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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A lower bound for sums of eigenvalues of the Laplacian


Author: Antonios D. Melas
Journal: Proc. Amer. Math. Soc. 131 (2003), 631-636
MSC (2000): Primary 58G25; Secondary 35P15, 58G05
Published electronically: September 25, 2002
MathSciNet review: 1933356
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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

\begin{displaymath}\lambda _{k}(\Omega )\thicksim C_{n}(k/V(\Omega ))^{2/n}.\end{displaymath}

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 $.


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Additional Information

Antonios D. Melas
Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
Email: amelas@math.uoa.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06834-X
PII: S 0002-9939(02)06834-X
Received by editor(s): August 28, 2001
Published electronically: September 25, 2002
Communicated by: Bennett Chow
Article copyright: © Copyright 2002 American Mathematical Society