The role of antisymmetric functions in nonlocal equations

TL;DR

This paper establishes a Hopf-type lemma for antisymmetric super-solutions of fractional Laplacian problems and uses it to prove symmetry of solutions in bounded domains, highlighting the role of antisymmetric functions in nonlocal equations.

Contribution

It introduces a Hopf-type lemma for antisymmetric super-solutions and applies it with the method of moving planes to prove symmetry results for fractional Laplacian problems.

Findings

Proved a Hopf-type lemma for antisymmetric super-solutions.

Established symmetry of solutions in bounded domains with parallel level surfaces.

Discussed maximum principles and Harnack inequality limitations for antisymmetric functions.

Abstract

We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms. As an application, we use such a Hopf-type lemma in combination with the method of moving planes to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $\Omega \subset \mathbb R^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $\Omega$; in turn, $\Omega$ must be a ball. Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only `local' assumptions are imposed on the solutions.