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
[Homogénéisation d'une classe d'équations elliptiques non linéaires de croissance non standard]

Présenté par : 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 (VO)
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.

Reçu le : 2007-02-13
Accepté le : 2007-02-26
Publié le : 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

Affiliations des auteurs :

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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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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Hubert Nnang Deterministic homogenization of nonlinear degenerate elliptic operators with nonstandard growth, Acta Mathematica Sinica. English Series, Volume 30 (2014) no. 9, pp. 1621-1654 | DOI:10.1007/s10114-014-2131-x | Zbl:1304.35067
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B. Amaziane; S. Antontsev; L. Pankratov; A. Piatnitski Homogenization of $p$ -Laplacian in perforated domain, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 26 (2009) no. 6, pp. 2457-2479 | DOI:10.1016/j.anihpc.2009.06.004 | Zbl:1182.35017
Brahim Amaziane; Leonid Pankratov; Vladyslav Prytula Homogenization of pε(x)-Laplacian in perforated domains with a nonlocal transmission condition, Comptes Rendus. Mécanique, Volume 337 (2009) no. 3, p. 173 | DOI:10.1016/j.crme.2009.03.011
B. Amaziane; L. Pankratov; A. Piatnitski Nonlinear flow through double porosity media in variable exponent Sobolev spaces, Nonlinear Analysis. Real World Applications, Volume 10 (2009) no. 4, pp. 2521-2530 | DOI:10.1016/j.nonrwa.2008.05.008 | Zbl:1163.35410
O. M. Buhrii; R. A. Mashiyev Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 70 (2009) no. 6, pp. 2325-2331 | DOI:10.1016/j.na.2008.03.013 | Zbl:1162.35399
S. Antontsev; L. Consiglieri Elliptic boundary value problems with nonstandard growth conditions, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 71 (2009) no. 3-4, pp. 891-902 | DOI:10.1016/j.na.2008.10.109 | Zbl:1175.35048
B. Amaziane; L. Pankratov; V. Prytula Nonlocal effects in homogenization of $p_{ε} (x)$ -Laplacian in perforated domains, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 71 (2009) no. 9, pp. 4078-4097 | DOI:10.1016/j.na.2009.02.097 | Zbl:1201.35031

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