Halfspaces minimise nonlocal perimeter: a proof via calibrations

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 $J_K$.