Comptes Rendus
Averaging in variational inequalities with nonlinear restrictions along manifolds
Comptes Rendus. Mécanique, Volume 339 (2011) no. 6, pp. 406-410.

We consider variational inequalities for the Laplace operator in a domain Ω of Rn periodically perforated along a manifold, with nonlinear restrictions for the flux on the boundary of the cavities. We assume that the perforations are balls of radius O(εα) distributed along a (n1)-dimensional manifold γ with period ε. Here ε>0 is a small parameter, α>0 and n3. On the boundary of the perforations, we have the restrictions for the solution uε0, νuεεκσ(x,uε) and uε(νuε+εκσ(x,uε))=0, where κ0 and σ is a certain smooth function. For α1 and κ=(α1)(n2), we characterize the asymptotic behavior of uε as ε0 providing the homogenized problems. A critical size of the cavities is found when α=κ=(n1)/(n2) for which the corrector in the energy norm is constructed.

Nous considèrons inégalités variationnelles pour lʼopérateur de Laplace dans une domaine Ω de Rn périodiquement perforé, et avec des restrictions pour le flux sur la frontière des trous. On suppose que les perforations sont des boules de rayon O(εα) distribuées sur une variété de dimension (n1), γ, de période ε. Ici ε>0 est une petite paramètre, α>0 et n3. Sur la frontière des trous nous avons des restrictions pour la solution uε0, νuεεκσ(x,uε) et uε(νuε+εκσ(x,uε))=0, où κ0 et σ est une certaine fonction régulière. Pour α1 and κ=(α1)(n2), nous caractérisons le comportement asymptotique de uε pour ε0. On trouve les problèmes homogéneisés et une taille critique des trous pour α=κ=(n1)/(n2). Pour cette taille on construit le correcteur dans la norme de lʼénergie.

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DOI: 10.1016/j.crme.2011.04.002
Keywords: Porous media, Variational inequalities, Nonlinear flux, Boundary homogenization
Mot clés : Milieux poreux, Inégalités variationnelles, Flux non linéaire, Homogénéisation des frontières

Delfina Gómez 1; Miguel Lobo 1; M. Eugenia Pérez 2; Tatiana A. Shaposhnikova 3

1 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain
2 Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Avenida de las Castros s/n, 39005 Santander, Spain
3 Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, 119992, GSP-2, Moscow, Russia
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     title = {Averaging in variational inequalities with nonlinear restrictions along manifolds},
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Delfina Gómez; Miguel Lobo; M. Eugenia Pérez; Tatiana A. Shaposhnikova. Averaging in variational inequalities with nonlinear restrictions along manifolds. Comptes Rendus. Mécanique, Volume 339 (2011) no. 6, pp. 406-410. doi : 10.1016/j.crme.2011.04.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.04.002/

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The work has been partially supported by the Spanish MICINN: MTM2009-12628.

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