Two new weyl-type bounds for the dirichlet laplacian

Lotfi Hermi

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

Two new weyl-type bounds for the dirichlet laplacian

Lotfi Hermi

Mathematics

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

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/H_n(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 H_n=2n/j²_n/2-1,1J² _n/2(j_n/2-1,1) is a constant which depends on n, the dimension of the underlying space, and Bessel functions and their zeros.

Original language	English (US)
Pages (from-to)	1539-1558
Number of pages	20
Journal	Transactions of the American Mathematical Society
Volume	360
Issue number	3
DOIs	https://doi.org/10.1090/S0002-9947-07-04254-7
State	Published - Mar 2008

Keywords

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

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/S0002-9947-07-04254-7

Cite this

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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.",

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N2 - 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.

AB - 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.

KW - Dirichlet problem

KW - Eigenvalues of the Laplacian

KW - Kröger bounds

KW - Li-Yau bounds

KW - Neumann problem

KW - Weyl asymptotics

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

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ER -

Two new weyl-type bounds for the dirichlet laplacian

Abstract

Keywords

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