Overdetermined problems for the fractional Laplacian in exterior and annular sets

TL;DR

This paper proves that solutions to certain fractional elliptic equations in exterior and annular domains must be radially symmetric, extending symmetry results to non-convex and symmetric bounded regions.

Contribution

It establishes radial symmetry for solutions of fractional elliptic problems in unbounded and non-convex domains, generalizing classical symmetry results.

Findings

Solutions are radially symmetric in exterior and annular sets.

Radial symmetry holds in non-convex regions under certain conditions.

Symmetry results extend to sets with rotational symmetry.

Abstract

We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.