Symmetry properties for solutions of nonlocal equations involving nonlinear operators

TL;DR

This paper investigates one-dimensional symmetry of solutions to nonlinear nonlocal equations, including fractional p-Laplacian cases, using two independent methods involving Poincaré inequalities and Liouville theorems, with broader applicability to general kernels.

Contribution

It introduces two novel proofs for symmetry results in nonlocal equations, extending classical methods to a wider class of operators without relying on the extension problem.

Findings

Establishment of a Poincaré inequality for nonlocal operators

Proof of a linear Liouville theorem for solutions

Energy estimates for layer solutions and stable solutions

Abstract

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional $p-$Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincar\'e inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.