Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

Abstract

Using symmetrization techniques, we show that, for every N ≥ 2 N \geq 2 , any second eigenfunction of the fractional Laplacian in the N N -dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball.