Monotonicity of the first nonzero Steklov eigenvalue of regular $N$-gon with fixed perimeter

TL;DR

This paper proves that the first nonzero Steklov eigenvalue of perimeter-normalized regular N-gons increases with N, using an analytic approach and computer verification for small N.

Contribution

It establishes the monotonicity of the Steklov eigenvalue for all regular N-gons with fixed perimeter, combining asymptotic analysis and computational methods.

Findings

Monotonic increase of eigenvalue for N ≥ 20

Computer-assisted verification for 3 ≤ N ≤ 20

Analytic framework linking eigenvalues to Toeplitz operators

Abstract

We study the first nontrivial Steklov eigenvalue of perimeter-normalized regular \(N\)-gons and show that it is strictly increasing in \(N\). The proof mainly relies on an analytic framework that establishes a refined asymptotic expansion in three steps: first, identifying the Steklov eigenvalue as the maximal eigenvalue of a Toeplitz-type operator; second, deriving the eigenvalue and its associated eigenfunctions simultaneously via Schur reduction; and finally, obtaining the exact coefficients in the Schur moment expansion by evaluating Euler-type sums. The monotonicity is proved to be eventual, holding for \(N\ge 20\). For the remaining cases \(3\le N\le 20\), we provide complementary computer-assisted verification, confirming monotonicity across the full range of \(N\).