2014/03/26 by Carlo Nitsch, Nitsch, Carlo · 1 citation
Mathematics · #35J25 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Point processes and geometric inequalities #Primary: 35P15 #Secondary: 49R05
paper · doi:10.48550/arxiv.1403.6709
openalex publication_date 2014/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Faber-Krahn inequality in ℝ2 states that among all open bounded sets of given area the disk minimizes the first Dirichlet Laplacian eigenvalue. There are numerical evidences 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 property is also suggested by the fact that the shape of regular polygons becomes more and more "rounded" as N increases and, among sets of given area, disk minimize the eigenvalue. Aiming to settle such a conjecture, in this work we investigate possible ways to estimate the difference between eigenvalues of regular N-gons and (N+1)-gons.