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In this paper we extend, for the first time, part of the Weierstrass extremalfield theory in the Calculus of Variations to a nonlocal framework. Our modelcase is the energy functional for the fractional Laplacian (theGagliardo-Sobolev seminorm), for which such a theory was still unknown.  We build a null-Lagrangian and a calibration for nonlinear equationsinvolving the fractional Laplacian in the presence of a field of extremals.Thus, our construction assumes the existence of a family of solutions to theEuler-Lagrange equation whose graphs produce a foliation. Then, the minimalityof each leaf in the foliation follows from the existence of the calibration. Asan application, we show that monotone solutions to fractional semilinearequations are minimizers.  In a forthcoming work we generalize the theory to a wide class of nonlocalelliptic functionals and give an application to the viscosity theory.

A Weierstrass extremal field theory for the fractional Laplacian