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

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.