A Weierstrass extremal field theory for the fractional Laplacian

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.
This work is also supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M), as well as by the Catalan project 2021 SGR 00087
The three authors are supported by grants PID2021-123903NB-I00 and RED2018-102650-T funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. The second author has received founding from the MINECO grant MDM-2014-0445-18-1. The third author has been supported by the Academy of Finland and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 818437).
Peer Reviewed
Postprint (published version)