LOWER-BOUND ESTIMATES FOR EIGENVALUE OF THE LAPLACE OPERATOR ON SURFACES OF REVOLUTION

Abstract

In this paper, we estimate eigenvalues of the Laplace operator on surfaces of revolution. We first reduce our Laplace eigenvalue problems to the corresponding Sturm-Liouville eigenvalue problems. Two variational inequalities are then used to obtain lower-bound estimates for eigenvalues of the corresponding Sturm-Liouville problems. Based on the relationship between eigenvalues of the Laplace problems and the Sturm-Liouville problems, we obtain lower-bound estimates for eigenvalues of the mixed and Neumann problems of the Laplace operator (Theorem 1 and Theorem 2). Indeed, our estimate in the first case is optimal.