Averaging in variational inequalities with nonlinear restrictions along manifolds

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

doi:10.1016/j.crme.2011.04.002

Averaging in variational inequalities with nonlinear restrictions along manifolds

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

Comptes Rendus. Mécanique, Volume 339 (2011) no. 6, pp. 406-410.

Abstract
Résumé

We consider variational inequalities for the Laplace operator in a domain Ω of $R^{n}$ 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 $O (ε^{α})$ distributed along a $(n - 1)$ -dimensional manifold γ with period ε. Here $ε > 0$ is a small parameter, $α > 0$ and $n ⩾ 3$ . On the boundary of the perforations, we have the restrictions for the solution $u_{ε} ⩾ 0$ , $\partial_{ν} u_{ε} ⩾ - ε^{- κ} σ (x, u_{ε})$ and $u_{ε} (\partial_{ν} u_{ε} + ε^{- κ} σ (x, u_{ε})) = 0$ , where $κ ⩾ 0$ and σ is a certain smooth function. For $α ⩾ 1$ and $κ = (α - 1) (n - 2)$ , we characterize the asymptotic behavior of $u_{ε}$ as $ε \to 0$ providing the homogenized problems. A critical size of the cavities is found when $α = κ = (n - 1) / (n - 2)$ for which the corrector in the energy norm is constructed.

Nous considèrons inégalités variationnelles pour lʼopérateur de Laplace dans une domaine Ω de $R^{n}$ périodiquement perforé, et avec des restrictions pour le flux sur la frontière des trous. On suppose que les perforations sont des boules de rayon $O (ε^{α})$ distribuées sur une variété de dimension $(n - 1)$ , γ, de période ε. Ici $ε > 0$ est une petite paramètre, $α > 0$ et $n ⩾ 3$ . Sur la frontière des trous nous avons des restrictions pour la solution $u_{ε} ⩾ 0$ , $\partial_{ν} u_{ε} ⩾ - ε^{- κ} σ (x, u_{ε})$ et $u_{ε} (\partial_{ν} u_{ε} + ε^{- κ} σ (x, u_{ε})) = 0$ , où $κ ⩾ 0$ et σ est une certaine fonction régulière. Pour $α ⩾ 1$ and $κ = (α - 1) (n - 2)$ , nous caractérisons le comportement asymptotique de $u_{ε}$ pour $ε \to 0$ . On trouve les problèmes homogéneisés et une taille critique des trous pour $α = κ = (n - 1) / (n - 2)$ . Pour cette taille on construit le correcteur dans la norme de lʼénergie.

Received: 2011-03-25
Accepted: 2011-04-01
Published online: 2011-05-06

DOI: 10.1016/j.crme.2011.04.002

Keywords: Porous media, Variational inequalities, Nonlinear flux, Boundary homogenization
Mots-clés : Milieux poreux, Inégalités variationnelles, Flux non linéaire, Homogénéisation des frontières

Author's affiliations:

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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     author = {Delfina G\'omez and Miguel Lobo and M. Eugenia P\'erez and Tatiana A. Shaposhnikova},
     title = {Averaging in variational inequalities with nonlinear restrictions along manifolds},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {406--410},
     publisher = {Elsevier},
     volume = {339},
     number = {6},
     year = {2011},
     doi = {10.1016/j.crme.2011.04.002},
     language = {en},
}

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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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