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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 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
Proc. Amer. Math. Soc. 131 (2003), 631-636
DOI: https://doi.org/10.1090/S0002-9939-02-06834-X
Published electronically: September 25, 2002

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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Bibliographic 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