## 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
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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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## 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