Upper bounds for the Neumann eigenvalues on a bounded domain in euclidean space

Abstract

Let μ k be the the kth eigenvalue for the Neumann boundary value problem with respect to the Laplace operator on a bounded domain Ω with piecewise smooth boundary in R n . Weyl's asymptotic formula states that μ k ~ C n( k V(Ω) ) 2 n . We aim to give an upper bound for partial sums ∑ j k = 1 μ j of eigenvalues. Our bound depends only on the volume of Ω and is to some extent asymptotically correct for any domain Ω. Because of the analogy between the problems of obtaining lower bounds for Dirichlet eigenvalues and upper bounds for Neumann eigenvalues (cf. G. Pólya, Proc. London Math. Soc. (3) 11 (1961), 419–433) we can adapt the technique used by Li and Yau ( Commun. Math. Phys. 88 (1983), 309–318) for the task of estimating Dirichlet eigenvalues.