Comptes Rendus
no. 9
Mathematical analysis/Functional analysis
A note on the fractional perimeter and interpolation
[Une note sur le périmètre fractionnaire et l'interpolation]

Présenté par : Haïm Brézis

Augusto C. Ponce1 ; Daniel Spector2, 3
1 Université catholique de Louvain, Institut de recherche en mathématique et physique, chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium
2 National Chiao Tung University, Department of Applied Mathematics, Hsinchu, Taiwan
3 National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei 106, Taiwan
Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 960-965.

Nous présentons le périmètre fractionnaire en tant que fonction d'ensemble qui interpole la mesure de Lebesgue et le périmètre au sens de De Giorgi. Notre motivation provient d'une inégalité fractionnaire que nous avons récemment démontrée dans l'esprit de la Boxing inequality de W. Gustin reliant le périmètre fractionnaire et le contenu de Hausdorff. Cette nouvelle inégalité permet de retrouver des propriétés de la semi-norme de Gagliardo dans le cadre des espaces de Sobolev Wα,1 d'ordre 0<α<1.

We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces Wα,1 of order 0<α<1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.09.001
Augusto C. Ponce 1 ; Daniel Spector 2, 3

1 Université catholique de Louvain, Institut de recherche en mathématique et physique, chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium
2 National Chiao Tung University, Department of Applied Mathematics, Hsinchu, Taiwan
3 National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei 106, Taiwan 
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Augusto C. Ponce; Daniel Spector. A note on the fractional perimeter and interpolation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 960-965. doi : 10.1016/j.crma.2017.09.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.001/

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