Monotonicity of the first Dirichlet eigenvalue of regular polygons

TL;DR

The work resolves the monotonicity question for the first Dirichlet eigenvalue of regular polygons by combining a large-$N$ asymptotic framework with rigorous computer-assisted bounds and a small-$N$ MPS-based verification. It proves $\lambda_1(\mathcal{P}_N)$ is strictly decreasing in $N$ for all $N\ge3$ and that the ratio $q_N=\frac{\lambda_1(\mathcal{P}_N)}{\lambda_1(\mathcal{P}_{N+1})}$ decreases as well, thereby addressing the Antunes–Freitas conjecture. The approach hinges on precise Schwarz–Christoffel conformal maps, validated numerics to bound defects and norms, and high-accuracy eigenpair approximations with controlled errors. This yields rigorous enclosures for eigenvalues across the full range of $N$, linking spectral properties of polygons to area-normalized geometry and providing robust insights into polygonal Faber–Krahn-type phenomena in spectral geometry.

Abstract

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].

Figures (5)

• Figure 1: A regular $28$-gon of area $\pi$ centered at the origin shown only in the first quadrant. The red circle of radius $0.95$ lies entirely inside the polygon, and the blue circle of radius $1.01$ fully contains the polygon. Neither circle touches the polygon.
• Figure 2: An octagon ($8$-gon) of area $\pi$ centered at the origin. The polygon contains an inscribed disk of radius $I$ (red circle) and is contained within a circumscribed disk of radius $C$ (blue circle). The figure also highlights the areas of the triangles, into which the polygon is decomposed, to derive two equivalent formulas for its total area.
• Figure 3: Coefficients $a_{2}$, $b_{1}$ and $b_{2}$ used in the approximate eigenfunction \ref{['eq:approximation-small-N']} and their dependence on $N$. For $a_{2}$ we scale the coefficients by $J_{N}(1)$ to take into account the term's dependence on $N$. For low values of $N$ the domains are relatively different from each other and there are more oscillations in the coefficients, to better see the asymptotic behavior for $b_{1}$ and $b_{2}$ the graph therefore starts at $N = 10$.
• Figure 4: Distance between successive approximations and the computed error bounds. As long as the error is smaller than the distance, the monotonicity can be verified. For the eigenvalues the error bounds are computed as the sum of the radii for the enclosures of $\lambda_{1}(\mathcal{P}_{N})$ and $\lambda_{1}(\mathcal{P}_{N + 1})$, for the $q_{N}$'s it is the sum of the radii for the enclosures for $q_{N}$ and $q_{N + 1}$.
• Figure 5: Geometric bounds for $\log(1 - tz)$ when $|1 - tz| \leq C < 1$ and $\mathop{\mathrm{Im}}\nolimits z \leq 0$. The values are confined to the blue semi-infinite strip $(-\infty, \log(C)] \times i[0, \pi]$. This region is enclosed by a cone of angle $\theta$ relative to the negative real axis, where $\theta = \arctan(\pi / |\log C|)$.

Theorems & Definitions (59)

• Theorem 1.1
• proof
• Proposition 2.1
• Lemma 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof