set, $u:\Omega \subset
\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$
and therefore $Du\in \mathbb{R}^{m\times n}$
(if $m=1$ we say that the problem is \textit{scalar} and otherwise we say
that it is \textit{vectorial}),
$F_{i}:\Omega \times \mathbb{R}^{m}\times \mathbb{R}^{m\times n}\rightarrow
\mathbb{R},$
$F_{i}=F_{i}(x,s,\xi ),$ $i=1,\ldots ,I,$ are given.\\
The equations that we will consider are called of \textit{implicit type},
since they exclude the quasilinear case, i.e., the case when the functions $%
F_{i}$ are linear with respect to the gradient $Du$.\\
We present some work in collaboration with Bernard Dacorogna; some
references are:\newpage
$\,$\\
B.Dacorogna and P.Marcellini, \textit{General existence theorems for
Hamilton-Jacobi equations in the scalar and vectorial case}, Acta
Mathematica, 178 (1997), 1--37.\smallskip\\
B.Dacorogna and P.Marcellini, \textit{Implicit partial differential equations%
}, Progress in Nonlinear Differential Equations and their Applications, 37,
Birkh\"{a}user, Boston, 1999.\smallskip\\
and more recently:\smallskip\\
B.Dacorogna and P.Marcellini, \textit{Viscosity solutions, almost everywhere
solutions and explicit formulas}, December 2002, preprint.
\newpage
\begin{center}
\begin{Large}
{\bf Abstracts - Short Communications}\bigskip\bigskip\\
Y. Altundas,\\
University of Minnesota
\smallskip\\
Parallel Computation of Phase Field Model and Comparision with 
the Experiment
\end{Large}
\end{center}
In this talk, we will present the results obtained from 3D
         calculation of a phase field model along with a movie of
         dendritic growth of a single needle crystal. The properties
         of dendritic growth such as interface growth velocity and the
         influence of anisotropy on the dendritic growth will be
         compared with the microgravity experiment.
			      tip radius of the dendrite will be compared\\
			      The results will be compared with microgravity
				     experiments for Succinonitrile.
\na
\begin{center}
\begin{Large}
M. Camar-Eddine,\\
University of Utah
\smallskip\\
Determination of the closure of the set of elasticity functionals
\end{Large}
\end{center}
We determine the closure for the Mosco-convergence 
in ${\rm L}^2(\Omega,{{\rm I}\!{\rm R}^3)}$ of the set of elasticity 
functionals. We prove that
this closure coincides with the set of all non-negative
lower-semicontinuous quadratic functionals which are objective,
i.e. which vanish for rigid motions. The result is still valid if we
consider only the set of isotropic elasticity functionals which have a
prescribed Poisson coefficient. This shows that a very large family of
materials can be reached when homogenizing a composite material with
highly contrasted rigidity coefficients.
\newpage
\begin{center}
\begin{Large}
A. Domokos,\\
University of Pittsburgh
\smallskip\\
$C^{1,\alpha}$-regularity
for the p-Laplacian\\in the Heisenberg group
\end{Large}
\end{center}
Let $\Omega$ be a bounded domain in the Heisenberg group
${\mathbb H}^n$ and let $p > 1$. Consider the p-Laplace equation:
\begin{equation}
\sum_{i=1}^{2n} X_i \left( |Xu|^{p-2} X_i u \right) = 0
 \, , \; \; \mbox{in} \; \; \Omega
\end{equation}
where $Xu = (X_1 u, ... , X_{2n} u)$ is the horizontal gradient of
$u$ and for $1 \leq i \leq n$
$$ X_i = \frac{\partial}{\partial x_i} - \frac{x_{n+i}}{2}
\frac{\partial}{\partial t} \, ,$$
$$ X_{n+i} = \frac{\partial}{\partial x_{n+i}} + \frac{x_i}{2}
\frac{\partial}{\partial t} \, $$
 are the left invariant vector fields corresponding to the
 canonical basis of the Lie algebra of  the Heisenberg group.
