Halfspaces minimise nonlocal perimeter: a proof via calibrations

Abstract

AbstractWe consider a nonlocal functional $$J_K$$ J K that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $$u:\mathbb {R}^d\rightarrow \mathbb {R}$$ u : R d → R , we define $$J_K(u)$$ 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 $$J_K$$ J K in a ball, provided they are admissible competitors. Finally, we outline how to exploit the optimality of hyperplanes to recover a $$\varGamma $$ Γ -convergence result concerning the scaling limit of $$J_K$$ J K .