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{\Large
Contemporary Developments in Partial Differential Equations and in the
Calculus of Variations}
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{\large{Organizers: Irene Fonseca  and Paolo Marcellini}}
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{\bf {\Large Schedule}
}
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\large{

9:00 -  9:35 --  D. Kinderlehrer

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9:40 - 10:15 -- G. Buttazzo

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10:50 - 11:25 -- G. Dolzmann

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11:30 - 12:05 -- G. Leoni

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12:10 - 12:45 -- G. Bellettini

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15:00 - 15:35 -- G. Dal Maso

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15:40 - 16:15 -- I. Fonseca

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16:50 - 17:25 -- C. Larsen

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17:30 - 18:05 -- P. Tilli


}


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\begin{center}
{\Large
Contemporary Developments in Partial Differential Equations and in the
Calculus of Variations}
\hfill\\
\hfill\\
{\large{Organizers: Irene Fonseca  and Paolo Marcellini}}
\hfill\\
\hfill\\
{\bf {\Large Abstracts}}
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\begin{center}
{\bf Some results on anisotropic geometric evolution problems\\
\hfill\\
{\it Giovanni Bellettini}}
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\noindent We will deal with the problem of motion by mean curvature
of a surface in presence of a possibly nonsmooth anisotropy.
In particular, we will focus the attention
on the evolution by crystalline mean curvature in three
dimensions. The breaking and bending of a facet under
this geometric flow will be discussed.

\bigskip

Giovanni Bellettini

Dipartimento di Matematica

Universit\'a di Roma ``Tor Vergata''

via della Ricerca Scientifica

00133 Roma, Italy

\email{belletti@mat.uniroma2.it}



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\begin{center}
{\bf Some results and questions in mass optimization problems\\
\hfill\\
{\it Giuseppe Buttazzo}
}
\end{center}


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\noindent We consider the optimization problem
$$\max\big\{{\cal E}(f,\Sigma,\mu)\ :
\ \mu\in{\cal M}^+(m,\overline\Omega)\big\}$$
where ${\cal E}(f,\Sigma,\mu)$ is the energy associated to $\mu$:
$${\cal E}(f,\Sigma,\mu)=\inf\big\{{1\over2}\int|Du|^2\,d\mu-\langle
f,u\rangle\ :\ u\in{\cal D}({\bf R}^n),\ u=0\hbox{ on }\Sigma\big\}.$$
The datum $f$ is a signed measure with finite total variation (and zero
average if $\Sigma$ is empty), while $\Omega$ is an open subset of 
${\bf R}^n$, $\Sigma$ is a closed subset of $\overline\Omega$, and
${\cal M}^+(m,\overline\Omega)$ is the class of nonnegative measures on
$\overline\Omega$ with mass $m$. We show that the optimization problem
above admits a solution which is a measure, not in $L^1(\Omega)$ in 
general.
This solution comes out by solving a mass transportation problem, for a
suitable distance $d_{\Omega,\Sigma}$ and with the associated
Monge-Kantorovich PDE equation of the form
$$
\left\{
\begin{array}{l}
-{\rm div}\big(\mu D_\mu w\big)=f
\quad\hbox{on }{\bf R}^n\setminus\Sigma\\
 w\in{\rm Lip}_1(\Omega,\Sigma),
\quad|D_\mu w|=1\hbox{ $\mu$-a.e.},\quad\mu(\Sigma)=0
\end{array}
\right.
$$
where ${\rm Lip}_1(\Omega,\Sigma)$ is the class of all Lipschitz functions
on $\Omega$ with constant 1, which vanish on $\Sigma$.

We also address some problems occurring in the optimization problems
which arise when we consider the Dirichlet region $\Sigma$ as an unknown. 


\bigskip

Giuseppe Buttazzo

Dipartimento di Matematica

Via Buonarroti, 2

56127 Pisa, Italy

\email{buttazzo@vaxsns.sns.it}

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\begin{center}
{\bf Neumann problems in domains with cracks and applications to fracture mechanics\\
\hfill\\
{\it Gianni Dal Maso}
}
\end{center}


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\noindent We consider solutions of nonlinear elliptic equations in domains of the form
${\Omega\setminus K}$, where $\Omega$ in
a two dimensional smooth domain and $K$ is a compact subset
of $\overline \Omega$ with a finite number of connected components.
The solutions are required to satisfy a homogeneous
Neumann boundary condition on $K$ and a nonhomogeneous Dirichlet
condition on ${\partial\Omega\setminus K}$. The main result is the
continuous dependence of the solution on $K$, with respect to the
Hausdorff metric, provided that the number of connected components
of $K$ is uniformly bounded.

This stability result is used to give a rigorous mathematical
formulation of a variational quasistatic
model for the slow growth of  brittle fractures, introduced
by Francfort and Marigo. Starting from a discrete-time formulation,
a more satisfactory continuous-time formulation is
obtained, with full justification of the convergence arguments.


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Gianni Dal Maso

SISSA, Classe di Matematica

Via Beirut 2-4

34014 Miramare Grignano, Trieste, Italy

\email{dalmaso@neumann.sissa.it}

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\begin{center}
{\bf Nematic elastomers\\
\hfill\\
{\it Georg Dolzmann}
}
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\noindent A modern trend in the calculus of variations which originates from
applications in physics and materials science is to try to identify in a
mathematically rigorous way limiting theories for complex systems. Often a
lot of information about a given system can be obtained by analyzing a
coarse-grained energy which describes the macroscopic behavior without
resolving the finest length scale in the Integral representation and $\Gamma$-convergence of variational integrals
with $p(x)$-growth

written in collaboration with Domenico Mucci, and submitted for
publication in ESAIM:COCV, has entered the refereeing process.
system. In this talk we discuss
nematic elastomers as a model system for which the coarse-grained energy
can be found analytically. We discuss the surprising predictions that
result from this description.


