Title: Grain boundaries in perimeter minimizing partitions with many cells
Abstract: Given a domain $E$ in the plane, I consider a minimal $N$-partition of $E$,
that is, a partition that minimizes the total perimeter among all
partitions of $E$ made of $N$ cells with equal area. T.C. Hales proved in
2001 that if $E$ is a flat $2$-dimensional torus, then the only minimal
N-partition is the regular hexagonal one (assuming that such partition
exists).
But what happens for if $E$ that does not admit a regular hexagonal
partition?
One can show that, as expected, cells look more and more hexagonal as $N$
tends to infinity, and thus the next question concerns the "regularity"
of such almost hexagonal partition: is it rigid, in the sense that the
orientation of the cells is (essentially) the same through the domain?
I will present evidence that the answer is negative, and that the domain
splits in large blocks (grains) where the orientation of the cells is
essentially constant, separated by comparatively thin regions (grain
boundaries) containing many non hexagonal cells.
The similarity with grain structures in the theory of dislocation in
continuum mechanics is not accidental.
This is an ongoing research project with Marco Caroccia (Politecnico di
Milano), Giacomo Del Nin (MPI Leipzig), Adriana Garroni and Emanuele
Spadaro (Roma Sapienza).
Title: Heat diffusion and electrical conduction in composites
with imperfect contact conditions
Abstract: The purpose of this talk is to present some models with imperfect interfaces which origin in the
description of composites made by a
hosting medium containing a periodic array of inclusions of size $\varepsilon$, where the inclusions
are coated by a thin layer consisting of two sublayers of different materials (with thickness of the order $\varepsilon\eta$
and
$\varepsilon\delta$, respectively), disposed in such a way that one of them is encapsulated into the other.
This two-phase coating material is such that one of the two components has a low diffusivity in the orthogonal
direction and the other one has a high diffusivity in the tangential direction. All the parameters
$\varepsilon$, $\delta$ and $\eta$
are supposed to be very small, but with different orders. In particular, the smallness of
$\delta$ and $\eta$, with respect to
$\varepsilon$, leads us to perform, for fixed $\varepsilon$, a two-step concentration procedure, which
produces on the resulting interface between the hosting material and the inclusions some conditions
involving the jump and the mean value of the two bulk potentials and the jump of their fluxes. Moreover,
also the appearance of a new surface heat potential can happen. The concentrated problems thus obtained are
then homogenized, i.e. we let $\varepsilon$ tend to zero, and we briefly discuss the different resulting models.
Title: Large deviations with respect to motion by curvature
Abstract: Consider the Allen-Cahn equation. Under diffusive rescaling of space
and time, for suitably prepared initial data the limiting dynamics is
described by the motion by mean curvature of the interface between the
two stable phases.
We consider, in space dimension $d\le 3$, a stochastic perturbation of
the Allen-Cahn equation and analyze its asymptotic in such sharp
interface limit proving the large deviations upper bound. The
corresponding rate function is finite only when there exists a $d-1$
interface between the two stable phases and can be written as the sum
of two terms: the first takes into account "wrong" motions of the
interface while the second is due to the possible occurrence of
nucleation events. In particular, the zero level set of this rate
function is given by the evolution by mean curvature in the Brakke
formulation.
Our results relies on previous analysis on the variational convergence
of the action functional associated to the Allen-Cahn equation.
Title: Discrete-to-continuum crystalline curvature flow
Abstract: We consider a discrete Almgren-Taylor-Wang implicit scheme,
on a finite cubic grid (in any dimension) with a scale parameter h.
The discrete surface tension involves only finite interactions, so that the
limiting surface tension is the support function of a zonotope. We use the
natural mobility defined by its polar.
We show that if the spatial and time scale are identical, and with an
appropriate definition of the distance to the discrete moving set, we
can recover in the limit the crystalline curvature flow corresponding
to the limiting surface tension.
Title: Diffuse approximation of the Willmore functional and a conjecture of De Giorgi
Abstract: We will discuss a conjecture of De Giorgi, dating back to 1991,
where he proposed a possible approximation, in the sense of Gamma-convergence, of the
Willmore functional based on the first variation of the Modica-Mortola functionals.
Since then several authors have investigated this problem and some modifications of the
approximating functionals have been considered. For some of these modifications it has been
proven that they actually approximate the Willmore functional on smooth sets.
After reviewing some of the most relevant contributions originated from this conjecture over
time we will present a recent result obtained in collaboration with G. Bellettini and N. Picenni
where we provide a negative answer to the original conjecture.
If time permits we will also discuss some other properties of the original functionals introduced
by De Giorgi.
Title: Connectivity of the phases and fine structure of minimizers in the theory of phase transitions
Abstract: We study minimizers of the Allen Cahn energy with a phase transition potential,
a nonnegative potential that vanishes only on a finite set of points that model the phases.
