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Article overview
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Bounding Eigenvalues with Packing Density | Neal Coleman
; | Date: |
28 Aug 2015 | Abstract: | We prove a lower bound on the eigenvalues $lambda_k$, $kinmathbb{N}$, of
the Dirichlet Laplacian of a bounded domain $Omegasubsetmathbb{R}^n$ of
volume $V$: $$ lambda_k geq C_nigg( deltafrac{k}{V}igg)^{2/n} $$ where
$delta$ is a constant that measures how efficiently $Omega$ can be packed
into $mathbb{R}^n$ and $C_n$ is the constant found in Weyl’s law.
If $delta^{2/n} > n/(n+2)$, this bound is stronger than the eigenvalue bound
proven by Li and Yau in 1983. For example, in the case of convex planar
domains, we have for all $kinmathbb{N}$, $$ lambda_k geq frac{2sqrt{3}pi
k}{V}. $$ | Source: | arXiv, 1508.7346 | Services: | Forum | Review | PDF | Favorites |
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