Two new Weyl-type bounds for the Dirichlet Laplacian

Hermi, Lotfi

doi:10.1090/S0002-9947-07-04254-7

Two new Weyl-type bounds for the Dirichlet Laplacian
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Trans. Amer. Math. Soc. 360 (2008), 1539-1558 Request permission

Abstract:

In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following lower bounds for its counting function. For $\lambda \ge \lambda _1$, one has \begin{equation} N(\lambda ) > \dfrac {2}{n+2} \ \dfrac {1}{H_n} \ \left (\lambda -\lambda _1\right )^{n/2} \ \lambda _1^{-n/2} \notag \end{equation} and \begin{equation} N(\lambda ) > \left (\dfrac {n+2}{n+4}\right )^{n/2} \ \dfrac {1}{H_n} \ \left (\lambda -(1+4/n) \ \lambda _1\right )^{n/2} \ \lambda _1^{-n/2}, \notag \end{equation} where \begin{equation} H_n=\dfrac {2 \ n}{j_{n/2-1,1}^2 J_{n/2}^2(j_{n/2-1,1})} \notag \end{equation} is a constant which depends on $n$, the dimension of the underlying space, and Bessel functions and their zeros.

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