The lower bound property of the Morley element eigenvalues

In this paper, we prove that the Morley element eigenvalues approximate the exact ones from below on regular meshes, including adaptive local refined meshes, for the fourth-order elliptic eigenvalue problems with the clamped boundary condition in any dimension. And we implement the adaptive computation with the Morley element to obtain lower bounds of eigenvalues for the vibration problem of clamped plate under tension and a fourth-order elliptic eigenvalue problem with variable coefficients.