Pólya's Conjecture Up to ε‐loss and Quantitative Estimates for the Remainder of Weyl's Law

ABSTRACT Let be a bounded Lipschitz domain. For any we show that for any Dirichlet eigenvalue , it holds, where, is given explicitly. This reduces the ‐loss version of Pólya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigenvalues. Our arguments in deriving such uniform estimates yield also, in all dimensions , classes of domains that may even have rather irregular shapes or boundaries but satisfy Pólya's conjecture. Another key observation is that on strip‐tiling domains (and therefore any triangles for instance) one actually has better eigenvalue estimates than Pólya conjectured.