Monotonicity of the first Dirichlet eigenvalue of regular polygons

TL;DR

This paper proves that the first Dirichlet eigenvalue of regular polygons decreases monotonically as the number of sides increases, confirming a long-standing conjecture and analyzing eigenvalue ratios.

Contribution

It establishes the monotonicity of the first Dirichlet eigenvalue for regular polygons with fixed area, settling a conjecture from 2006.

Findings

First Dirichlet eigenvalue decreases with increasing number of sides.

Monotonicity of the eigenvalue ratios between polygons with N and N+1 sides.

Confirmation of the Antunes-Freitas conjecture.

Abstract

In this paper we prove that the first Dirichlet eigenvalue $\lambda_1^N$ 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 $\displaystyle \frac{\lambda_1^{N}}{\lambda_1^{N+1}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006].