Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

Abstract

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

14 pages, 2 figures. Minor improvements according to the referee's suggestions. In particular, a couple of references were added and Lemma 2.1 was strengthened. Accepted to Proceedings of the AMS

References in corpus (1)

• The Poisson equation from non-local to local

Cited by in corpus (1)

• On fractional Schrödinger equations with Hartree type nonlinearities