Comptes Rendus
no. 1
Calculus of Variations
Homogenization of nonlinear integrals via the periodic unfolding method
[Homogénéisation non linéaire par éclatement périodique]

Présenté par : Philippe G. Ciarlet

Doina Cioranescu1 ; Alain Damlamian2 ; Riccardo De Arcangelis3
1 Université Pierre et Marie Curie (Paris VI), laboratoire J.-L. Lions CNRS UMR 7598, 175, rue du Chevaleret, 75013 Paris, France
2 Université Paris XII Val de Marne, laboratoire d'analyse et de mathématiques appliquées, CNRS UMR 8050, 94010 Créteil cedex, France
3 Università di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy
Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 77-82.

Cette Note présente un résultat d'homogénéisation périodique pour des énergies intégrales à croissance polynômiale. On utilise la méthode d'éclatement périodique qui réduit la démonstration à de la convergence faible dans un espace de Lebesgue.

We consider the periodic homogenization of nonlinear integral energies with polynomial growth. The study is carried out by the periodic unfolding method which reduces the homogenization process to a weak convergence problem in a Lebesgue space.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.03.028
Doina Cioranescu 1 ; Alain Damlamian 2 ; Riccardo De Arcangelis 3

1 Université Pierre et Marie Curie (Paris VI), laboratoire J.-L. Lions CNRS UMR 7598, 175, rue du Chevaleret, 75013 Paris, France
2 Université Paris XII Val de Marne, laboratoire d'analyse et de mathématiques appliquées, CNRS UMR 8050, 94010 Créteil cedex, France
3 Università di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy 
@article{CRMATH_2004__339_1_77_0,
     author = {Doina Cioranescu and Alain Damlamian and Riccardo De~Arcangelis},
     title = {Homogenization of nonlinear integrals via the periodic unfolding method},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {77--82},
     publisher = {Elsevier},
     volume = {339},
     number = {1},
     year = {2004},
     doi = {10.1016/j.crma.2004.03.028},
     language = {en},
}
TY  - JOUR
AU  - Doina Cioranescu
AU  - Alain Damlamian
AU  - Riccardo De Arcangelis
TI  - Homogenization of nonlinear integrals via the periodic unfolding method
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 77
EP  - 82
VL  - 339
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2004.03.028
LA  - en
ID  - CRMATH_2004__339_1_77_0
ER  - 
%0 Journal Article
%A Doina Cioranescu
%A Alain Damlamian
%A Riccardo De Arcangelis
%T Homogenization of nonlinear integrals via the periodic unfolding method
%J Comptes Rendus. Mathématique
%D 2004
%P 77-82
%V 339
%N 1
%I Elsevier
%R 10.1016/j.crma.2004.03.028
%G en
%F CRMATH_2004__339_1_77_0
Doina Cioranescu; Alain Damlamian; Riccardo De Arcangelis. Homogenization of nonlinear integrals via the periodic unfolding method. Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 77-82. doi : 10.1016/j.crma.2004.03.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.028/

[1] G. Allaire Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992), pp. 1482-1518

[2] T. Arbogast; J. Douglas; U. Hornung Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., Volume 21 (1990), pp. 823-836

[3] N.S. Bakhvalov; G.P. Panasenko Homogenization: Averaging Processes in Periodic Media, Kluwer Academic, 1989

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

[5] A. Braides; A. Defranceschi Homogenization of Multiple Integrals, Oxford University Press, 1998

[6] L. Carbone; R. De Arcangelis Unbounded Functionals in the Calculus of Variations. Representation, Relaxation, and Homogenization, Chapman & Hall/CRC, 2001

[7] L. Carbone; C. Sbordone Some properties of Γ-limits of integral functionals, Ann. Mat. Pura Appl. (4), Volume 122 (1979), pp. 1-60

[8] J. Casado-Dı́az; M. Luna-Laynez; J.D. Martı́n An adaptation of the multi-scale methods for the analysis of bery thin reticulated structures, C. R. Acad. Sci. Paris, Sér. I Math., Volume 332 (2001), pp. 223-228

[9] C. Castaing; M. Valadier Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., vol. 580, Springer, 1977

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

[11] D. Cioranescu; P. Donato An Introduction to Homogenization, Oxford University Press, 1999

[12] G. Dal Maso An Introduction to Γ-Convergence, Birkhäuser, 1993

[13] E. De Giorgi; S. Spagnolo Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), Volume 8 (1973), pp. 391-411

[14] P. Marcellini Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., Volume 4 (1978) no. 117, pp. 139-152

[15] F. Murat; L. Tartar H-Convergence (A. Cherkaev; R. Kohn, eds.), Topics in the Mathematical Modelling of Composite Materials, Birkhäuser, 1997, pp. 21-44

[16] G. Nguetseng A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., Volume 20 (1989), pp. 608-629

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Periodic unfolding and homogenization

Doina Cioranescu; Alain Damlamian; Georges Griso

C. R. Math (2002)

Periodic unfolding and Robin problems in perforated domains

Doina Cioranescu; Patrizia Donato; Rachad Zaki

C. R. Math (2006)

Reiterated homogenization for elliptic operators

Nicolas Meunier; Jean Van Schaftingen

C. R. Math (2005)