# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Two new Weyl-type bounds for the Dirichlet LaplacianHTML articles powered by AMS MathViewer

by Lotfi Hermi
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 $$N(\lambda ) > \dfrac {2}{n+2} \ \dfrac {1}{H_n} \ \left (\lambda -\lambda _1\right )^{n/2} \ \lambda _1^{-n/2} \notag$$ and $$N(\lambda ) > \left (\dfrac {n+2}{n+4}\right )^{n/2} \ \dfrac {1}{H_n} \ \left (\lambda -(1+4/n) \ \lambda _1\right )^{n/2} \ \lambda _1^{-n/2}, \notag$$ where $$H_n=\dfrac {2 \ n}{j_{n/2-1,1}^2 J_{n/2}^2(j_{n/2-1,1})} \notag$$ is a constant which depends on $n$, the dimension of the underlying space, and Bessel functions and their zeros.
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