Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory

TL;DR

This paper develops null-Lagrangians and calibrations for nonlocal elliptic functionals, demonstrating minimality of solutions and linking minimizers to viscosity solutions, with applications to fractional Laplacian energies.

Contribution

It introduces a new calibration method for nonlocal functionals assuming a foliation of solutions, connecting minimality and viscosity solutions.

Findings

Monotone solutions are minimizers in translation invariant nonlocal equations.

The foliation framework proves minimizers are viscosity solutions.

Null-Lagrangian for fractional Laplacian energy was constructed.

Abstract

In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequence of the calibration, we show the minimality of each leaf in the foliation. Our model case is the energy functional for the fractional Laplacian, for which such a null-Lagrangian was recently discovered by us. As a first application of our calibration, we show that monotone solutions to translation invariant nonlocal equations are minimizers. Our second application is somehow surprising, since here ``minimality'' is assumed instead of being concluded. We will see that the foliation framework is broad enough to provide a proof which establishes that minimizers of nonlocal elliptic functionals are viscosity solutions.