Homogenization of $ {p}_{\epsilon }(x)$-Laplacian in perforated domains with a nonlocal transmission condition

Brahim Amaziane; Leonid Pankratov; Vladyslav Prytula

doi:10.1016/j.crme.2009.03.011

Homogenization of

p_{ε} (x)

-Laplacian in perforated domains with a nonlocal transmission condition
[Une condition de transmission non locale dans l'homogénéisation du

p_{ε} (x)

-Laplacian dans des domaines perforés]

Présenté par : Évariste Sanchez-Palencia

Brahim Amaziane ¹ ; Leonid Pankratov ^1,² ; Vladyslav Prytula ³

¹ Laboratoire de mathématiques et de leurs applications, CNRS–UMR 5142, Université de Pau, avenue de l'Université, 64000 Pau, France
² Département de mathématiques, B. Verkin Institut des Basses Températures, 47, avenue Lénine, 61103, Kharkov, Ukraine
³ Departamento de Matemáticas, Universidad de Castilla-La Mancha, av. Camilo Jose Cela, 13071 Ciudad Real, España

Comptes Rendus. Mécanique, Volume 337 (2009) no. 3, pp. 173-178.

Résumé
Abstract

On étudie le comportement asymptotique, lorsque $ε \to 0$ , des solutions $u^{ε}$ d'une équation elliptique non linéaire de croissance non standard dans des domains qui contiennent une microstructure ayant la forme d'une grille. Cette microstructure est concentrée dans un petit voisinage arbitraire d'une hypersurface Γ. On suppose que $u^{ε} = A^{ε}$ sur $\partial F^{ε}$ , où $A^{ε}$ est une constante inconnue. L'équation macroscopique et une condition de transmission non locale sur Γ sont obtenues par la technique de l'homogénéisation variationnelle dans le cadre des espaces de Sobolev avec des exposants variables. On présente un exemple périodique pour illustrer le résultat obtenu.

We study the asymptotic behavior, as $ε \to 0$ , of $u^{ε}$ solutions to a nonlinear elliptic equation with nonstandard growth condition in domains containing a grid-type microstructure $F^{ε}$ that is concentrated in an arbitrary small neighborhood of a given hypersurface Γ. We assume that $u^{ε} = A^{ε}$ on $\partial F^{ε}$ , where $A^{ε}$ is an unknown constant. The macroscopic equation and a nonlocal transmission condition on Γ are obtained by the variational homogenization technique in the framework of Sobolev spaces with variables exponents. This result is then illustrated by a periodic example.

Reçu le : 2008-05-24
Accepté le : 2009-03-20
Publié le : 2009-03-01

DOI : 10.1016/j.crme.2009.03.011

Keywords: Fluid mechanics, Homogenization, Nonlinear variational problem, Nonstandard growth
Mots-clés : Mécanique des fluides, Homogénéisation, Problème variationnel non linéaire, Croissance non standard

Affiliations des auteurs :

Brahim Amaziane ¹ ; Leonid Pankratov ^{1, 2} ; Vladyslav Prytula ³

¹ Laboratoire de mathématiques et de leurs applications, CNRS–UMR 5142, Université de Pau, avenue de l'Université, 64000 Pau, France
² Département de mathématiques, B. Verkin Institut des Basses Températures, 47, avenue Lénine, 61103, Kharkov, Ukraine
³ Departamento de Matemáticas, Universidad de Castilla-La Mancha, av. Camilo Jose Cela, 13071 Ciudad Real, España

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     author = {Brahim Amaziane and Leonid Pankratov and Vladyslav Prytula},
     title = {Homogenization of $ {p}_{\epsilon }(x)${-Laplacian} in perforated domains with a nonlocal transmission condition},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {173--178},
     publisher = {Elsevier},
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     number = {3},
     year = {2009},
     doi = {10.1016/j.crme.2009.03.011},
     language = {en},
}

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Brahim Amaziane; Leonid Pankratov; Vladyslav Prytula. Homogenization of $ {p}_{\epsilon }(x)$-Laplacian in perforated domains with a nonlocal transmission condition. Comptes Rendus. Mécanique, Volume 337 (2009) no. 3, pp. 173-178. doi : 10.1016/j.crme.2009.03.011. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.03.011/

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