On principal frequencies and isoperimetric ratios in convex sets
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 4, pp. 977-1005.

On a convex set, we prove that the Poincaré–Sobolev constant for functions vanishing at the boundary can be bounded from above by the ratio between the perimeter and a suitable power of the N-dimensional measure. This generalizes an old result by Pólya. As a consequence, we obtain the sharp Buser’s inequality (or reverse Cheeger inequality) for the p-Laplacian on convex sets. This is valid in every dimension and for every 1<p<+. We also highlight the appearing of a subtle phenomenon in shape optimization, as the integrability exponent varies.

Pour un ensemble convexe, on démontre que la constante de Poincaré–Sobolev pour les fonctions qui s’annulent au bord, peut être majorée par le rapport entre le perimétre et une puissance opportune de la mesure N-dimensionnelle. Ceci généralise un vieux résultat de Pólya. En consequence de ce résultat, on obtient l’inégalité de Buser (ou inégalité inverse de Cheeger) sous forme optimale, pour le p-Laplacian sur les ensembles convexes. Cela est valable pour toute dimension et tout 1<p<+. On souligne aussi l’apparition d’un phénomène subtil en optimisation de formes, lorsque l’exposant d’intégrabilité varie.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1653
Classification: 35P15, 49J40, 35J70
Keywords: Buser’s inequality, convex sets, $p-$Laplacian, Cheeger constant, shape optimization
Brasco, Lorenzo 1

1 Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121 Ferrara (Italy)
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Brasco, Lorenzo. On principal frequencies and isoperimetric ratios in convex sets. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 4, pp. 977-1005. doi : 10.5802/afst.1653. http://www.numdam.org/articles/10.5802/afst.1653/

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