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 annuli-like 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<1-|a| $. 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.