Two new Weyl-type bounds for the Dirichlet Laplacian

Hermi, Lotfi

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

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Two new Weyl-type bounds for the Dirichlet Laplacian

Author: Lotfi Hermi
Journal: Trans. Amer. Math. Soc. 360 (2008), 1539-1558
MSC (2000): Primary 35P15; Secondary 47A75, 49R50, 58J50
Posted: September 25, 2007
MathSciNet review: 2357704
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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

$\displaystyle N(\lambda) > \dfrac{2}{n+2} \dfrac{1}{H_n} \ \left(\lambda-\lambda_1\right)^{n/2} \lambda_1^{-n/2}$

and

$\displaystyle N(\lambda) > \left(\dfrac{n+2}{n+4}\right)^{n/2} \dfrac{1}{H_n} \ \left(\lambda-(1+4/n) \lambda_1\right)^{n/2} \lambda_1^{-n/2},$

where

$\displaystyle H_n=\dfrac{2 n}{j_{n/2-1,1}^2 J_{n/2}^2(j_{n/2-1,1})}$

is a constant which depends on

, the dimension of the underlying space, and Bessel functions and their zeros.

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