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

Evans M. Harrell, Lotfi Hermi

Research output: Contribution to journalArticle

20 Citations (Scopus)

### Abstract

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 λ, 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) .

Original language English (US) 3173-3191 19 Journal of Functional Analysis 254 12 https://doi.org/10.1016/j.jfa.2008.02.016 Published - Jun 15 2008

### Fingerprint

Riesz Means
Differential Inequalities
Eigenvalue
Difference Inequalities
Dirichlet Laplacian
Counting Function
Bounded Domain
Corollary
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### Keywords

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

• Analysis

### Cite this

In: Journal of Functional Analysis, Vol. 254, No. 12, 15.06.2008, p. 3173-3191.

Research output: Contribution to journalArticle

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abstract = "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) .",
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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) .

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