On the asymptotic behaviour of nonlocal perimeters

¹ Institut für Analysis und Numerik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany.
² Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy.

^* Corresponding author: pagliari@mail.dm.unipi.it

Received: 6 February 2018
Accepted: 18 June 2018

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 that has been recently introduced by A. Chambolle et al. [Archiv. Rational Mech. Anal. 218 (2015) 1263–1329]. Also, we establish existence of minimisers for the corresponding Plateau’s problem and, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. A Γ-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 Γ-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.