On the polygonal Faber-Krahn inequality - Archive ouverte HAL
Article Dans Une Revue Journal de l'École polytechnique — Mathématiques Année : 2023

On the polygonal Faber-Krahn inequality

Résumé

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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Dates et versions

hal-03625471 , version 1 (30-03-2022)

Identifiants

Citer

Beniamin Bogosel, Dorin Bucur. On the polygonal Faber-Krahn inequality. Journal de l'École polytechnique — Mathématiques, 2023, 11, pp.19-105. ⟨10.5802/jep.250⟩. ⟨hal-03625471⟩
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