Abstract. Classical capillarity theory is based on the study of volume-constrained critical points and local/global minimizers of the Gauss free energy of a liquid droplet occupying a region inside a container. In 2016 Maggi and Valdinoci introduced a family of nonlocal capillarity models where surface tension energies are replaced with fractional interaction energies. I will discuss a nonlocal capillarity problem involving kernels which are possibly anisotropic and have different homogeneity in order to take into account the possibility that the liquid/air interaction and the liquid/surface one are different.
Abstract. Existence and regularity of minimal surfaces (i.e. stationary
points for area functional) with various boundary conditions has been an active
topic of research for the last decades.
Although minimizing methods usually provide existence of solutions for some kinds of
boundary constraints, they may produce just trivial solutions when dealing with different
topological/boundary conditions; in that cases, one has to rely on different ways to
obtain a critical point for the given functional. Min-max methods have been successfully
used for some of these problems.
In this talk I will explain the basic ideas of min-max theory applied to the problem of
finding, in a container in the 3-dimensional euclidean space, a minimal surface
which meets the boundary with a fixed angle.
This is based on a joint work with Guido De Philippis.
Abstract. We will introduce the space of Vector Fields with Integer Valued Fluxes (VFIVF) and discuss some of its analytical properties. In particular we will address the weak and strong closure of this space in and the relationship of VFIVFs with integer rectifiable 1-currents. We will also see how VFIVFs arise naturally in some problems in the Calculus of Variations.
Abstract. We introduce the generalized principal frequencies and their connection with the Lane-Emden equation for the p−Laplacian. Then we prove, throughout a comparison principle, a variety of results, as the uniqueness of solutions under minimal assumptions on the set, sharp pointwise two-sided estimates for positive solutions in convex sets, a “hierarchy” of sign-changing solutions of the equation and a sharp geometric estimate on the generalized principal frequencies of convex sets. The results presented in this talk are obtained in collaboration with Lorenzo Brasco (Ferrara) and Francesca Prinari (Pisa).
Abstract. In this talk, we describe some monotonicity formulas holding along the level sets of suitable p-harmonic functions in asymptotically flat 3–manifolds with a single end, either with or without boundary, having nonnegative scalar curvature. Using such the formulas, we obtain a simple proof of the positive mass theorem and the Riemannian Penrose inequality. The results discussed are obtained in collaboration with Virginia Agostiniani, Carlo Mantegazza and Lorenzo Mazzieri.