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

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 als