Let us denote by
$$ T = \frac{\partial}{\partial t} \, .$$
 We will discuss some recent results regarding
the regularity of the weak solutions of equation (1),
improving related results of H\"{o}rmander, Capogna and Marchi. \\
Our main contribution is an iteration scheme in the T-direction.
Along the way we prove a result on fractional difference quotients
that allows a more direct proof than the use of  Besov spaces and
opens the way to further improvements of our results.
\newpage
\begin{center}
\begin{Large}
L. Freddi,\\
University of Udine
\smallskip\\
A Young measure theory for martensitic thin films
via dimension reduction
\end{Large}
\end{center}
A variational limit defined on
the space of bi-dimensional gradient Young measures is
obtained from three-dimensional elasticity via dimension
reduction.  The obtained limit problem uniquely determines
the energy density of the thin film.  Our  result might be
used to compute the microstructure in membranes  made of
phase transforming material.\\
Preprint available
at the web address http://cvgmt.sns.it/papers/frepar03
\na
\begin{center}
\begin{Large}
N. Jung,\\
University of Illinois at Urbana-Champaign
\smallskip\\
 Continuity Properties of Distributional Determinant of the Hessian
\end{Large}
\end{center}
Motivated by the statisticians' problem of minimizing a functional
involving $\int|\det D^2u|$ in higher dimensional spaces,
I define the distributional determinant
of the Hessian on $BV^2\cap W^{1,\infty}(\R^3)$ which includes
possible minimizers of the functional, that is, piecewise linear
functions, and prove its continuity
by introducing the notion of strict convergence and minimal liftings.
\na
\begin{center}
\begin{Large}
V. Kucher,\\
University of San Diego
\smallskip\\
The second variation of a two-phase potential and some 
 necessary conditions for local minimizers at
  an interface
\end{Large}
\end{center}
  This study is motivated by a problem of classification of coexisting
    equilibrium deformations in nonlinear elasticity. A variational
    problem for the functional with two-phase energy density is
    considered. It is well-known that if a piecewise smooth local
    minimizer exists then, from the vanishingfirst variation, the
    minimizer satisfies the Weierstrass-Erdmann conditions at the
    surface of discontinuty of the gradient. A natural question is
    whether the second order necessary conditions at the interface
    exist. To take into account a variation of the interface both
    dependent and independent variables are altered. The corresponding
    formula for the second variation of the functional is
    obtained. The localization theorem leads to the integral
    inequality at internal points of the interface.  Some consequences
    of this inequality are discussed.  In particular, connection
    between the Weierstrass-Erdmann conditions and necessary
    conditions for the positivity of quadratic functionals obtained by
    H.Simpson and S.Spector is shown.  I am very grateful to Professor
    A.Freidin and Professor V.Osmolovskii for the advice and numerous
    discussions.
\na
\begin{center}
\begin{Large}
S. Levine,\\
Duquesne University, Pittsburgh 
\smallskip\\
Nonstandard Growth Functionals in Image Restoration
\end{Large}
\end{center}
We present a functional of $p(x)$ growth ($p(x)\geq 1$) for which the
    corresponding minimization problem provides a model for image
    restoration.  The model results in total variation based diffusion
    along object boundaries and isotropic diffusion in homogeneous
    regions.  At all other locations, the type of anisotropy varies
    according to the local image information.  Existence, uniqueness,
    and asymptotic behavior of the minimization problem and it's
    related flow are established.Experimental results show the
    effectiveness of the model in image restoration.
\na
\begin{center}
\begin{Large}
H. Merdan,\\
University of Pittsburgh
\smallskip\\
Renormalization group methods and regimes in interface
problems
\end{Large}
\end{center}
In this talk, we will discuss the temporal evolution of an interface
separating two phases for its large time behavior. The calculation
involves adaptation of methodology known in the physics literature as
renormalization group and scaling theory.\\
We will consider a full two-phased model in the quasi-static regime.
