Gagliardo–Nirenberg inequalities involving the gradient $ {L}^{2}$-norm

Martial Agueh

doi:10.1016/j.crma.2008.05.015

Differential Geometry

Gagliardo–Nirenberg inequalities involving the gradient

L^{2}

-norm

Presented by: Thierry Aubin

Martial Agueh ¹

¹ Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045 STN CSC, Victoria B.C., V8W 3P4, Canada

Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 757-762.

Abstract
Résumé

We present a method giving the sharp constants and optimal functions of all the Gagliardo–Nirenberg inequalities involving the $L^{2}$ -norm of the gradient. We show that the optimal functions can be explicitly derived from a specific non-linear ordinary differential equation which appears to be linear for a subclass of the Gagliardo–Nirenberg inequalities or when the space dimension reduces to 1. In these cases, we give the explicit expressions of the optimal functions, along with the sharp constants of the corresponding Gagliardo–Nirenberg inequalities. Our method extend to the $L^{p}$ -Gagliardo–Nirenberg and $L^{p}$ -Nash's inequalities, for all $p > 1$ .

Nous présentons une méthode donnant les constantes et fonctions optimales de toutes les inégalités de Gagliardo–Nirenberg dépendant de la norme $L^{2}$ du gradient. Nous montrons que les fonctions optimales se calculent explicitement à partir d'une équation différentielle ordinaire nonlinéaire, qui devient linéaire pour une sous-classe de ces inégalités ou quand la dimension de l'espace est réduite a 1. Dans ces cas, nous obtenons explicitement les fonctions et constantes optimales des inégalités de Gagliardo–Nirenberg correspondantes. Notre méthode se généralise aux inégalités de Gagliardo–Nirenberg et de Nash dependant de la norme $L^{p}$ du gradient, pour tout $p > 1$ .

Received: 2007-07-11
Accepted: 2008-05-23
Published online: 2008-06-24

DOI: 10.1016/j.crma.2008.05.015

Author's affiliations:

Martial Agueh ¹

¹ Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045 STN CSC, Victoria B.C., V8W 3P4, Canada

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     title = {Gagliardo{\textendash}Nirenberg inequalities involving the gradient $ {L}^{2}$-norm},
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Martial Agueh. Gagliardo–Nirenberg inequalities involving the gradient $ {L}^{2}$-norm. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 757-762. doi : 10.1016/j.crma.2008.05.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.015/

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