Comptes Rendus
no. 2
Potential theory/Calculus of variations
Optimal geometric estimates for fractional Sobolev capacities
[Estimées géométriques optimales pour les capacités fractionnelles de Sobolev]

Présenté par : Haïm Brézis

Jie Xiao1
1 Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 149-153.

Cette note révèle de nouvelles inégalités exactes mettant la capacité fractionnelle de Sobolev d'un ensemble en relation avec son volume standard et son périmètre fractionnel, respectivement, et démontre, par conséquence, que l'inégalité fractionnelle exacte de Sobolev est equivalente, soit à l'inégalité fractionnelle isocapacitaire exacte, soit à l'inégalité fractionnelle isopérimétrique exacte.

This note discovers new sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter, respectively, and consequently proves that the sharp fractional Sobolev inequality is equivalent to either the sharp fractional isocapacitary inequality or the sharp fractional isoperimetric inequality.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.014
Mots clés : Sharpness, Volume, Fractional Sobolev capacity, Fractional perimeter

Jie Xiao 1

1 Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada 
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Jie Xiao. Optimal geometric estimates for fractional Sobolev capacities. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 149-153. doi : 10.1016/j.crma.2015.10.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.014/

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