### Abstract

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian,R_{ρ} (z) : = under(∑, k) (z - λ_{k})_{+}^{ρ} . Here {λ_{k}}_{k = 1}^{∞} are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ R^{d}, and x_{+} : = max (0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λ_{k}, on averages such as over(λ_{k}, -) : = frac(1, k) ∑_{ℓ ≤ k} λ_{ℓ}, and on the eigenvalue counting function. For example, we prove that for all domains and all k ≥ j frac(1 + frac(d, 2), 1 + frac(d, 4)),frac(over(λ_{k}, -), over(λ_{j}, -)) ≤ 2 (frac(1 + frac(d, 4), 1 + frac(d, 2)))^{1 + frac(2, d)} (frac(k, j))^{frac(2, d)} .

Original language | English (US) |
---|---|

Pages (from-to) | 3173-3191 |

Number of pages | 19 |

Journal | Journal of Functional Analysis |

Volume | 254 |

Issue number | 12 |

DOIs | |

State | Published - Jun 15 2008 |

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### Keywords

- Dirichlet problem
- Laplacian
- Riesz means
- Universal bounds
- Weyl law

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*254*(12), 3173-3191. https://doi.org/10.1016/j.jfa.2008.02.016

**Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues.** / Harrell, Evans M.; Hermi, Lotfi.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 254, no. 12, pp. 3173-3191. https://doi.org/10.1016/j.jfa.2008.02.016

}

TY - JOUR

T1 - Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

AU - Harrell, Evans M.

AU - Hermi, Lotfi

PY - 2008/6/15

Y1 - 2008/6/15

N2 - We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian,Rρ (z) : = under(∑, k) (z - λk)+ρ . Here {λk}k = 1∞ are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ : = max (0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as over(λk, -) : = frac(1, k) ∑ℓ ≤ k λℓ, and on the eigenvalue counting function. For example, we prove that for all domains and all k ≥ j frac(1 + frac(d, 2), 1 + frac(d, 4)),frac(over(λk, -), over(λj, -)) ≤ 2 (frac(1 + frac(d, 4), 1 + frac(d, 2)))1 + frac(2, d) (frac(k, j))frac(2, d) .

AB - We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian,Rρ (z) : = under(∑, k) (z - λk)+ρ . Here {λk}k = 1∞ are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ : = max (0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as over(λk, -) : = frac(1, k) ∑ℓ ≤ k λℓ, and on the eigenvalue counting function. For example, we prove that for all domains and all k ≥ j frac(1 + frac(d, 2), 1 + frac(d, 4)),frac(over(λk, -), over(λj, -)) ≤ 2 (frac(1 + frac(d, 4), 1 + frac(d, 2)))1 + frac(2, d) (frac(k, j))frac(2, d) .

KW - Dirichlet problem

KW - Laplacian

KW - Riesz means

KW - Universal bounds

KW - Weyl law

UR - http://www.scopus.com/inward/record.url?scp=43049110988&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43049110988&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2008.02.016

DO - 10.1016/j.jfa.2008.02.016

M3 - Article

AN - SCOPUS:43049110988

VL - 254

SP - 3173

EP - 3191

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 12

ER -