Calculus of Variations and Free Boundary Problems III

    The Steiner problem in its classical formulation reads as follows: given a finite collection of points S in the plane, find the connected set that contains S with minimal length. Although existence and regularity of minimizers is well known, in general finding explicitly a solution is extremely challenging, even numerically. For this reason every method to determine solutions is welcome. A possible tool is the notion of calibrations. In this talk I will define calibrations for the Steiner problem within the framework of covering spaces. I will also give some example of both existence and non– existence of calibrations and to overcome this second unlucky case I will introduce the notion of calibration in families. I will also show how with the same tools one can deal with the minimal partition problem. If time allows, I will consider generalizations of the problem to higher dimensions and to the case of nonlocal perimeters.

    We study the positive principal eigenvalue of a weighted problem associated with the Neumann Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is led to consider a shape optimization problem, which is known to admit spherical optimal shapes only in very specific cases. We investigate whether spherical shapes can be recovered in general situations, in some singular perturbation limit. These are joint works with Dario Mazzoleni and Benedetta Pellacci.

    We study the Gran Canonical Optimal Transport, which can be seen a natural generalization of multimarginal optimal transport, where we do fix the average density of the electrons but we not restrict ourselves to a fixed number of electron. This comes up naturally in localization arguments and far interactions. We will discuss briefly the classical aspects (existence, duality, c-monotonicity) as well as new phenomena that appear (bounds on the number of electrons, choice of a stable cost function).

    In this talk we present a notion of irreversibility for the evolution of cracks in presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models. We investigate its applicability to the construction of a quasistatic evolution in a simple one-dimensional model. This is a joint work with M. Bonacini and S. Conti.

    A Blaschke-Santalo' diagram is the range of a vector shape functional (F₁,F₂) in R².
The determination of such attainable set amounts to completely characterize the relation between F₁ and F₂. In this talk I will present some recent results obtained in collaboration with D. Zucco, in the case of F₁ the first Dirichlet eigenvalue and F₂ the inverse of the torsional rigidity, defined on convex shapes with unit volume, and, as a variant, on convex sets with volume at most 1. The study led us to address some very deep questions, whose answers are still open problems: in the last part of the talk, I will list them, together with our conjectures.

    We study minimizers for a variational model describing the shape of charged liquid droplets. The surface tension forces the particles to stay together whereas the electric charge causes repulsion. Thus, one expects a certain transition to happen when increasing the charge. This heuristic is indeed confirmed by the experiments first conducted by Zeleny in the beginning of the previous century. A spherical droplet, when exposed to an electric field, remains stable until the charge reaches a certain critical value. The droplet then starts developing singularities.
There are several models trying to capture this phenomenon. We work with the variational model proposed by Muratov and Novaga, as the most commonly used Rayleigh’s one is ill-posed. Using the recent regularity result by De Philippis, Hirsch, and Vescovo we are able to prove that the only minimizers in the case of small charge are balls.
This is a joint work with Giulia Vescovo.

    The first part of this talk will be addressed to the problem of minimizing the first eigenvalue of the Dirichlet Laplacian with drift in a box. If the drift is the gradient of a function, we can prove a regularity result for the optimal shapes. In a second time, we will consider the minimization of the k first eigenvalues for an operator in divergence form. We can then prove the first step in the regularity theory, that is, the Lipschitz continuity of the eigenfunctions.

    We provide a sharp double-sided estimate for Poincare-Sobolev constants on a convex set, in terms of its inradius and N−dimensional measure. Our results extend and unify previous works by Hersch and Protter (for the first eigenvalue) and of Makai, Polya and Szego (for the torsional rigidity), by means of a single proof. This is a joint work with Lorenzo Brasco (Ferrara).

    In the last decades, and particularly in the last years, there has been a lot of effort to study the problem of minimizing the Riesz functional among sets of given volume in R^N. The Riesz functional of a set is given by the sum of its perimeter and the double integral of a repulsive potential of the form |y-x|^p where p is a negative number between 0 and -N. While it is more or less obvious that for small volumes the minimizer should look like a ball, and for big volumes the mass should be spread away, a remarkable known result is that minimizers are exactly balls for volume small enoungh. We will describe the problem and present a new proof of this result, which is valid in a more general call of functionals which in particular contains the negative powers.