2007/05/24 by Evans M. Harrell, Evans M. Harrell II, Harrell, Evans M. +2
Computer Science · Mathematics · Physics and Astronomy · #49R50 #58J50 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical methods in inverse problems #Primary 35P15 #Secondary 47A75 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:35P15 #msc:47A75 #msc:49R50 #msc:58J50
paper · pdf · doi:10.48550/arxiv.0705.3673
21 pages, 3 figures
arxiv created 2007/05/24 · openalex publication_date 2007/05/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑k(z -λk)+σ. Here λkk=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 λk := \frac 1 k∑ℓ ≤ kλ_ℓ, and on the eigenvalue counting function. For example, we prove that for all domains and all k ≥ j (1+\frac d 2)/(1+\frac d 4), λk/λj ≤ 2 ((1+\frac d 4)/(1+\frac d 2))1+\frac 2 d(\frac k j)\frac 2 d.