On the asymptotic behaviour of nonlocal perimeters

TL;DR

This paper investigates the properties of nonlocal perimeters, proving their perimeter-like nature, existence of minimisers, and their convergence to classical perimeters under certain conditions.

Contribution

It establishes that nonlocal perimeters are generalized perimeters, proves minimiser existence, and shows their Gamma-convergence to classical perimeters with explicit constants.

Findings

Nonlocal perimeters are valid generalized perimeters.

Existence of minimisers for the Plateau problem is proven.

Nonlocal perimeters Gamma-converge to classical perimeters under specific rescalings.

Abstract

We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe flat boundary conditions. A Gamma-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-L1 decay at infinity and we show that the Gamma-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.