Monotonicity of the first nonzero Steklov eigenvalue of regular $N$-gon with fixed perimeter
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915891862568960 |
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| author | Cheng, Zhuo Gui, Changfeng Hu, Yeyao Li, Qinfeng Yao, Ruofei |
| author_facet | Cheng, Zhuo Gui, Changfeng Hu, Yeyao Li, Qinfeng Yao, Ruofei |
| contents | 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 \(N\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25116 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monotonicity of the first nonzero Steklov eigenvalue of regular $N$-gon with fixed perimeter Cheng, Zhuo Gui, Changfeng Hu, Yeyao Li, Qinfeng Yao, Ruofei Analysis of PDEs 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 \(N\). |
| title | Monotonicity of the first nonzero Steklov eigenvalue of regular $N$-gon with fixed perimeter |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.25116 |