Nonradiality of second fractional eigenfunctions of thin annuli
Abstract
In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annulilike domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius $R$ and outer radius $R+1$. We show that for $R$ sufficiently large any corresponding second eigenfunction of this annulus is nonradial. In the second part, we investigate the second eigenvalue in domains of the form $B_1(0)\setminus \overline{B_{\tau}(a)}$, where $a$ is in the unitary ball and $0<\tau<1a$. We show that this value is maximized for $a=0$, if the set $B_1(0)\setminus \overline{B_{\tau}(0)}$ has no radial second eigenfunction. We emphasize that the first part of our paper implies that this assumption is indeed nonempty.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.04907
 arXiv:
 arXiv:2201.04907
 Bibcode:
 2022arXiv220104907D
 Keywords:

 Mathematics  Analysis of PDEs