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

TL;DR

This paper establishes differential and difference inequalities for Riesz means of Laplacian eigenvalues, leading to Weyl-type bounds on individual eigenvalues, averages, and counting functions, applicable to all bounded domains.

Contribution

It introduces novel differential inequalities for Riesz means of eigenvalues, enabling new bounds on eigenvalues and their averages for the Dirichlet Laplacian.

Findings

Derived inequalities for Riesz means of eigenvalues.

Established Weyl-type bounds for eigenvalues and their averages.

Provided bounds on eigenvalue ratios for all domains.

Abstract

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, R_{\sigma}(z) := \sum_k{(z -\lambda_k)_+^{\sigma}}. Here ${\lambda_k}_{k=1}^{\infty}$ are the ordered eigenvalues of the Laplacian on a bounded domain $\Omega \subset \R^d$, and $x_+ := \max(0, x)$ denotes the positive part of the quantity $x$. As corollaries of these inequalities, we derive Weyl-type bounds on $\lambda_k$, on averages such as $\bar{\lambda_k} := {\frac 1 k}\sum_{\ell \le k}\lambda_\ell$, 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{\lambda_{k}}}/{\bar{\lambda_{j}}} \le 2 (\frac{1+\frac d 4}{1+\frac d 2})^{1+\frac 2 d}({\frac k j})^{\frac 2 d}.