A lower bound for sums of eigenvalues of the Laplacian

Melas, Antonios

doi:10.1090/S0002-9939-02-06834-X

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 $\Omega$.

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