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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.