Implementing a renormalization procedure we will calculate the
characteristic length, $R(t)$, of a self-similar system that is the time
dependent length scale characterizing the pattern growth. When the dynamical
undercooling is non-zero $(\alpha \neq 0),$ we will show that $R(t)$
increases as $t^{-1/\lambda }$ while the total surface area of the
interface, $S\left( t\right) $, increases as $t^{(1-d)/\lambda }$, where 
$\lambda $ can take on values in the continuous spectrum, $\ [-3,-2]$. For 
$\alpha =0$ the spectrum is $[-3,0)$ so that the single value of $\lambda =-1$
is selected by the plane wave imposed by Jasnow and Vinals. It will be also
shown that the capillarity length arising from surface tension is irrelevant
for the large time behavior even though it has a crucial role at the early
stage evolution of an interface. At the end, the results will be compared
with those of Jasnow and Vinals, and Caginalp.
\na
\def\res{\mathbin{\vrule height9pt width.1pt\vrule height.1pt width9pt}}
\begin{center}
\begin{Large}
J. S. Moll,\\
University of Valencia
\smallskip\\
The Total Variation Flow with Measure Initial Conditions
\end{Large}
\end{center}
We consider the minimizing
total variation flow in $\R^N$
\begin{equation}
\label{evol_curvatura0} \displaystyle \frac{\partial u}{\partial
t} = {\rm div} \left(\frac{Du}{|Du|}\right) \qquad {\rm in} ~ Q_T
= ]0,T[ \times \R^N,
\end{equation}
coupled with the initial condition
\begin{equation}
\label{evol_curvatura0datoin} \displaystyle u(0) = \mu, \qquad \;
\; \hbox{$\mu$ \ being a Radon measure in $\R^N$.}
\end{equation}
This PDE appears (in a bounded domain $\Omega$) in the steepest descent
method for minimizing the total variation, a method introduced by
L. Rudin, S. Osher and E. Fatemi in the context
of image denoising and reconstruction.\medskip\\
 In previous papers thes gradient descent flow
(\ref{evol_curvatura0})  has been studied in a bounded domain
under Neumann  and Dirichlet boundary conditions, obtaining existence and uniqueness results
for initial data in $L^1(\Omega)$. Our aim
is to continue the study of the flow (\ref{evol_curvatura0}) when the initial conditions are
Radon measures in $\R^N$. We shall not consider general measures, instead we shall restrict
ourselves to the case of measures
\begin{equation}
\label{quinesmesures} \mu = h + \alpha {\cal H}^k\res S,
\end{equation}
where $h \in L^1(\R^N) \cap L^{\infty}(\R^N)$, $\alpha \geq 0$,
 ${\cal H}^k$ is the $k$-dimensional Hausdorff measure in
$\R^N$ and $S$ is a $k$-manifold in $\R^N$ of class  $W^{3,
\infty}$.\medskip\\
We study limit
solutions obtained by weakly approximating the initial measure
$\mu$ by functions in $L^1(\R^N)$, characterizing these
limit solutions when the initial condition $\mu = h + \mu_s$ where
$h \in L^1(\R^N) \cap L^{\infty}(\R^N)$, and $\mu_s = \alpha {\cal
H}^k\res S$, $\alpha \geq 0$, $k$ is an integer and $S$ is a
$k$-dimensional manifold with bounded principal curvatures. In
case $k < N-1$ we prove that the singular part of the solution
does not move, it remains equal to $\mu_s$ for all $t\geq 0$ and its
absolutely continuous part is the unique strong solution of the Cauchy problem associated
with (\ref{evol_curvatura0}) with initial datum $h$.  In particular, $u(t) = \delta_0$ when
$u(0) =
\delta_0$. In case
$k=N-1$ we prove that the singular part of the limit solution is
$(1-\frac{2}{\alpha}t)^+ \mu_s$ and we also characterize its
absolutely continuous part.\\
This is joint work with F. Andreu (University of Valencia), V. Caselles (University of Pompeu-Fabra) and
