An isoperimetric inequality for eigenvalues of the bi-harmonic operator

TL;DR

This paper establishes an isoperimetric inequality for the first non-zero Neumann eigenvalue of the bi-harmonic operator on smooth domains, showing it is maximized by the ball of equal volume.

Contribution

It proves a Szeg"o-Weinberger type inequality for the bi-harmonic operator's eigenvalues, extending classical results to higher-order operators and Neumann boundary conditions.

Findings

First non-zero eigenvalue is maximized by the ball of equal volume.

The inequality extends to higher even-multi-Laplacian operators.

Provides a new isoperimetric inequality for bi-harmonic Neumann eigenvalues.

Abstract

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $\Delta^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero eigenvalue $\Upsilon_1(\Om)$ admits the isoperimetric inequality of Szeg\"o-Weinberger type: $\Upsilon_1(\Om)\le \Upsilon_1(B_{\Om})$, where $B_{\Om}$ is a ball in ${\bf R}^n$ with the same volume of $\Om$. The isoperimetric inequality of Szeg\"o-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $\Delta^{2m}$ ($m\ge1$) on $\Om$ is also exploited.