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 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 Laplacian sur les ensembles convexes. Cela est valable pour toute dimension et tout . On souligne aussi l’apparition d’un phénomène subtil en optimisation de formes, lorsque l’exposant d’intégrabilité varie.
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 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 Laplacian on convex sets. This is valid in every dimension and for every . We also highlight the appearing of a subtle phenomenon in shape optimization, as the integrability exponent varies.
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Mots clés : Buser’s inequality, convex sets, $p-$Laplacian, Cheeger constant, shape optimization
Lorenzo Brasco 1
CC-BY 4.0
@article{AFST_2020_6_29_4_977_0,
author = {Lorenzo Brasco},
title = {On principal frequencies and isoperimetric ratios in convex sets},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {977--1005},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 29},
number = {4},
year = {2020},
doi = {10.5802/afst.1653},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1653/}
}
TY - JOUR AU - Lorenzo Brasco TI - On principal frequencies and isoperimetric ratios in convex sets JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2020 SP - 977 EP - 1005 VL - 29 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1653/ DO - 10.5802/afst.1653 LA - en ID - AFST_2020_6_29_4_977_0 ER -
%0 Journal Article %A Lorenzo Brasco %T On principal frequencies and isoperimetric ratios in convex sets %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2020 %P 977-1005 %V 29 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1653/ %R 10.5802/afst.1653 %G en %F AFST_2020_6_29_4_977_0
Lorenzo Brasco. On principal frequencies and isoperimetric ratios in convex sets. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 4, pp. 977-1005. doi : 10.5802/afst.1653. https://afst.centre-mersenne.org/articles/10.5802/afst.1653/
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