Symmetry of fractional Neumann eigenfunctions in the ball

TL;DR

This paper studies the symmetry of the first nontrivial eigenfunctions of the fractional Laplacian with Neumann boundary conditions in a ball, showing near-classical behavior as s approaches 1.

Contribution

It proves that for s close to 1, the eigenspace is generated by N antisymmetric eigenfunctions with two nodal domains, revealing symmetry properties of fractional Neumann eigenfunctions.

Findings

Eigenfunctions are antisymmetric with two nodal domains near s=1.

Spectral stability results connect fractional and classical cases.

Eigenfunction structure depends on the parameter s.

Abstract

We investigate symmetry properties of the first nontrivial eigenfunctions of the fractional Laplacian $(-\Delta)^s$, where $s \in (0,1)$, in an $N$-dimensional ball with nonlocal Neumann boundary conditions. By means of a spectral stability result, we prove that, when $s$ is sufficiently close to $1$, the eigenspace associated to the first nontrivial eigenvalue is generated by $N$ antisymmetric eigenfunctions with exactly two nodal domains in the ball.