On critical points of the relative fractional perimeter

TL;DR

This paper investigates the properties and localization of sets with constant nonlocal mean curvature, analyzing their proximity to critical points of a non-local potential, and studies minimizers of fractional perimeter in half-spaces.

Contribution

It establishes the closeness of constant mean curvature sets to critical points of a non-local potential and proves existence and properties of fractional perimeter minimizers in half-spaces.

Findings

Sets with constant nonlocal mean curvature are close to critical points of a non-local potential.

Existence of minimizers of fractional perimeter under volume constraint in half-spaces.

Minimizers are smooth, symmetric, and can be represented as graphs in the vertical direction.

Abstract

We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the existence of a minimizer under fixed volume constraint, showing some of its properties such as smoothness and symmetry, being a graph in the $x_N$-direction, and characterizing its intersection with the hyperplane $\{x_N=0\}$.