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 difference between eigenvalues of regular N-gons and (N + 1)-gons.
Citation
Carlo Nitsch. "On the first Dirichlet Laplacian eigenvalue of regular polygons." Kodai Math. J. 37 (3) 595 - 607, October 2014. https://doi.org/10.2996/kmj/1414674611
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