Calculus of Variations and Geometric Measure Theory


On reverse Faber-Krahn inequalities

created by k on 28 Oct 2020
modified on 18 May 2021


Published Paper

Inserted: 28 oct 2020
Last Updated: 18 may 2021

Journal: J. Math. Anal. Appl.
Volume: 485
Number: 1
Pages: 20
Year: 2020

ArXiv: 1806.11491 PDF


Payne-Weinberger showed that `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.

Keywords: isoperimetric inequality, Elasticity problem, First eigenvalue of $p$-Laplacian, Interior parallels, Nagy's inequality, Non-radiality