Symmetry of fractional Neumann eigenfunctions in the ball

We investigate symmetry properties of the first nontrivial eigenfunctions of the fractional Laplacian $(-\mathrm{\Delta }{)}^s$, 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.