J. M. Maz\'on (University of Valencia).
\na
\begin{center}
\begin{Large}
I. Nikolova,\\
University of Pittsburgh 
\smallskip\\
 Level Set-like Method Applied to Solving Markstein (Eiconal) Equation
\end{Large}
\end{center}
Phenomena of moving surfaces arises in many areas, such as fluid mechanics, combustion, material science [1-3]. Recently, a paper on methods to study interfaces, [1] drew our attention. Among the methods describing moving surfaces are different geometry methods in combination with functional analysis approach, which is the one adopted in [2]. The method treats the flame as a surface of discontinuity, given by the function $(x,y,t)=const$. The surface propagates (advances) in a normal direction with a speed F. The formulation of the idea, given in [1] is very close, if not the same as the one given in [2], but was published  8 years later.
    In this paper we derive the eiconal equation, which  is known as Markstein equation in combustion  community. This equation describes the flame front propagation, $=const.$ Equation for the evolution of 
 is solved by upwind finite differences method. In modern literature [3, 4]  the method is referred to as ``level set method''. The front position at the time $t$ is given by the zero level set of evolving function. 
    Results on the flame front position in vortex structured flow fields are presented illustrating the features of the method applied to this problem. Depending on the type of the flow field, possibility for the flame to be ``trapped'' arises.\smallskip\\
References:\\
1. Seithian, J.A, J. of Computational Physics, 169, 503-555(2001)\\
2. Nikolova, I.P. in 25 Symp. Int. on Combustion, 1994, Irvine, CA. Proceedings\\
3. Osher, S, and Sethian, J.A., J. Computational Physisc, 79, 12 (1988).\\
4. Sethian, J.A., Level set methods, Cambridge University Press, UK, 1999\smallskip\\
Acknowledgements: This study was inspired by my visit to the University of Bristol in the frame of the EC program ``Go West''.  For the references and discussions on the topic I would like to thank my former colleague and friend Sonia Tabakova from the Technical University of Plovdiv.
\na
\begin{center}
\begin{Large}
G. Pisante,\\
University of Lausanne
\smallskip\\
Existence of viscosity solution -- Sufficient conditions for the existence of viscosity solutions for non convex Hamiltonians
\end{Large}
\end{center}
We study the Dirichlet problem for the Hamilton-Jacobi
equation
\begin{equation}\label{uno}
\left\{
  \begin{array}{ccccc}
    F(Du) & = & 0 & \mbox{in} & \Omega \\
    u & = & \varphi & \mbox{on} & \partial\Omega
  \end{array}
\right.
\end{equation}
where $\Omega \subset \Real^n$ is a bounded open set, $F:\Real^n\to \Real$ is
continuous and $\varphi:\partial\Omega \to \Real$ is Lipschitz. \\
Under some technical assumptions on $F$, we give a sufficient geometric condition for the existence of Lipschitz viscosity solutions of problem (\ref{uno}). In particular we show an Hopf-Lax type extension formula in the case where we drop the classical hypotheses of convexity and regularity on $F$ and $\Omega$.
\na
\begin{center}
\begin{Large}
C. Rios,\\
Mc Master University
\smallskip\\
A higher dimensional partial Legendre transform, and regularity of
degenerate Monge-Amp\'ere equations
\end{Large}
\end{center}
In dimension $n\geq 3$, we define a generalization of the classical two
dimensional partial Legendre transform, that reduces interior regularity
of the generalized Monge-Amp\'ere equation $\det D^{2}u=k\left(
x,u,Du\right) $ to regularity of a divergence form quasilinear system of
special form. This is then used to obtain smoothness of $C^{2,1}$
solutions, having $n-1$ nonvanishing principal curvatures, to certain
subelliptic Monge-Amp\'ere equations in dimension $n\geq 3$. A
corollary is that if $k\geq 0$ vanishes only at nondegenerate critical
points, then a $C^{2,1}$ convex solution $u$ is smooth if and only if the
symmetric function of degree $n-1$ of the principal curvatures of $u$ is
positive, and moreover, $u$ fails to be $C^{3,1-\frac{n}{2}+\varepsilon }$
when not smooth.\\
This is joint work with Eric Sawyer, McMaster University, and Richard Wheeden,
Rutgers University.
