On eigenvalues of a system of elliptic equations and of the biharmonic operator

Let Ω be a bounded domain in an n  -dimensional Euclidean space RnRn. We study eigenvalues of an eigenvalue problem of a system of elliptic equations:{Δu+αgrad(divu)=−σu,in Ω,u|∂Ω=0. Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the (k+1)(k+1)th eigenvalue σk+1σk+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.