  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.
