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

## TL;DR

This paper develops a calibration method for nonlocal perimeters using null-Lagrangians, proving minimality of certain sets and functions, and connects minimizers to viscosity solutions in the context of fractional perimeters.

## Contribution

It introduces a calibration approach for nonlocal perimeters based on null-Lagrangians, establishing minimality and uniqueness results, and providing a new proof linking minimizers to viscosity solutions.

## Key findings

- Calibration constructed for nonlocal perimeter in foliated solutions.
- Minimality of nonlocal minimal graphs proved.
- Minimizers of fractional perimeter are viscosity solutions.

## 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.