Calculus of Variations and Geometric Measure Theory

D. Bucur - B. Bogosel

On the Polygonal Faber-Krahn Inequality

created by bucur on 31 Mar 2022



Inserted: 31 mar 2022

Year: 2022

ArXiv: arXiv.2203.16409 PDF


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.