\na
\begin{center}
\begin{Large}
A. Tudorascu,\\
Carnegie Mellon University
\smallskip\\
General Fokker-Planck Equations via Monge-Kantorovich-Theory
\end{Large}
\end{center}
The generic Fokker-Planck equation describes the evolution of the
probability density for a stochastic process associated with an Ito
stochastic differential equation. Here we employ elements of the
Monge-Kantorovich mass transfer theory to study general (possibly
degenerate) Fokker-Planck equations.
\na
\begin{center}
\begin{Large}
C. Wang,\\
University of Kentucky
\smallskip\\
The Aronsson equation for absolute minimizer of L-infinity functional associated with
vector fields satisfying Hormander's condition
\end{Large}
\end{center}
In this talk, I will discuss the Aronsson
equation for absolute minimizer, i.e.\ local minimizers,
of certain $L^\infty$ functionals with respect to the
CC metric space generated by Hormander's vector fields and
its Euler-Aronsson equation and its uniqueness
when the CC metric is induced from a Carnot group.
\na
\begin{center}
\begin{Large}
S. Watson,\\
Northwestern University, Chicago
\smallskip\\
Coarsening dynamics of faceted crystal surfaces
\end{Large}
\end{center}
The coarsening properties of a faceted crystal surface {\em annealing} 
		    (in equilibrium) with it's melt is known
		    experimentally  
		    to change dramatically if,	
		    instead, the crystal is subject to net growth. 
		    We present a theory which explains this transition,
		    and exhibit the main ideas in the setting where
		    {\em attachment kinetics} is the dominant mass
		    transfer mechanism.  Here, the {\em
		    faceting-Cahn-Hilliard} ({\bf $\mathcal{ FCH}$})
		    equation is the annealing model, while the {\em
		    faceting-eikonal-Cahn-Hilliard} ({\bf
		    $\mathcal{FECH}$}) equation is the associated
		    growth model.  We identify the sharp-interface
		    theory of {\bf $\mathcal{FECH}$} through a matched
		    asymptotic analysis.  The result is a novel
		    coarsening dynamical system ({\bf
		    $\mathcal{CDS}$}) for the edge network of the
		    faceted surface, where coarsening occurs through
		    the merging and annihilation of facets.  In the
		    case of square symmetry, we prove for {\bf
		    $\mathcal{CDS}$} that
		    \begin{itemize} 
		    \item The characteristic length $\mathcal{L}(t)$ of
		    the faceted surface 
		    scales like $t^{1/2}$. 
		    \item {\em Pyramids} are stable and {\em
		    anti-pyramids} are unstable.  
		    \end{itemize} 
		    Direct numerical simulations of {\bf
		    $\mathcal{FECH}$} also yield this scaling 
		    law and confirm the stability-instability dichotomy
		    through the absence of  
		    anti-pyramids in the surface evolution.  
		    Hence, our results capture the  {\em annealing to
		    growth transition} 
		    of the coarsening dynamics since, for {\bf
		    $\mathcal{FCH}$},  
		    $\mathcal{L}(t) \sim t^{1/4}$ and also both  
		    pyramids and anti-pyramids appear in the evolution.\\
		    The one-dimensional {\bf $\mathcal{FECH}$}, which
		    is equivalent to the  
		    convective Cahn-Hilliard  {\bf $\mathcal{CCH}$}
		    equation, 
		    displays unusual coarsening mechanisms. 
