On the polygonal Faber-Krahn inequality

Bogosel, Beniamin; Bucur, Dorin

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 19-105

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 $n\ge 5$ 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.

Il y a soixante-dix ans, Pólya et Szegö ont conjecturé que l’ensemble du plan qui minimise la première valeur propre du laplacien avec conditions de Dirichlet au bord parmi les polygones de $n$ côtés et aire fixée est le polygone régulier. Malgré sa simplicité apparente, cette conjecture a été démontrée seulement pour les triangles et les quadrilatères. Dans cet article, nous démontrons que pour chaque $n\ge 5$ la preuve de la conjecture peut être réduite à un nombre fini de calculs numériques certifiés. En particulier, la minimalité locale du polygone régulier est réduite à un seul calcul certifié. Pour $n=5,6,7,8$ nous faisons ce calcul et nous certifions l’approximation par éléments finis, aux erreurs d’arrondi près.

Classification : 35P15, 49Q10
Keywords: Faber-Krahn inequality, polygons, shape optimization, numerical approximations
Mots-clés : Inégalité de Faber-Krahn, polygones, optimisation de forme, approximations numériques

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