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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Two new Weyl-type bounds for the Dirichlet Laplacian
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by Lotfi Hermi PDF
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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Additional Information
  • Lotfi Hermi
  • Affiliation: Department of Mathematics, University of Arizona, 617 Santa Rita, Tucson, Arizona 85721
  • Email: hermi@math.arizona.edu
  • Received by editor(s): April 15, 2004
  • Received by editor(s) in revised form: February 3, 2006
  • Published electronically: September 25, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1539-1558
  • MSC (2000): Primary 35P15; Secondary 47A75, 49R50, 58J50
  • DOI: https://doi.org/10.1090/S0002-9947-07-04254-7
  • MathSciNet review: 2357704