Estimates for eigenvalues of a system of of elliptic equations and of the biharmonic operator

TL;DR

This paper derives bounds for eigenvalues of elliptic systems and the biharmonic operator, providing new estimates and bounds relevant for mathematical physics and differential geometry.

Contribution

It introduces new eigenvalue estimates for elliptic systems and sharp bounds for biharmonic eigenvalues on curved manifolds.

Findings

Upper bounds for the (k+1)-th eigenvalue of the elliptic system.

Sharp lower bounds for the first eigenvalue of biharmonic problems.

Eigenvalue estimates depend on domain geometry and curvature.

Abstract

Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &\Delta {\mathbf u}+ \alpha{\rm grad}(\text{div}{\mathbf u})=-\sigma {\mathbf u}, \ \text{in $\Omega$}, &{\mathbf u}|_{\partial \Omega}={\mathbf 0}. \aligned . $$ Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the $(k+1)^{\text{th}}$ eigenvalue $\sigma_{k+1}$. We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive Ricci curvature.