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.

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.

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.

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.

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.

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.

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).