A Weierstrass extremal field theory for the fractional Laplacian

TL;DR

This paper extends Weierstrass extremal field theory to the nonlocal fractional Laplacian setting, constructing calibrations and proving minimality of solutions, including monotone solutions to fractional semilinear equations.

Contribution

It introduces the first nonlocal Weierstrass extremal field theory for the fractional Laplacian, including null-Lagrangians and calibrations for nonlinear equations.

Findings

Constructed a null-Lagrangian and calibration for fractional Laplacian equations.

Proved that monotone solutions to fractional semilinear equations are minimizers.

Established a framework for extremal field theory in nonlocal variational problems.

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.