On the first eigenvalue of the Laplacian for polygons

TL;DR

This paper investigates the principal frequency of polygons, proving the existence of counterexamples to a long-standing conjecture for polygons with five or more sides, and provides explicit formulas and stability results.

Contribution

It constructs explicit polygonal manifolds and proves the existence of polygons that minimize the principal frequency, addressing conjectures and stability problems in spectral geometry.

Findings

Existence of polygons with minimal principal frequency for large n

Explicit formula for convex polygons' principal frequency

Solution to the polygonal Faber-Krahn stability problem for triangles

Abstract

In 1947, P\'olya proved that if $n=3,4$ the regular polygon $P_n$ minimizes the principal frequency of an n-gon with given area $\alpha>0$ and suggested that the same holds when $n \ge 5$. In $1951,$ P\'olya & Szeg\"o discussed the possibility of counterexamples in the book "Isoperimetric Inequalities In Mathematical Physics." This paper constructs explicit $(2n-4)$--dimensional polygonal manifolds $\mathcal{M}(n, \alpha)$ and proves the existence of a computable $N \ge 5$ such that for all $n \ge N$, the admissible $n$-gons are given via $\mathcal{M}(n, \alpha)$ and there exists an explicit set $ \mathcal{A}_{n}(\alpha) \subset \mathcal{M}(n,\alpha)$ such that $P_n$ has the smallest principal frequency among $n$-gons in $\mathcal{A}_{n}(\alpha)$. Inter-alia when $n \ge 3$, a formula is proved for the principal frequency of a convex $P \in \mathcal{M}(n,\alpha)$ in terms of an equilateral $n$-gon with the same area; and, the set of equilateral polygons is proved to be an $(n-3)$--dimensional submanifold of the $(2n-4)$--dimensional manifold $\mathcal{M}(n,\alpha)$ near $P_n$. If $n=3$, the formula completely addresses a 2006 conjecture of Antunes and Freitas and another problem mentioned in "Isoperimetric Inequalities In Mathematical Physics." Moreover, a solution to the sharp polygonal Faber-Krahn stability problem for triangles is given and with an explicit constant. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and $W^{2,p}/BMO$ estimates. Last, an application is given in the context of electron bubbles.