We work on two dimensional domains with Dirichlet
boundary conditions.
We show that the assumption of connectivity of the phases allows for a detailed description
of the fine structure of minimizers. In particular one can characterize the shape and the
size of the domain regions where a minimizers remains in a neighborhood of one or another
of the zeros of the potential and also how these regions depend on the surface tensions.
Title: Variational analysis of non local energies on periodically perforated domains
Abstract: In the last years non local functionals have attracted great interest
in view of various applications to different physical models. In this talk
I will focus on non local energies of convolution-type. This kind of energies
date back to the work by Bourgain, Brezis and Mironescu on fractional Sobolev
norms and have been recently widely studied as continuous energies depending on
finite differences appear in a natural way in the study of models of inhomogeneous
media with an underlying periodic microstructure. In this respect, I will present
some recent results on the asymptotics of such energies accounting for a Dirichlet
type condition imposed on a periodic perforation of the domain. More in detail, I will
show that the interplay among the periodicity, the size of the perforations, and the
approximation parameter affects deeply the asymptotic behaviour of the energies, highlighting
different phenomena.
Title: Stability for the surface diffusion flow
Abstract: We study the surface diffusion flow in the flat torus, that
is, smooth hypersurfaces moving with the outer normal velocity given
by the Laplacian of their mean curvature.
This model describes the evolution in time of interfaces between solid
phases of a system, driven by the surface diffusion of atoms under the
action of a chemical potential.
We show that if the initial set is sufficiently ''close'' to a
strictly stable critical set for the Area functional under a volume
constraint, then the flow actually exists for all times and
asymptotically converges
to a ''translated'' of the critical set. This generalizes the
analogous result in dimension three, by Acerbi, Fusco, Julin and
Morini.
Joint work with Antonia Diana e Nicola Fusco.
Title: Soap films and partial wetting
Abstract:In the context of soap films spanning a given 1D frame, an interesting feature,
unespectedly physically and mathematically feasible, show that there exist
nontrivial minimal surfaces (possibly comprising singularities of the types listed
by J. Taylor) that touches the wire frame only partially.
We show the results of some numerical simulations that should help in understanding
the structure and shape of the surface in the vicinity of a point at the boundary
of the "wetted" portion of the wire frame.
This work is done in collaboration with G. Bellettini.
Title: From the regularization of the Perona-Malik functional to a challenging free-discontinuity problem
Abstract: We consider the Perona-Malik functional, that is an
integral functional with nonconvex Lagrangian, whose formal gradient flow
is the celebrated forward-backward equation introduced by Perona and Malik
in the 90s. We regularize this functional either by space-discretization or
by adding a higher order term, and we discuss the asymptotic behavior of
minimizers when the regularization parameter tends to zero.
When the space dimension is equal to one, we can show that minimizers of the
regularized functionals develop a microstructure that looks like a piecewise
constant function at a suitable scale, and more precisely that the blow-ups
at a suitable scale of any sequence of minimizers converge to a local minimizer
of a free-discontinuity problem. In this one-dimensional case, such local
minimizers can be easily characterized, and turn out to be staircase-like functions.
After a brief discussion of these one-dimensional results, we describe the
higher-dimensional version of the limiting free-discontinuity problem, which
turns out to be much more challenging. In particular, in the planar case,
we show that there exists at least one symmetry-breaking local minimizer.
The talk is based on joint works with Massimo Gobbino.
Title: Uniqueness of blowups for the motion by curvature of networks
Abstract: A network is a finite union of embedded open curves in the plane
such that their endpoints are joined at triple junctions. A motion by curvature
is a one-parameter family of networks evolving so that the normal driving velocity
at any point is given by the curvature vector. Hence, a motion by curvature corresponds
to the $L^2$-gradient flow of the length functional on networks.
Like higher dimensional mean curvature flows, the motion by curvature of networks may
develop singularities, whose behavior can be understood by studying blowups at points
where the singularity occurs. A blowup is a flow obtained as the limit of a sequence
of dilations centered at the singularity point.
In this talk we discuss a result that proves uniqueness of blowups for the motion by
curvature of networks, that is the fact that the blowup at a singularity does not depend
on the chosen sequence of rescaling factors.
The proof is based on the application of a Lojasiewicz--Simon gradient inequality.
The talk is based on a work in collaboration with C. Mantegazza and A. Pluda.
Title: Recent developments in the relaxation of area functional
Abstract: In the recent years I collaborated with Giovanni Bellettini
on the project of trying to understand some properties of the relaxed area functional
in dimension and codimension greater than 1. I will summarize some results obtained
in the last, say, 5 years, and will discuss some recent developments. Precisely, I will
introduce a notion of weak Jacobian determinant for BV maps, and show how it pops up in
the analysis of the relaxed area functional of $R^2$ valued maps from a 2-dimensional domain.
I will also present some perspectives and open problems.