### 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/2}λ1^{-n/2} where H_{n}=2n/j^{2}_{n/2-1,1}J^{2} _{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) |
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Pages (from-to) | 1539-1558 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 360 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2008 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Two new weyl-type bounds for the dirichlet laplacian.** / Hermi, Lotfi.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 360, no. 3, pp. 1539-1558. https://doi.org/10.1090/S0002-9947-07-04254-7

}

TY - JOUR

T1 - Two new weyl-type bounds for the dirichlet laplacian

AU - Hermi, Lotfi

PY - 2008/3

Y1 - 2008/3

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

UR - http://www.scopus.com/inward/record.url?scp=43049116937&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43049116937&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-07-04254-7

DO - 10.1090/S0002-9947-07-04254-7

M3 - Article

AN - SCOPUS:43049116937

VL - 360

SP - 1539

EP - 1558

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -