### 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) |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis

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## Cite this

*Journal of Functional Analysis*,

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