On the first eigenvalue of the Laplacian for polygons

A 2006 conjecture of Antunes and Freitas is addressed connecting the scaling-invariant polygonal isoperimetric and principal frequency deficits for triangles. This yields a quantitative polygonal Faber–Krahn inequality for triangles with an explicit constant. Furthermore, a problem mentioned in the 1951 book “Isoperimetric Inequalities In Mathematical Physics” by Pólya and Szegö is addressed: a formula is given for the principal frequency of a triangle. Moreover, a space of polygons is constructed for the classical Pólya and Szegö problem: in 1947, Pólya proved that if n = 3, 4 the regular polygon P_n minimizes the principal frequency of an n-gon with given area α > 0 and suggested that the same holds when n ≥ 5. In 1951, Pólya and Szegö discussed the possibility of counterexamples. This paper constructs explicit (2n − 4)–dimensional polygonal manifolds $M(n,α)$ and proves the existence of a computable N ≥ 5 such that for all n ≥ N, the admissible n-gons are given via $M(n,α)$and there exists an explicit set $An(α)⊂M(n,α)$ such that P_n has the smallest principal frequency among n-gons in $An(α)$⁠. Inter-alia when n ≥ 3, a formula is proved for the principal frequency of a convex $P∈M(n,α)$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 $M(n,α)$near P_n. 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.