Estimates for Sums of Eigenvalues of the Laplacian

The aim of this paper is to give bounds for the eigenvalues of the Laplacian on a domain in Euclidean space and on a compact Riemannian manifold. First, we consider the eigenvalue problem for the Laplacian on a bounded domain in Euclidean space under Dirichlet and Neumann boundary conditions. Our method for obtaining an upper bound for sums of eigenvalues 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 boundary conditions. On the other hand, we modify the method used by P. Li and S. T. Yau (Comm. Math. Phys.88, 1983, 309-318) for obtaining a lower bound for sums of eigenvalues under Dirichlet boundary conditions in order to get a lower bound for sums of eigenvalues under Neumann boundary conditions under the assumption that the domain under consideration is Lipschitz equivalent to a ball. Finally, we derive estimates for sums of squares of eigenvalues on a compact Riemannian manifold without boundary.