A Weierstrass extremal field theory for the fractional Laplacian

Abstract In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian 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 the minimality of each leaf in the foliation follows from the existence of the calibration. As an application, we show that monotone solutions to fractional semilinear equations are minimizers. In a forthcoming work, we generalize the theory to a wide class of nonlocal elliptic functionals and give an application to the viscosity theory.