On the reverse Faber-Krahn inequalities

TL;DR

This paper extends Payne-Weinberger's reverse Faber-Krahn inequality to higher dimensions and p-Laplacian, demonstrating geometric constraints on eigenfunctions' nodal sets in spherical domains.

Contribution

It generalizes the inequality to higher dimensions and p-Laplacian, and applies it to analyze the shape of nodal sets of second eigenfunctions.

Findings

Extended inequality to higher dimensions and p-Laplacian.

Proved nodal sets of second eigenfunctions cannot be concentric spheres.

Demonstrated geometric constraints on eigenfunctions in spherical domains.

Abstract

Payne-Weinberger showed that \textit{`among the class of membranes with given area $A$, free along the interior boundaries and fixed along the outer boundary of given length $L_0$, the annulus $\Omega^\#$ has the highest fundamental frequency,'} where $\Omega^\#$ is a concentric annulus with the same area as $\Omega$ and the same outer boundary length as $L_0$. We extend this result for the higher dimensional domains and $p$-Laplacian with $p\in (1,\infty),$ under the additional assumption that the outer boundary is a sphere. As an application, we prove that the nodal set of the second eigenfunctions of $p$-Laplacian (with mixed boundary conditions) on a ball and a concentric annulus cannot be a concentric sphere.