Monotonicity of the first Dirichlet eigenvalue of regular polygons
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908783245000704 |
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| author | Dahne, Joel Gómez-Serrano, Javier Pech-Alberich, Joana |
| author_facet | Dahne, Joel Gómez-Serrano, Javier Pech-Alberich, Joana |
| contents | In this paper we prove that the first Dirichlet eigenvalue $λ_1^N$ of an $N$-sided regular polygon of fixed area is a monotonically decreasing function of $N$ for all $N \geq 3$, as well as the monotonicity of the quotients $\displaystyle \frac{λ_1^{N}}{λ_1^{N+1}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16285 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monotonicity of the first Dirichlet eigenvalue of regular polygons Dahne, Joel Gómez-Serrano, Javier Pech-Alberich, Joana Spectral Theory Analysis of PDEs In this paper we prove that the first Dirichlet eigenvalue $λ_1^N$ of an $N$-sided regular polygon of fixed area is a monotonically decreasing function of $N$ for all $N \geq 3$, as well as the monotonicity of the quotients $\displaystyle \frac{λ_1^{N}}{λ_1^{N+1}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006]. |
| title | Monotonicity of the first Dirichlet eigenvalue of regular polygons |
| topic | Spectral Theory Analysis of PDEs |
| url | https://arxiv.org/abs/2601.16285 |