Nonradiality of second fractional eigenfunctions of thin annuli

Sidy M. Djitte; Sven Jarohs

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arXiv · 2201.04907

Nonradiality of second fractional eigenfunctions of thin annuli

Sidy M. Djitte·Sven Jarohs

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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.

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Sidy M. Djitte, Sven Jarohs. 2022-01-13. Nonradiality of second fractional eigenfunctions of thin annuli. https://arxiv.org/abs/2201.04907

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