# Halfspaces minimise nonlocal perimeter: a proof via calibrations

## TL;DR

This paper introduces a nonlocal functional as a generalization of total variation, establishes a calibration method for minimization, and proves that halfspaces uniquely minimize this energy in a ball, with implications for scaling limits.

## Contribution

It develops a calibration framework for nonlocal functionals and proves halfspaces are the unique minimizers under certain conditions, extending classical results to nonlocal settings.

## Key findings

- Halfspaces are the unique minimizers of the nonlocal functional in a ball.
- A calibration method is introduced as a sufficient condition for minimality.
- The approach facilitates a $ ext{Gamma}$-convergence analysis for the scaling limit.

## Abstract

We consider a nonlocal functional $J_K$ that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $u\colon \mathbb{R}^d \to \mathbb{R}$, we define $J_K(u)$ as the integral of weighted differences of $u$. The weight is encoded by a positive kernel $K$, possibly singular in the origin. We study the minimisation of this energy under prescribed boundary conditions, and we introduce a notion of calibration suited for this nonlocal problem. Our first result shows that the existence of a calibration is a sufficient condition for a function to be a minimiser. As an application of this criterion, we prove that halfspaces are the unique minimisers of $J_K$ in a ball, provided they are admissible competitors. Finally, we outline how to exploit the optimality of hyperplanes to recover a $\Gamma$-convergence result concerning the scaling limit of $J_K$.