Two new weyl-type bounds for the dirichlet laplacian

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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 λ ≥ λ1 one has N(λ)> 2/n+2 1/Hn(λ-λ1)n/2 λ1-n/2 and N(λ) >(n+2/n+4)n/2 1/H n (λ-(1+4/n)λ1)n/2λ1-n/2 where Hn=2n/j2n/2-1,1J2 n/2(jn/2-1,1) is a constant which depends on n, the dimension of the underlying space, and Bessel functions and their zeros.

Original languageEnglish (US)
Pages (from-to)1539-1558
Number of pages20
JournalTransactions of the American Mathematical Society
Volume360
Issue number3
DOIs
StatePublished - Mar 1 2008

Keywords

  • Dirichlet problem
  • Eigenvalues of the Laplacian
  • Kröger bounds
  • Li-Yau bounds
  • Neumann problem
  • Weyl asymptotics

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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