Periodic unfolding and nonhomogeneous Neumann problems in domains with small holes

Amar Ould Hammouda

doi:10.1016/j.crma.2008.07.001

Partial Differential Equations

Periodic unfolding and nonhomogeneous Neumann problems in domains with small holes

Presented by: Philippe G. Ciarlet

Amar Ould Hammouda ^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

Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 963-968.

Abstract (Original language)
Résumé

We consider elliptic problems in periodically perforated domains in $R^{N}$ , 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 $r (ε) / ε \to 0$ . 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 $r (ε) = \exp (N / N - 1)$ , 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 $R^{N}$ , 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 $r (ε) / ε \to 0$ 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 $ε \to 0$ , 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 $r (ε) = \exp (N / N - 1)$ , on a un terme supplémentaire qui apparait dans le second membre de l'équation limite.

Received: 2008-05-09
Accepted: 2008-06-16
Published online: 2008-08-13

DOI: 10.1016/j.crma.2008.07.001

Author's affiliations:

Amar Ould Hammouda ^{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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     title = {Periodic unfolding and nonhomogeneous {Neumann} problems in domains with small holes},
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Amar Ould Hammouda. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.001/

References
Cited by

[1] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978

[2] D. Cioranescu; A. Damlamian; G. Griso Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 99-104

[3] C. Conca; P. Donato Non-homogeneous Neumann problems in domains with small holes, RAIRO Modél. Math. Anal. Numér., Volume 4 (1988) no. 22, pp. 561-608

[4] D. Cioranescu; P. Donato; R. Zaki The periodic unfolding method in perforated domains, Portugaliae Mathematica, Volume 63 (2006) no. 4, pp. 467-496

[5] D. Cioranescu; F. Murat Un terme étrange venu d'ailleurs (H. Brezis; J.L. Lions, eds.), Nonlinear Partial Differential Equations and their Applications, College de France Seminar, II & III, Research Notes in Math., vols. 60–70, Pitman, Boston, 1982, pp. 98-138 (154–178)

[6] A. Damlamian An elementary introduction to periodic unfolding (A. Damlamian; D. Lukkassen; A. Meidell; A. Piatnitski, eds.), Proc. of the Narvik Conference 2004, Gakuto Int. Series, Math. Sci. Appl., vol. 24, Gakkokotosho, 2006, pp. 119-136

[7] A. Ould Hammouda, Homogenization of a class of Neumann problems in perforated domains, in press

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