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arXiv · 1208.5226

Lower bound estimates for eigenvalues of the Laplacian

Qing-Ming Cheng·Xuerong Qi
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Abstract

For an $n$-dimensional polytope $Ω$ in $\mathbb{R}^{n}$, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau (1983) obtained the first term with the order $k^{\frac2n}$, which is optimal. The next landmark goal is to give the second term with the order $k^{\frac1n}$ in the asymptotic formula. For this purpose, Kovař\'ık, Vugalter and Weidl (2009) have made an important breakthrough in the case of dimension 2. It is our purpose to study the $n$-dimensional case for arbitrary dimension $n$. We obtain the second term in the asymptotic sense.

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Qing-Ming Cheng, Xuerong Qi. 2012-08-26. Lower bound estimates for eigenvalues of the Laplacian. https://arxiv.org/abs/1208.5226

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