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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
DOI: https://doi.org/10.1090/S0002-9939-02-06834-X
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 \[ \lambda _{k}(\Omega )\thicksim C_{n}(k/V(\Omega ))^{2/n}.\] 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 $\Omega$.


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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
Article copyright: © Copyright 2002 American Mathematical Society