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arXiv · 2610.13139

Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains

Quanyu Tang·Zuoqin Wang
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Abstract

We establish the strict Dirichlet Pólya inequality on every bounded simply connected planar domain. The proof uses a unitary boundary transport formulation of the Helmholtz equation and relates Dirichlet resonances to unitary eigenphase crossings through the Dirichlet-to-Neumann map. An exact area identity yields the sharp Weyl coefficient, while a Ky Fan estimate leaves a strictly positive remainder in the counting estimate. The general case follows by conformal exhaustion.

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Quanyu Tang, Zuoqin Wang. 2026-10-09. Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains. https://arxiv.org/abs/2610.13139

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