Some upper bounds for sums of eigenvalues of the Neumann Laplacian
Liangpan Li, Lan Yu Tang · Proceedings of the American Mathematical Society · 2006
Let μ k ( Ω ) \mu _{k}(\Omega ) be the k k th Neumann eigenvalue of a bounded domain Ω \Omega with piecewisely smooth boundary in R n \textbf {R}^{n} . In 1992, P. Kröger proved that k − n + 2 n ∑ j = 1 k μ j ≤ 4 n π 2 n + 2 ( ω n V ) − 2 / n k^{-\frac {n+2}{n}}\sum _{j=1}^{k}\mu _{j}\leq {4n\pi ^{2}\over n+2}( \omega _{n}V)^{-2/n} , where the upper bound is sharp in view of Weyl’s asymptotic formula. The aim of this paper is twofold. First, we will improve this estimate by multiplying a factor in terms of k k to its right-hand side which approaches strictly from below to 1 as k k tends to infinity. Second, we will generalize Kröger’s estimate to the case when Ω