On the polygonal Faber-Krahn inequality

Abstract

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.Résumé (Sur l'inégalité Faber-Krahn polygonale). -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 ⩾ 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.

On second-order tensor representation of derivatives in shape optimization

“…They provide increased accuracy for numerical approximations, as evidenced for instance in [20,21,27,28], where both strong and weak forms of shape derivatives have been used and methodically compared for various numerical applications. Moreover, the weak form can be employed to study optimality conditions for shape optimization problems involving domains with low smoothness such as polygons [29,30]. In particular, shape optimization problems on polygons, polyhedrons and in general domains with corners and edges often appear in applications.…”