On the first eigenvalue of the Laplacian for polygons

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.