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arXiv · 2603.25116

Global Strict Monotonicity and Asymptotics of the First Steklov Eigenvalue for Regular Polygons

Zhuo Cheng·Changfeng Gui·Yeyao Hu·Qinfeng Li·Ruofei Yao
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Abstract

For a bounded Lipschitz planar domain $Ω$, let $σ_1(Ω)$ denote its first nonzero Steklov eigenvalue. Let $Ω_N$ be the regular $N$-gon normalized to have perimeter $2π$. We prove that \[ σ_1(Ω_{N+1})>σ_1(Ω_N),\qquad N\ge3. \] Consequently, $σ_1(Ω_N)$ increases strictly to the disk value $1$. We also show that the real first eigenspace has dimension two and derive the asymptotic expansion \[ σ_1(Ω_N) =1-\frac{2ζ(3)}{N^3} -\frac{8ζ(4)}{N^4} -\frac{26ζ(5)}{N^5} +O(N^{-6}). \] The proof combines an equivariant Schwarz--Christoffel pullback, a nodal selection of the two critical Fourier residue classes, and a positivity-improving Perron comparison for the associated reciprocal operators. The asymptotic expansion follows independently from a Schur reduction and the evaluation of Euler-type sums.

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Zhuo Cheng, Changfeng Gui, Yeyao Hu, Qinfeng Li, Ruofei Yao. 2026-09-17. Global Strict Monotonicity and Asymptotics of the First Steklov Eigenvalue for Regular Polygons. https://arxiv.org/abs/2603.25116

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