Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues
| dc.creator | Harrell II, Evans M. | |
| dc.creator | Hermi, Lotfi | |
| dc.date | 2007-05-24 | |
| dc.date.accessioned | 2026-07-07T08:03:12Z | |
| dc.date.available | 2026-07-07T08:03:12Z | |
| dc.description | We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, R_σ(z) := \sum_k{(z -λ_k)_+^σ}. Here ${λ_k}_{k=1}^{\infty}$ are the ordered eigenvalues of the Laplacian on a bounded domain $Ω\subset \R^d$, and $x_+ := \max(0, x)$ denotes the positive part of the quantity $x$. As corollaries of these inequalities, we derive Weyl-type bounds on $λ_k$, on averages such as $\bar{λ_k} := {\frac 1 k}\sum_{\ell \le k}λ_\ell$, and on the eigenvalue counting function. For example, we prove that for all domains and all $k \ge j \frac{1+\frac d 2}{1+\frac d 4}$, {\bar{λ_{k}}}/{\bar{λ_{j}}} \le 2 (\frac{1+\frac d 4}{1+\frac d 2})^{1+\frac 2 d}({\frac k j})^{\frac 2 d}. | |
| dc.description | 21 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/0705.3673 | |
| dc.identifier | http://arxiv.org/abs/0705.3673 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/129491 | |
| dc.subject | Spectral Theory | |
| dc.subject | Mathematical Physics | |
| dc.subject | Primary 35P15; Secondary 47A75, 49R50, 58J50 | |
| dc.title | Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues | |
| dc.type | text |