Homogenization of a class of nonlinear elliptic equations with nonstandard growth

Brahim Amaziane; Stanislav Antontsev; Leonid Pankratov

doi:10.1016/j.crme.2007.02.007

Homogenization of a class of nonlinear elliptic equations with nonstandard growth

Presented by: Évariste Sanchez-Palencia

Brahim Amaziane ¹ ; Stanislav Antontsev ²; Leonid Pankratov ^1,³

¹ Laboratoire de mathématiques appliquées, CNRS–UMR5142, université de Pau, avenue de l'université, 64000 Pau, France
² Departamento de Matématica Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
³ Département de Mathématiques, B. Verkin Institut des Basses Températures, 47, av. Lénine, 61103, Kharkov, Ukraine

Comptes Rendus. Mécanique, Volume 335 (2007) no. 3, pp. 138-143.

Abstract
Résumé

We study the homogenization of the Dirichlet variational problem of a class of nonlinear elliptic equations with nonstandard growth. Such equations arise in many engineering disciplines, such as electrorheological fluids, non-Newtonian fluids with thermo-convective effects, and nonlinear Darcy flow of compressible fluids in heterogeneous porous media. We derive the homogenized model by means of the variational homogenization technique in the framework of Sobolev spaces with variable exponents. This result is then illustrated with a periodic example.

On étudie l'homogénéisation du problème variationnel de Dirichlet d'une classe d'équations elliptiques non linéaires de croissance non standard. Ce genre d'équations apparaît dans la modélisation de certains problèmes de l'ingénierie, comme par exemple les fluides électrorhéologiques, les écoulements non Newtoniens thermoconvectifs, et les écoulements non linéaires de Darcy de fluides compressibles en milieux poreux hétérogènes. On obtient le problème homogénéisé par la technique de l'homogénéisation variationnelle dans le cadre des espaces de Sobolev avec des exposants variables. Enfin, on présente un exemple périodique pour illustrer le résultat obtenu.

Received: 2007-02-13
Accepted: 2007-02-26
Published online: 2007-03-01

DOI: 10.1016/j.crme.2007.02.007

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

Author's affiliations:

Brahim Amaziane ¹; Stanislav Antontsev ²; Leonid Pankratov ^{1, 3}

¹ Laboratoire de mathématiques appliquées, CNRS–UMR5142, université de Pau, avenue de l'université, 64000 Pau, France
² Departamento de Matématica Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
³ Département de Mathématiques, B. Verkin Institut des Basses Températures, 47, av. Lénine, 61103, Kharkov, Ukraine

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     author = {Brahim Amaziane and Stanislav Antontsev and Leonid Pankratov},
     title = {Homogenization of a class of nonlinear elliptic equations with nonstandard growth},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {138--143},
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     volume = {335},
     number = {3},
     year = {2007},
     doi = {10.1016/j.crme.2007.02.007},
     language = {en},
}

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Brahim Amaziane; Stanislav Antontsev; Leonid Pankratov. Homogenization of a class of nonlinear elliptic equations with nonstandard growth. Comptes Rendus. Mécanique, Volume 335 (2007) no. 3, pp. 138-143. doi : 10.1016/j.crme.2007.02.007. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.02.007/

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