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 R n . We study eigenvalues of an eigenvalue problem of a system of elliptic equations: { Δ u + α grad ( div u ) = − σ u , 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 ) th eigenvalue σ 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.