[Inégalites de Gagliardo–Nirenberg optimales]
Nous présentons une méthode donnant les constantes et fonctions optimales de toutes les inégalités de Gagliardo–Nirenberg dépendant de la norme 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 du gradient, pour tout .
We present a method giving the sharp constants and optimal functions of all the Gagliardo–Nirenberg inequalities involving the -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 -Gagliardo–Nirenberg and -Nash's inequalities, for all .
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@article{CRMATH_2008__346_13-14_757_0,
author = {Agueh, Martial},
title = {Gagliardo{\textendash}Nirenberg inequalities involving the gradient $ {L}^{2}$-norm},
journal = {Comptes Rendus. Math\'ematique},
pages = {757--762},
publisher = {Elsevier},
volume = {346},
number = {13-14},
year = {2008},
doi = {10.1016/j.crma.2008.05.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.05.015/}
}
TY - JOUR
AU - Agueh, Martial
TI - Gagliardo–Nirenberg inequalities involving the gradient $ {L}^{2}$-norm
JO - Comptes Rendus. Mathématique
PY - 2008
SP - 757
EP - 762
VL - 346
IS - 13-14
PB - Elsevier
UR - http://www.numdam.org/articles/10.1016/j.crma.2008.05.015/
DO - 10.1016/j.crma.2008.05.015
LA - en
ID - CRMATH_2008__346_13-14_757_0
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%T Gagliardo–Nirenberg inequalities involving the gradient $ {L}^{2}$-norm
%J Comptes Rendus. Mathématique
%D 2008
%P 757-762
%V 346
%N 13-14
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.05.015/
%R 10.1016/j.crma.2008.05.015
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Agueh, Martial. Gagliardo–Nirenberg inequalities involving the gradient $ {L}^{2}$-norm. Comptes Rendus. Mathématique, Tome 346 (2008) no. 13-14, pp. 757-762. doi : 10.1016/j.crma.2008.05.015. http://www.numdam.org/articles/10.1016/j.crma.2008.05.015/
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