An isoperimetric inequality for eigenvalues of the bi-harmonic operator

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 ${\mathbf{R}}^n$ ($n\ge 2$) and then prove that the corresponding first non-zero eigenvalue $Υ_1(\Om)$ admits the isoperimetric inequality of Szegö-Weinberger type: $Υ_1(\Om)\le Υ_1(B_{\Om})$, where $B_{\Om}$ is a ball in ${\mathbf{R}}^n$ with the same volume of $\Om$. The isoperimetric inequality of Szegö-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators ${\Delta }^{2m}$ ($m\ge 1$) on $\Om$ is also exploited.

12 pages