Non-local planelike minimizers and $\Gamma$-convergence of periodic energies to a local anisotropic perimeter

TL;DR

This paper studies how non-local periodic energies behave under homogenization, showing they converge to a local anisotropic perimeter through the analysis of planelike minimizers and the stable norm.

Contribution

It establishes the existence of planelike minimizers and proves the $ ext{Gamma}$-convergence of rescaled energies to a local anisotropic perimeter, advancing understanding of non-local to local energy limits.

Findings

Existence of planelike minimizers for non-local energies.

$ ext{Gamma}$-convergence of rescaled energies to a local anisotropic perimeter.

Explicit bounds and estimates for minimizers and their level sets.

Abstract

We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies $\Gamma$-converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large.