ESTIMATES FOR SUMS OF EIGENVALUES OF THE LAPLACIAN

The aim of this paper is to give bounds for the eigenvalues of the Lap lacian on a domain in Euclidean space and on a compact Riemannian mani fold. First, we consider the eigenvalue problem for the Laplacian on a bounded domain in Euclidean space under Dirichlet and Neumann boundar y conditions. Our method for obtaining an upper bound for sums of eige nvalues under Dirichlet boundary conditions is closely related to the method used earlier (J. Funct. Anal, 106, 1992, 353-357) for the task of getting an upper bound for sums of eigenvalues under Neumann bounda ry conditions. On the other hand, we modify the method used by P. Li a nd S. T. Yau (Comm. Math. Phys. 88, 1983, 309-318) for obtaining a low er bound for sums of eigenvalues under Dirichlet boundary conditions i n order to get a lower bound for sums of eigenvalues under Neumann bou ndary conditions under the assumption that the domain under considerat ion is Lipschitz equivalent to a ball. Finally, we derive estimates fo r sums of squares of eigenvalues on a compact Riemannian manifold with out boundary. (C) 1994 Academic Press, Inc.