On the first Dirichlet Laplacian eigenvalue of regular polygons

Abstract

The Faber-Krahn inequality in R 2 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.2010 Mathematics Subject Classification. Primary: 35P15; Secondary: 49R05, 35J25.

On mixed Dirichlet-Neumann eigenvalues of triangles

“…Even though right triangles are a rather special case, they are of interest in studying other polygonal domains. In particular, recent paper by Nitsch [23] studies regular polygons via eigenvalue perturbations on right triangles. Similar approach is taken in the author's upcoming collaboration [22].…”

On Dirichlet eigenvalues of regular polygons

“…By the general theory developed by Vekua [23, (13.5), p. 58] any function ϕ that satisfies ∆ϕ + λ (N ) ϕ = 0 in P N can be represented as (19) ϕ…”

“…Table 2. The functions V n (z) for n ≤ 4 |z| = 1 back into (19) we obtain a sequence of functions ϕ N,r : P N → R, and numbers λ N,r . The numbers λ N,r converge to λ k as N → ∞ for each fixed r and the functions ϕ N,r satisfy ∆ϕ N,r (z) + λ N,r ϕ N,r (z) = 0 , z ∈ P N , together with ϕ N,r 2 ≫ 1 and ϕ N,r | ∂P N ∞ ≪ r N −r−1 .…”

“…Note, however, that since ζ(3) > 0, equation (2) implies this inequality for all sufficiently large N. For further results along the theme of Faber-Krahn and eigenvalue optimization via regular polygons under the presence of various constraints (in-and circumradius normalization, etc.) we refer to [19], [21] and the references therein.…”

Riesz-type inequalities and overdetermined problems for triangles and quadrilaterals

“…Besides the above mentioned conjecture by Pólya and Szegő, solved only for the logarithmic capacity in [21], the problem of optimality of regular N -gons for variational functionals has been the object of several contributions. Among these, we mention the papers [5,10,17], dealing with various shape optimization problems on polygons involving spectral functionals, and [4], where it is proved that the regular polygon minimizes the Cheeger constant among polygons with fixed area and number of sides.…”