# Lisbon WADE — Webinar in Analysis and Differential Equations

## Planned sessions

### 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.

### Homogenisation of discrete dynamical optimal transport

Many stochastic systems can be viewed as gradient flow ('steepest descent') in the space of probability measures, where the driving functional is a relative entropy and the relevant geometry is described by a dynamical optimal transport problem. In this talk we focus on these optimal transport problems and describe recent work on the limit passage from discrete to continuous.
Surprisingly, it turns out that discrete transport metrics may fail to converge to the expected limit, even when the associated gradient flows converge. We will illustrate this phenomenon in examples and present a recent homogenisation result.

This talk is based on joint work with Peter Gladbach, Eva Kopfer, and Lorenzo Portinale.

### A proof of the Caffarelli contraction theorem via entropic interpolation

The Caffarelli contraction theorem states that optimal transport maps (for the quadratic cost) from a Gaussian measure onto measures that satisfy certain convexity properties are globally Lipschitz, with a dimension-free estimate. It has found many applications in probability, such as concentration and functional inequalities. In this talk, I will present an alternative proof, using entropic interpolation and variational arguments. Joint work with Nathael Gozlan and Maxime Prod'homme.

### 13/05/2021, 14:00 — 15:00 Europe/Lisbon — Online Simone DiMarino, Università di Genova

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.