The isoperimetric problem for nonlocal perimeters

TL;DR

This paper establishes a general isoperimetric inequality for nonlocal perimeters, including fractional and Riesz potentials, and discusses applications such as the existence of an isoperimetric profile.

Contribution

It introduces a broad class of nonlocal perimeter functionals and proves a general isoperimetric inequality for them, extending classical results to nonlocal settings.

Findings

Proved a general isoperimetric inequality for nonlocal perimeters.

Established existence of an isoperimetric profile under certain conditions.

Included applications to fractional and Riesz type potentials.

Abstract

We consider a class of nonlocal generalized perimeters which includes fractional perimeters and Riesz type potentials. We prove a general isoperimetric inequality for such functionals, and we discuss some applications. In particular we prove existence of an isoperimetric profile, under suitable assumptions on the interaction kernel.