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

Abstract

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}.