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

Zhuo Cheng, Changfeng Gui, Yeyao Hu, Qinfeng Li, Ruofei Yao

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 \(N\).

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

Abstract

We study the first nontrivial Steklov eigenvalue of perimeter-normalized regular -gons and show that it is strictly increasing in . 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 . For the remaining cases , we provide complementary computer-assisted verification, confirming monotonicity across the full range of .

Paper Structure

This paper contains 46 sections, 89 theorems, 551 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Equivalent formulation: a Steklov eigenvalue problem on the unit disk with polygonal weight
  3. Boundary regularity and conformal invariance of the Dirichlet form
  4. Weak formulation of the Steklov eigenvalue problem
  5. Weighted Rayleigh--Ritz Quotient
  6. Isospectral reduction to a weighted DtN problem on $\mathbb S^1$
  7. Block Decomposition and The Eigenvalue of an Infinite Dimensional Symmetric Matrix
  8. Fourier coefficients of the boundary weight: symmetry and sparsity
  9. The Toeplitz operator induced by $w_N$
  10. Matrix form of the weighted eigenproblem and residue-class block decomposition
  11. Normalized block operators and the reciprocal parameter $\lambda=1/\sigma$
  12. The Separation Lemma and the Identification of the First Non‑Zero Steklov Eigenvalue: the Schur Reduction
  13. Spectral separation of non-critical blocks
  14. The first cluster as a critical-block minimum
  15. Critical blocks and reduction to the $r=1$ block
  16. ...and 31 more sections

Key Result

Theorem 1.1

Recall that $\Omega_N$ is the regular $N$-gon normalized to have perimeter $2\pi$. Then, for all $N\ge 20$, where $\sigma_1(\Omega_N)$ denotes the first nonzero Steklov eigenvalue of $\Omega_N$ with multiplicity two.

Theorems & Definitions (196)

  • Conjecture 1.1: Existence and non-degeneracy of the maximizer
  • Conjecture 1.2: The Pólya--Szegő conjecture for the Steklov eignvalue
  • Theorem 1.1: Eventual monotonicity
  • Theorem 1.2: Asymptotic expansion with effective remainder estimate
  • Remark 1.1
  • Theorem 1.3: Global monotonicity
  • Remark 1.2
  • Lemma 2.1: Boundary pullback measure and perimeter normalization
  • proof
  • Lemma 2.2: Conformal invariance of the Dirichlet form
  • ...and 186 more