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

Abstract

For a bounded Lipschitz planar domain $\Omega$, let ${\sigma }_1(\Omega )$ denote its first nonzero Steklov eigenvalue. Let ${\Omega }_N$ be the regular $N$-gon normalized to have perimeter $2\pi$. We prove that \[ σ_1(Ω_{N+1})>σ_1(Ω_N),\qquad N\ge3. \] Consequently, ${\sigma }_1({\Omega }_N)$ 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.