On the Polygonal Faber-Krahn Inequality

03/30/2022
by   Beniamin Bogosel, et al.
0

It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with n sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each n ≥ 5 the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For n=5, 6,7, 8 we perform this computation and certify the numerical approximation by finite elements, up to machine errors.

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