Aula Magna del Dipartimento di Matematica

*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.

with integer valued fluxes

*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).

and nonlinear potential theory

*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.