Comptes Rendus
A sharp inequality for Sobolev functions
[Une inégalité dans un espace de Sobolev]
Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 105-108.

Nous considérons N⩾5, a>0, Ω un ouvert borné régulier de N , 2 * =2N N-2, 2 # =2(N-1) N-2 et ||u||2=|∇u|22+a|u|22. Nous prouvons qu'il existe α0>0 tel que, pour toute fonction uH 1 (Ω){0},

S 2 2/N u 2 |u| 2 * 2 1+α 0 |u| 2 # 2 # u·|u| 2 * 2 * /2 .
Cette inégalité implique l'inégalité de Cherrier.

Let N⩾5, a>0, Ω be a smooth bounded domain in N , 2 * =2N N-2, 2 # =2(N-1) N-2 and ‖u2=|∇u|22+a|u|22. We prove there exists an α0>0 such that, for all uH 1 (Ω){0},

S 2 2/N u 2 |u| 2 * 2 1+α 0 |u| 2 # 2 # u·|u| 2 * 2 * /2 .
This inequality implies Cherrier's inequality.

Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02215-X

Pedro M. Girão 1

1 Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
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Pedro M. Girão. A sharp inequality for Sobolev functions. Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 105-108. doi : 10.1016/S1631-073X(02)02215-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02215-X/

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