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Georg Dolzmann

Mathematics Department

University of Maryland

College Park, MD 20742-4015, USA


\email{dolzmann@math.umd.edu}

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\begin{center}
{\bf On Det versus det\\
\hfill\\
{\it Irene Fonseca}
}
\end{center}


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\noindent Sharp results for weak convergence of the determinant jacobian are given using carefully crafted isoperimetric inequalities. It is shown that if $u_n \in W^{1,N}(\Omega;\mathbb R^N)$, $u_n \rightharpoonup u$ in $W^{1,N-1}(\Omega;\mathbb R^N)$, where $\Omega$ is a bounded, open subset of $\mathbb R^N$, and if $\{{\rm det}\, \nabla u_n\}$ converges weakly-* in the sense of measures to a Radon measure $\mu$, then $\frac{d \mu}{d \mathcal L^N}={\rm det}\, \nabla u$ a.e. in $\Omega$.
 
This is joint work with Giovanni Leoni and Jan Mal\'y.



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Irene Fonseca

Department of Mathematical sciences

Carnegie Mellon University

Pittsburgh, USA

\email{fonseca@andrew.cmu.edu}



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\begin{center}
{\bf Diffusion mediated transport and the brownian motor
\\ \hfill\\
{\it David Kinderlehrer}
}
\end{center}


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\noindent We discuss a system of equations understood to describe the function of 
a typical protein motor, as described by Oster, Ermentrout, and Peskin 
and Adjari and Prost and their collaborators.  An example of transport 
in an environment of diffusion, this mechanism governs many molecular 
level transduction processes and often falls under the rubric of the 
brownian motor or molecular rachet.  What ingredients are essential in 
the function of these systems and why?  We give an interpretation of the 
system, relating them to a variational principle involving 
Monge-Kantorovich mass transfer and the Wasserstein metric and describe 
our progress toward understanding the transport properties.  This is 
joint work with Michel Chipot and Michal Kowalczyk.



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David Kinderlehrer

Department of Mathematical sciences

Carnegie Mellon University

Pittsburgh, USA

\email{davidk@andrew.cmu.edu}





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\begin{center}
{\bf Transfer of jump sets for convergent sequences in SBV\\
\hfill\\
{\it Christopher Larsen}
}
\end{center}


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\noindent In proving existence for mathematical models of brittle fracture
growth, the following problem arises:

Suppose $u_n\in SBV(\Omega)$ minimizes
\[E(v):=\int_\Omega |\nabla v|^2 dx + {\cal{H}}^{N-1}(S_v\backslash
S_{u_n})\]
subject to $v=u_n$ on $\partial \Omega$, where $\Omega\subset I\!\!R^N$.
If $u_n\rightarrow u$ (in the sense of SBV compactness), does the corresponding
minimality hold for $u$?

A partial answer has been provided by Dal Maso and Toader, who proved this
minimality for $N=2$ assuming that the number of components of $S_{u_n}$
is bounded.  For the general result, our proof relies on a method of
transferring jump sets.  In particular, for any $\phi\in SBV$, we
construct a sequence $\phi_n\rightarrow \phi$
such that
\[\lim_{n\rightarrow\infty}{\cal{H}}^{N-1}(S_{\phi_n}\backslash S_{u_n})\leq
{\cal{H}}^{N-1}(S_{\phi}\backslash S_{u}).\]
In this talk, I will explain the proof and its connection to the fracture
problem.  This is part of joint work with G. Francfort.


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Christopher Larsen

Department of Mathematical Sciences

Worcester Polytechnic Institute

100 Institute Rd.

Worcester, MA  0160, USA

\email{cjlarsen@wpi.edu}



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\begin{center}
{\bf New lower semicontinuity results in the Calculus of Variations\\
\hfill\\
{\it Giovanni Leoni}
}
\end{center}


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\noindent We discuss lower semicontinuity properties for integral functionals
 which are non-coercive or satisfy non-standard growth conditions.


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Giovanni Leoni


Dipartimento di Scienze e Tecnologie Avanzate

Corso Borsalino 54

15100 Alessandria, Italy

\email{leoni@al.unipmn.it}



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\begin{center}
{\bf A variational approach to the Hele-Shaw flow with injection\\
\hfill\\
{\it Paolo Tilli}
}
\end{center}


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\noindent We describe a variational approach to the Hele-Shaw flow with
injection.

After giving a suitable weak formulation of the evolution
equation, we construct e sequence of approximating solutions to
the Hele-Shaw flow, depending on a discretization step $\delta t$,
by iteratively solving a variational problem.

As the time step tends to zero, the approximating solutions converge
increasingly (with respect to set inclusion, at fixed time) to a
weak solution of the problem. When the latter is not unique, the
solution thus obtained is characterized by a minimality property, with
respect to set  inclusion, at fixed time.

We also prove several monotonicity results of the weak solutions, with
respect to both the initial initial set and  the forcing term.
In particular, the monotonicity property with respect to the initial set
implies that the interface has finite perimeter at every time, provided
the starting set has finite perimeter.


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Paolo Tilli

Scuola Normale Superiore

Piazza dei Cavalieri 7

56100 Pisa

Italy

\email{tilli@cibs.sns.it}


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