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 $\Omega \subset {\mathbb{R}}^n$, we built a calibration for the nonlocal perimeter inside $\Omega \subset {\mathbb{R}}^n$. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in $\Omega$ 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.

To appear in Annali di Matematica Pura ed Applicata

References in corpus (2)

• Halfspaces minimise nonlocal perimeter: a proof via calibrations
• Minimization problems involving nonlocal functionals: nonlocal minimal surfaces and a free boundary problem

Cited by in corpus (3)

• The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions
• Enhanced boundary regularity of planar nonlocal minimal graphs, and a butterfly effect
• Asymptotic behaviour of rescaled nonlocal functionals and evolutions