An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions

TL;DR

This paper investigates nonlinear elliptic equations involving sums of fractional Laplacians of various orders, establishing regularity and energy estimates, and proving a 1-D symmetry result for solutions related to phase transitions driven by non-stable Lévy diffusions.

Contribution

It extends the extension problem framework to sums of fractional Laplacians and proves symmetry results for solutions of phase transition equations with non-stable Lévy operators.

Findings

Established regularity and sharp energy estimates for solutions.

Proved 1-D symmetry of monotone solutions to Allen-Cahn type equations.

Demonstrated local PDE realization of non-stable Lévy operators.

Abstract

We study nonlinear elliptic equations for operators corresponding to non-stable L\'evy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) L\'evy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen-Cahn type equations with a non-stable L\'evy diffusion. These operators may still be realized as local operators using a system of PDEs ---in the spirit of the extension problem of Caffarelli and Silvestre.