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Some upper bounds for sums of eigenvalues of the Neumann Laplacian

Authors: Liangpan Li and Lan Tang
Journal: Proc. Amer. Math. Soc. 134 (2006), 3301-3307
MSC (2000): Primary 35P15; Secondary 58G25
Published electronically: May 12, 2006
MathSciNet review: 2231915
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Abstract: Let $\mu _{k}(\Omega )$ be the $k$th Neumann eigenvalue of a bounded domain $\Omega$ with piecewisely smooth boundary in $\textbf {R}^{n}$. In 1992, P. Kröger proved that $k^{-\frac {n+2}{n}}\sum _{j=1}^{k}\mu _{j}\leq {4n\pi ^{2}\over n+2}( \omega _{n}V)^{-2/n}$, 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 $k$ to its right-hand side which approaches strictly from below to 1 as $k$ tends to infinity. Second, we will generalize Kröger’s estimate to the case when $\Omega$ is a compact Euclidean submanifold.

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Additional Information

Liangpan Li
Affiliation: Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China

Lan Tang
Affiliation: Department of Mathematics, Xidian University, Xi’an 710071, People’s Republic of China
Address at time of publication: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712

Keywords: Eigenvalue, Neumann Laplacian
Received by editor(s): November 1, 2004
Received by editor(s) in revised form: May 28, 2005
Published electronically: May 12, 2006
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.