Strict Dirichlet Pólya inequality on simply connected planar domains

Theorem 1.1 of arXiv:2610.13139: for every bounded nonempty simply connected planar domain $\Omega$, with no boundary regularity assumed, $|\Omega|\lambda_j(\Omega)>4\pi j$ for all $j\ge1$, equivalently $N_D(E)<|\Omega|E/(4\pi)$ for $E>0$ (eigenvalues counted with multiplicity, endpoint included). This is the strict form of Pólya's Dirichlet inequality on the whole simply connected planar class. Previously it was known for tiling domains (Pólya 1961), disks and sectors (Filonov, Levitin, Polterovich and Sher 2023), annuli and a few further families. Multiply connected planar domains and higher dimensions remain open. The companion strict Neumann result for simply connected Lipschitz domains is a separate entry.

Theorem 1.1 of arXiv:2610.13139: for every bounded nonempty simply connected planar domain $\Omega$, with no boundary regularity assumed, $|\Omega|\lambda_j(\Omega)>4\pi j$ for all $j\ge1$, equivalently $N_D(E)<|\Omega|E/(4\pi)$ for $E>0$ (eigenvalues counted with multiplicity, endpoint included). This is the strict form of Pólya's Dirichlet inequality on the whole simply connected planar class. Previously it was known for tiling domains (Pólya 1961), disks and sectors (Filonov, Levitin, Polterovich and Sher 2023), annuli and a few further families. Multiply connected planar domains and higher dimensions remain open. The companion strict Neumann result for simply connected Lipschitz domains is a separate entry.