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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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A lower bound for sums of eigenvalues of the Laplacian
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by Antonios D. Melas PDF
Proc. Amer. Math. Soc. 131 (2003), 631-636 Request permission

Abstract:

Let $\lambda _{k}(\Omega )$ be the $k$th Dirichlet eigenvalue of a bounded domain $\Omega$ in $\mathbb {R}^{n}$. According to Weyl’s asymptotic formula we have \[ \lambda _{k}(\Omega )\thicksim C_{n}(k/V(\Omega ))^{2/n}.\] The optimal in view of this asymptotic relation lower estimate for the sums $\sum _{j=1}^{k}\lambda _{j}(\Omega )$ has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its right-hand side a term of the order of $k$ that depends on the ratio of the volume to the moment of inertia of $\Omega$.
References
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Additional Information
  • Antonios D. Melas
  • Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
  • MR Author ID: 311078
  • Email: amelas@math.uoa.gr
  • Received by editor(s): August 28, 2001
  • Published electronically: September 25, 2002
  • Communicated by: Bennett Chow
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 631-636
  • MSC (2000): Primary 58G25; Secondary 35P15, 58G05
  • DOI: https://doi.org/10.1090/S0002-9939-02-06834-X
  • MathSciNet review: 1933356