Payne-Pólya-Weinberger Type Inequalities

Abstract

We develop the spectral theory of the Laplacian on domains Ω ⊂ R n, including the classical result showing completeness of the Laplacian spectrum and the variational characterization of eigenvalues via the Rayleigh-Ritz formulation. These results are meant as an introduction to the Payne–P´olya– Weinberger Inequality, which states that the fundamental ratio, λ2/λ1, as a function of a domain Ω ∈ R n is maximized by the n-ball. We consider the Polygonal Payne–P´olya–Weinberger Conjecture, which states that among planar k-polygons, the regular k-polygon maximizes λ2/λ1. We show that among the class of triangles, λ2/λ1 is uniquely maximized by the equilateral triangle, and that among quadrilaterals λ2/λ1 is uniquely maximized by the square. To do so, we use perturbative techniques to establish local maximality together with a continuity estimate to reduce the problem to finitely many computations accomplished by a computer. This provides an alternative proof of the k = 3 case, a proof for the convex k = 4 case, and the sketch of a proof for the non-convex k = 4 case.