Some uniform elliptic estimates in a porous medium

Nader Masmoudi

doi:10.1016/j.crma.2004.10.007

Partial Differential Equations

Some uniform elliptic estimates in a porous medium
[Estimations elliptiques uniformes dans un milieu poreux.]

Présenté par : Pierre-Louis Lions

Nader Masmoudi ^1,²

¹ Courant Institute of Mathematical Sciences, New York University, Warren Weaver Hall, 251, Mercer Street, New York, NY 10012-1185, USA
² Ceremade, UMR CNRS 7534, université de Paris IX, place du Maréchal DeLattre de Tassigny, 75775 Paris cedex 16, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 849-854.

Résumé
Abstract

Dans cette Note, nous présentons des estimations uniformes en ɛ pour des problèmes elliptiques écrits dans un milieu poreux. Le domaine $ω_{ɛ}$ est un domaine perforé obtenu aprés l'elimination de trous de largeure ɛ. Nous étudions en particulier le problème de Dirichlet, l'opérateur de projection sur les vecteurs de divergence nulle, et l'opérateur de Stokes. Nous donnons aussi des estimations sur le problème $div v = f$ .

In this Note, we give uniform elliptic estimates (uniform in ɛ) for some elliptic problems written in a periodic porous medium (of period ɛ) in $L^{p}$ spaces. The domain $ω_{ɛ}$ is obtained by removing a grid of holes of size ɛ from a smooth domain ω. In particular, we study the Dirichlet problem, the projection operator onto divergence-free vector fields as well as the Stokes operator. We also give estimates for the problem $div v = f$ .

Reçu le : 2004-07-07
Accepté le : 2004-10-05
Publié le : 2004-11-18

DOI : 10.1016/j.crma.2004.10.007

Affiliations des auteurs :

Nader Masmoudi ^{1, 2}

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     title = {Some uniform elliptic estimates in a porous medium},
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Nader Masmoudi. Some uniform elliptic estimates in a porous medium. Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 849-854. doi : 10.1016/j.crma.2004.10.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.10.007/

Bibliographie
Cité par

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[6] N. Masmoudi, Uniform estimates for some elliptic problems in a porous medium, 2004, in preparation

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