Inserted: 28 oct 2020
Last Updated: 18 may 2021
Journal: J. Math. Anal. Appl.
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