		    We prove that  
		    \begin{itemize} 
		    \item Binary coalescence of {\em phase boundaries}
		    is impossible 
		    \item Coarsening may only occur through a specific 
		    coalescence of three phase boundaries; the {\em
		    kink-ternary}. 
		    \end{itemize} 
		    These properties stand in marked contrast with 
		    with the generic binary coalescence of the 1-d
		    Cahn-Hilliard equation.\\
		    Motivated by a linear stability analysis we also
		    propose  
		    a length-scale-doubling coarsening ansatz for the  
		    {\bf $\mathcal{FECH}$} evolution, from which we
		    identify the  
		    scaling constant $C$ of the $t^{1/2}$ scaling
		    regime; e.g., 
		     $\mathcal{L}(t) = C t^{1/2} $. This work is, in
		    part,  
		    joint with Felix Otto (Bonn)
\newpage
\begin{center}
\begin{Large}
Q. Xia,\\
University of Texas
\smallskip\\
Variational problems in the intersection homology theory of singular varities
\end{Large}
\end{center}
In this talk, I am going to discuss ``soap films'' or ``minimal
surfaces'' that live in singular spaces such as singular complex
projective varieties.
On complex projective varieties, the intersection homology theory
introduced by MacPherson and Goresky is more suitable than ordinary
homology theory.  I will first discuss how to express the intersection
homology groups in terms of integer multiplicity rectifiable currents.
These are then isomorphic to the usual intersection homology groups
defined by geometric or subanalytic chains with the corresponding
perversity conditions. After that, I will discuss the existence and
regularity of modified mass minimizer in every intersection homology
class on a compact stratified subanalytic pseudomanifold. My modified
mass minimizers may intersect the singular locus of the singular variety
in a controlled fashion.
\na
\begin{center}
\begin{Large}
X. Yao,\\
Texas A\&M Univ.
\smallskip\\
A Minimax Method for Finding the Eigenpairs of Nonlinear Operators
\end{Large}
\end{center}
A minimax algorithm for finding the eigenpairs of nonlinear operators in Banach
spaces will be given in this talk.  As a preliminary result, the following type
of variational eigenpair problems will be considered. Find eigenpairs
$(\lambda, u)\in {\cal R}\times B$ such that $\|u\|=1$
and
\[
F'u=\lambda G'u
\]
where $B$ is a Banach space and $F,G \in C^{1}(B,R)$
satisfy the homogeneous conditions
\[
F(tu)=t^{k}F(u)\quad\mbox{and}\quad G(tu)=t^{k}G(u)\quad\forall t>0,\;
\mbox{for some }\;k\neq 0.
\]
As examples, the eigenpairs of the nonlinear {\em p-Laplacian} and
{\em pseudo-p-Laplacian} operators will be studied and their numerical results
 will be presented.\\
 This is a joint work with Professor J.~Zhou of Texas A\&M University.
\na
\begin{center}
\begin{Large}
J. Zimmer,\\
California Institute of Technology\smallskip\\
On the computation of nonconvex hulls
\end{Large}
\end{center}
An open problem in the calculus of variations is the computation of
the quasiconvex envelope of a given set of matrices. Also, main
results in relaxation theory rely on the quasiconvex envelope of a
function. Such a envelope or hull, however, is known only for very
specific cases.\\
Therefore, we want to study two related, but simpler, notions of
convexity, namely separate convexity and rank-1-convexity. We present
a graph-theoretical method for the computation of the separate convex
envelope of a finite set of matrices.\\
For rank-1-convexity, we want to answer the following question: Given
a finite set of matrices, does it contain a Tartar square? Tartar
squares are known to be a main obstruction for nontrivial
rank-1-convex hulls, and indeed new results indicate that the
algorithm presented in the talk indeed can show triviality of the
rank-1-convex hull for a large class of configurations. Unlike
previous approaches, we do not build on numerical methods (in the
sense of discretization). Rather, we use methods of computational
algebraic geometry to avoid mistakes due to rounding errors and get
exact results in a few seconds.
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