Partial Differential Equations
Periodic unfolding and nonhomogeneous Neumann problems in domains with small holes
Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 963-968.

We consider elliptic problems in periodically perforated domains in RN, with nonhomogeneous Neumann conditions on the boundary of the holes. We are interested in the asymptotic behavior of the solutions as the period ε goes to zero. In a first case all the holes are “small”, i.e., are of size r(ε) with r(ε)/ε0. In the second case, there are again small holes but also holes of size ε. We use the periodic unfolding method introduced in Cioranescu et al. (2002), which allows us to study second order operators with highly oscillating coefficients and so, to generalize here the results of Conca and Donato (1988). In both cases, if r(ε)=exp(N/N1), an additional term appears in the right-hand side of the limit equation.

L'objet de cette Note est l'homogénéisation d'une classe de problèmes élliptiques dans des domaines de RN, périodiquement perforés par des petits trous, avec des conditions de Neumann non homogènes sur le bord des trous. Dans un premier temps, les trous de taille r(ε) avec r(ε)/ε0 et dans un second, on a des trous de taille r(ε) mais aussi des trous de taille ε. Le premier cas, pour le Laplacien, a été étudié dans Conca et Donato (1988). Pour étudier le comportement asymptotique des solutions lorsque ε0, on utilise ici la méthode de l'éclatement périodique introduite par Cioranescu et al. (2002), ce qui permet de considérer des opérateurs de second ordre à coefficients oscillants. Dans les deux situations, pour r(ε)=exp(N/N1), on a un terme supplémentaire qui apparait dans le second membre de l'équation limite.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2008.07.001
Ould Hammouda, Amar 1, 2

1 Laboratoire Jacques-Louis Lions-CNRS, boîte courrier 187, Université Pierre-et-Marie-Curie, 4 place Jussieu, 75005 Paris, France
2 Departement of Mathematics, ENS, P.O. Box 92, 16050 Kouba, Algiers
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Ould Hammouda, Amar. Periodic unfolding and nonhomogeneous Neumann problems in domains with small holes. Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 963-968. doi : 10.1016/j.crma.2008.07.001. http://www.numdam.org/articles/10.1016/j.crma.2008.07.001/

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