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.

Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods

“…Using this last equality and the bilinearity of E, we obtain that 1 + α 2 0 = 2α 0 (as in the proof of [4,Lemma 2.1]). Then α 0 = 1 and this concludes the proof.…”

“…In the rest of the proof, following some ideas in [4], we show that w a , the polarization of w with respect to H a (see (4.11) for the definition of w a ), is a non-radial second eigenfunction for a > 0 sufficiently small, which yields a contradiction with Corollary 3.1. As a consequence, it follows that problem (2.2) admits only the trivial solution, namely we obtain the nondegeneracy of u.…”

“…The proof of the nondegeneracy stated in Theorem 1.1 is therefore achieved by exhibiting a non-radial second eigenfunction for (1.5). This is done by using the polarization of a second radial eigenfunction, following the arguments in [4] to prove the non radiality of the second eigenfunction of the fractional Laplacian (−∆) s in the ball.…”

“…Inspired by the ideas in [4], we introduce the following characterization for the second eigenvalue µ 2 of the eigenvalue problem (2.6).…”