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 -