# Lisbon Webinar in Analysis in Differential Equations

Transmissão através de Videoconferência

Asymptotic stability for the gradient flow of some nonlocal energies | 14h00-15h00
Nicola Fusco (Università di Napoli "Federico II")

I will start by discussing some recent results on the asymptotic stability of the $H^{-1}$-gradient flow of the perimeter, the so called surface diffusion. Then I will consider the $H^{-1}$-gradient flow of some energy functionals given by the area of an interface plus a non local volume term.

This is a joint work with E. Acerbi, V. Julin and M. Morini

Nonlocality features of the area functional and the Plateau problem | 15h00-16h00
Riccardo Scala (Università degli Studi di Siena)

We briefly discuss the definition of relaxation of the area functional. The relaxed area functional, denoted by $A$, extends the classical area functional, which, for any "regular" map $v:U\subset \mathbb{R}^n\rightarrow \mathbb{R}^N$ evaluates the $n$-dimensional area of its graph over $U$. The problem of determining the domain and the expression of $A$ is open in codimension greater than 1. Specifically, this relaxed functional turns out to be nonlocal and cannot be expressed by an integral formula. We discuss how it is related to classical and nonclassical versions of the Plateau problem. As a main example, we try to understand what is the relaxed graph of the function $x/|x|$, a question that surprisingly remained open for decades.