![[Logo] Instituto Superior Técnico](/img/Tecnico_logo.svg "Instituto Superior Técnico"){kind=logo}
08/04/2021, 14:00 — 15:00 Europe/Lisbon —
Online
Valeria Chiadò Piat, Politecnico di Torino
An extension theorem from connected sets and homogenization of non-local functionals
Extensions operators are a classical tool to provide uniform estimates and gain compactness in the homogenization of integral functionals over perforated domains. In this talk we discuss the case of non-local functionals. The results are obtained in collaboration with Andrea Braides and Lorenza D'Elia.
15/04/2021, 14:00 — 15:00 Europe/Lisbon —
Online
Nicola Fusco, Università di Napoli "Federico II"
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
15/04/2021, 15:00 — 16:00 Europe/Lisbon —
Online
Riccardo Scala, Università degli Studi di Siena
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.
