TY - JOUR

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

AU - Harrell, Evans M.

AU - Hermi, Lotfi

N1 - Funding Information:
We would like to acknowledge the crucial assistance and guidance of Robert Haggerty during all aspects of photometric film development in the AS&E Solar Physics Program. Anna Franco and Daniel O'Mara provided significant laboratory support to this project. We would like to thank S. Kahler, A. Krieger and D. Webb for useful discussions. This work builds on the strong tradition of quantitative image analysis that is a hallmark of AS&E. This work was supported by NASA contract NAS5-25496 and NASA GSRP Grant NGT-50308.

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

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U2 - 10.1016/j.jfa.2008.02.016

DO - 10.1016/j.jfa.2008.02.016

M3 - Article

AN - SCOPUS:43049110988

SN - 0022-1236

VL - 254

SP - 3173

EP - 3191

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 12

ER -