## “Boundary stickiness of nonlocal minimal surfaces”

Lunedì 22 Giugno 2020, ore 10:00 - Zoom - Luca Lombardini (University of Western Australia, Perth, Australia)

Abstract

The nonlocal minimal surfaces were introduced in 2010, in a seminal paper by Caffarelli, Roquejoffre and Savin, as the boundaries of those sets that minimize the so-called s-perimeter.

We will give an introduction to the s-perimeter, which is, roughly speaking, the $W^{s,1}$-seminorm of the characteristic function of a set. We will explain in which sense this perimeter functional can be thought of as a fractional and nonlocal analogue of the perimeter introduced by De Giorgi and Caccioppoli. Then we will focus on the qualitative and regularity properties of the nonlocal minimal surfaces.

On the one hand, when the fractional parameter s is close to 1, the s-perimeter and its minimizers are close (in an appropriate sense) to the usual perimeter and the classical minimal surfaces.

On the other hand, we will show that when s is close to 0 the nonlocal character of the s-perimeter becomes more and more evident, and its minimizers exhibit some surprising, typically nonlocal, boundary behavior.

Part of the results that I will present come from a joint paper with Claudia Bucur and Enrico Valdinoci.

Seminari di equazioni differenziali e applicazioni