Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory

Abstract

For nonnegative even kernels K, we consider the K-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated K-nonlocal mean curvature equation in an open set O¿Rn, we built a calibration for the nonlocal perimeter inside O¿Rn. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in O of each leaf of the foliation. As an application, we prove the minimality of K-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions.