Second order expansion for the nonlocal perimeter functional

TL;DR

This paper develops a second order expansion for nonlocal perimeter functionals, deriving their Gamma-limit and analyzing minimizers, with applications to thin ferromagnetic films and geometric optimization.

Contribution

It introduces a second order expansion for nonlocal perimeters and characterizes their Gamma-limit, extending classical perimeter approximation results.

Findings

Derived the Gamma-limit of the second order nonlocal perimeter functional.

Proved existence of minimizers with prescribed volume fractions.

Identified the ball as the unique minimizer for small volume fractions.

Abstract

The seminal results of Bourgain, Brezis, Mironescu and D\'avila show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a corresponding second order expansion for the nonlocal perimeter functional. In a special case, the considered family of energies is also relevant for a variational model for thin ferromagnetic films. We derive the Gamma--limit of these functionals. We also show existence for minimizers with prescribed volume fraction. For small volume fraction, the unique, up to translation, minimizer of the limit energy is given by the ball. The analysis is based on a systematic exploitation of the associated symmetrized autocorrelation function.