Lower bound estimates for eigenvalues of the Laplacian

Abstract

For an $n$-dimensional polytope $\Omega$ 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 $k^{\frac{2}{n}}$, which is optimal. The next landmark goal is to give the second term with the order $k^{\frac{1}{n}}$ 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.

15 pages

References in corpus (1)

• Two dimensional Berezin-Li-Yau inequalities with a correction term