# Lisbon WADE — Webinar in Analysis and Differential Equations

## Next session

### Asymptotic stability for the gradient flow of some nonlocal energies

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

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.

All seminars will take place in the Zoom platform which you need to install (although you don't need to register). In order to get the password to access the seminars, please subscribe the announcements or contact the organizers.

Organizers: Hugo Tavares, James Kennedy and Nicolas Van Goethem

Joint iniciative of the research centers CAMGSD, CMAFcIO and GFM.