Monotonicity of the first Dirichlet eigenvalue of regular polygons

Abstract

In this paper we prove that the first Dirichlet eigenvalue $\hat{I}{»}_1^N$ of an $N$-sided regular polygon of fixed area is a monotonically decreasing function of $N$ for all $N\ge 3$, as well as the monotonicity of the quotients $\frac{\hat{I}{»}_1^N}{\hat{I}{»}_1^{N+1}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006].

46 pages, 5 figures