On the polygonal Faber-Krahn inequality

Beniamin Bogosel; Dorin Bucur

doi:10.5802/jep.250

On the polygonal Faber-Krahn inequality
[Sur l’inégalité Faber-Krahn polygonale]

Beniamin Bogosel ¹ ; Dorin Bucur ²

¹ CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
² Laboratoire de Mathématiques UMR CNRS 5127, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France

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

Résumé
Abstract

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 $n \geq 5$ 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.

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 $n \geq 5$ 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.

Reçu le : 2022-03-31
Accepté le : 2023-12-12
Publié le : 2023-12-21

DOI : 10.5802/jep.250

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

Affiliations des auteurs :

Beniamin Bogosel ¹ ; Dorin Bucur ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Beniamin Bogosel and Dorin Bucur},
     title = {On the polygonal {Faber-Krahn} inequality},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {19--105},
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Beniamin Bogosel; Dorin Bucur. On the polygonal Faber-Krahn inequality. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 19-105. doi : 10.5802/jep.250. https://jep.centre-mersenne.org/articles/10.5802/jep.250/

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