Mathematics – Differential Geometry
Scientific paper
2010-05-17
Mathematics
Differential Geometry
19 pages. This is a revised version. Since the third and forth authors also prove similar results to the first version. We dec
Scientific paper
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.
Chen Daguang
Cheng Qing-Ming
Wang Qiaoling
Xia Changyu
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