  In this paper, we give some lower bounds for several eigenvalues. Firstly, we
investigate the eigenvalues $\lambda_i$ of the Laplace operator and prove a
sharp lower bound. Moreover, we extent this estimate of the eigenvalues to
general cases. Secondly, we study the eigenvalues $\Gamma_i$ for the clamped
plate problem and deliver a sharp bound for the clamped plate problem for
arbitrary dimension.
