On the first dirichlet laplacian eigenvalue of regular polygons

Abstract

The Faber-Krahn inequality in R2 states that among all open bounded sets of given area the disk minimizes the first Dirichlet Laplacian eigenvalue. It was conjectured in [1] that for all N ≥ 3 the first Dirichlet Laplacian eigenvalue of the regular N -gon is greater than the one of the regular (N + 1)-gon of same area. This natural idea is suggested by the fact that the shape becomes more and more “rounded” as N increases and it is supported by clear numerical evidences. Aiming to settle such a conjecture, in this work we investigate possible ways to estimate the di¤erence between eigenvalues of regular N-gons and (N + 1)-gons.