$\Gamma$-convergence of some nonlocal perimeters in bounded subsets of $\mathbb{R}^n$ with general boundary conditions

TL;DR

This paper proves the $ ext{Gamma}$-convergence of nonlocal interaction energy functionals in bounded domains, showing how boundary conditions influence the limiting perimeter or fractional perimeter in the context of elliptic operators.

Contribution

It extends $ ext{Gamma}$-convergence results to nonlocal energies with general boundary conditions, including Robin, Dirichlet, and Neumann, affecting the limiting functional form.

Findings

Boundary conditions significantly influence the limiting perimeter functional.

The limiting functional can be a classical perimeter or a fractional perimeter.

Results encompass both local and nonlocal elliptic operators.

Abstract

We establish the $\Gamma$-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary interest of our results is to identify the effects that the boundary conditions imposed on the potential have on the limiting functional. We consider general Robin boundary conditions, which include Dirichlet and Neumann conditions as particular cases. Depending on the order of the elliptic operator the limiting functional involves the usual perimeter or some fractional perimeter. We also consider the $\Gamma$-convergence of a related energy functional combining the usual perimeter functional and the nonlocal repulsive interaction energy.