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

Evans M. Harrell, Lotfi Hermi

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

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 languageEnglish (US)
Pages (from-to)3173-3191
Number of pages19
JournalJournal of Functional Analysis
Volume254
Issue number12
DOIs
StatePublished - Jun 15 2008

Keywords

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

ASJC Scopus subject areas

  • Analysis

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