Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

TL;DR

This paper proves that second eigenfunctions of the fractional Laplacian in a ball are nonradial, with their nodal sets forming an equatorial section, using symmetrization techniques.

Contribution

It establishes the nonradiality of second eigenfunctions of the fractional Laplacian in a ball, a result previously unknown for this operator.

Findings

Second eigenfunctions are nonradial in the fractional Laplacian case.

Nodal sets are equatorial sections of the ball.

Result holds for all dimensions N ≥ 2.

Abstract

Using symmetrization techniques, we show that, for every $N \geq 2$, 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.