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

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