Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues
نویسنده
چکیده
We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. 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 λk := 1 k ∑ l≤k λl, and on the eigenvalue counting function. For example, we prove that for all domains and all k ≥ j 1+ d 2 1+ d 4 ,
منابع مشابه
. SP ] 2 4 M ay 2 00 7 Differential inequalities for Riesz means and Weyl - type bounds for eigenvalues 1
We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. 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 λ...
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