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

Abstract

Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space ${\mathbb{R}}^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &Δ{\mathbf u}+ α{\rm grad}(\text{div}{\mathbf u})=-σ{\mathbf u}, \ \text{in $Ω$}, &{\mathbf u}|_{\partial Ω}={\mathbf 0}. \aligned . $$ 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.

19 pages. This is a revised version. Since the third and forth authors also prove similar results to the first version. We decide to write this joint work

References in corpus (1)

• Universal bounds on eigenvalues of the buckling problem on spherical domains