Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law

Abstract: Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $\epsilon\in (0,1)$ we show that for any Dirichlet eigenvalue $\lambda_k(\Omega)>\Lambda(\epsilon,\Omega)$, it holds \begin{align*} k&\le (1+\epsilon)\frac{|\Omega|\omega(n)}{(2\pi)^n}\lambda_k(\Omega)^{n/2}, \end{align*} where $\Lambda(\epsilon,\Omega)$ is given explicitly. This reduces the $\epsilon$-loss version of P\'olya's conjecture to a computational problem. This estimate is based on a uniform estimate on the remainder of the Weyl law on Lipschitz domains, which appears to be the first quantitative estimate for the remainder of Weyl's law since Weyl's seminal work in the year 1911. We also provide in all dimensions $n\ge 2$, a class of domains that may even have rather irregular shapes or boundaries but satisfy P\'olya's conjecture.