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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Some upper bounds for sums of eigenvalues of the Neumann Laplacian
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by Liangpan Li and Lan Tang PDF
Proc. Amer. Math. Soc. 134 (2006), 3301-3307 Request permission

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
  • Email: liliangpan@yahoo.com.cn
  • 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
  • Email: ltang@math.utexas.edu
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3301-3307
  • MSC (2000): Primary 35P15; Secondary 58G25
  • DOI: https://doi.org/10.1090/S0002-9939-06-08355-9
  • MathSciNet review: 2231915