2022/03/30 by Beniamin Bogosel, Bogosel, Beniamin, Dorin Bucur +1 · 4 citations
Mathematics · #Mathematics and Applications #Point processes and geometric inequalities #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2203.16409
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.