A.A. 2020/21

Abstract

Rectifiable sets are widely studied in geometric measure theory and in several areas of application. Even more interesting is the higher order version of such sets which has been of current investigation in the Euclidean setting. In this talk, we shall consider such inquiry in a metric setting - the Heisenberg groups. In particular, we give a characterization of higher order ($C^{1,\alpha}$-) rectifiable sets of low codimension in the spirit of almost every where existence of \textit{approximate tangent paraboloids}.

Abstract

In this talk I present the problems that I have studied in my Ph.D. project.

We have considered a core-radius approach to non-local perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0<s<1, to the case s greater or equal to 1.

We show that, as the core-radius vanishes, such core-radius regularized s-fractional perimeters, suitably scaled, Gamma-converge to the standard Euclidean perimeter. We also show that the geometric flow of the core radius regularized s-fractional perimeters, suitably scaled, converge to the classical mean curvature flow.

We consider functionals of Riesz type and geometrical type, as for instance non-local perimeters, showing that it is possible to obtain them as Gamma limits of interaction functionals among rigid spheres

Abstract

Multiphase mean curvature flow is a widely studied system of geometric evolution equations. In the talk we introduce a notion of weak solution based on De Giorgi's general framework of gradient flows. In the first part we will recall the classical MBO scheme for mean curvature flow, we will recall the abstract framework of gradient flows in detail and we will show how it can be used to prove convergence of the classical MBO scheme to a De Giorgi solution. In the second part of the talk we introduce a new variant of the MBO scheme, due to Esedoglu and Salvador, which allows to handle general mobilities and surface tensions. In the same way as in the first part of the talk, we will be able to put this scheme in the general framework of gradient flows. Using a careful localization argument we will prove the convergence of the modified scheme to a De Giorgi solution.

Abstract

In the first part of this talk, we introduce some tools that are often used in optimization theory (such as Γ-convergence and relaxation) to deal with minimization problems. In the second part, we present a shape optimization problem arising from the optimal reinforcement of a membrane using one-dimensional stiffeners. In particular, we consider the maximization of the first eigenvalue on several admissible classes and discuss the corresponding relaxed problems, existence results and open problems.