Lower bounds for eigenvalues of Laplacian operator and the clamped plate problem

We first investigate the lower bound for higher eigenvalues λi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _i$$\end{document} of the Laplace operator on a bounded domain and obtain a sharp lower bound. Then, we extent this estimate of the eigenvalues to general cases. Finally, we study the eigenvalues Γi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _i$$\end{document} for the clamped plate problem and deliver a sharp bound for the clamped plate problem for arbitrary dimension.