Comptes Rendus
Calculus of Variations
Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent
[Analyse asymptotique dans l'étude de milieux perforés au voisinage d'un exposant critique]
Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 363-367.

On établit un résultat général de Γ-convergence d'énergies vectorielles nonlinéares définies sur des domaines perforés, dans le cas où l'intégrande est de croissance p, dans le cas critique p=n ; la limite est caractérisée par une formule de type homogénéisation. On démontre également que pour p voisin de n trois régimes sont possibles, deux avec une taille du perforation non triviale (exponentielle et polynomiale-exponentielle), et une taille pour laquelle la Γ-limite est toujours triviale.

We give a general Γ-convergence result for vector-valued nonlinear energies defined on perforated domains for integrands with p-growth in the critical case p=n. We characterize the limit extra term by a formula of homogenization type. We also prove that for p close to n there are three regimes, two with a nontrivial size of the perforation (exponential and mixed polynomial-exponential), and one where the Γ-limit is always trivial.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.01.010
Andrea Braides 1 ; Laura Sigalotti 2

1 Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, via della ricerca scientifica, 00133 Roma, Italy
2 Dipartimento di Matematica, Università di Roma ‘La Sapienza’, piazzale A.Moro, 00185 Roma, Italy
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Andrea Braides; Laura Sigalotti. Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 363-367. doi : 10.1016/j.crma.2008.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.010/

[1] G. Alberti; S. Baldo; G. Orlandi Variational convergence for functionals of Ginzburg–Landau type, Indiana Univ. Math. J., Volume 54 (2005), pp. 1411-1472

[2] N. Ansini; A. Braides Asymptotic analysis of periodically-perforated nonlinear media, J. Math. Pures Appl., Volume 81 (2002), pp. 439-451 (Erratum in J. Math. Pures Appl., 84, 2005, pp. 147-148)

[3] F. Bethuel; H. Brezis; F. Hélein Ginzburg–Landau Vortices, Birkhäuser, Boston, 1994

[4] A. Braides Γ-Convergence for Beginners, Oxford University Press, Oxford, 2002

[5] A. Braides, L. Truskinovsky, Asymptotic expansions by Γ-convergence, Cont. Mech. Therm., in press

[6] D. Cioranescu; F. Murat Un terme étrange venu d'ailleur, Nonlinear Partial Differential Equations and their Applications, Res. Notes in Math., vol. 60, Pitman, London, 1982, pp. 98-138

[7] A.V. Marchenko; Ya.E. Khruslov Boundary Value Problems in Domains with Fine-Granulated Boundaries, Naukova Dumka, Kiev, 1974 (in Russian)

[8] E. Sandier; S. Serfaty Vortices in the Magnetic Ginzburg–Landau Model, Birkhäuser, 2007

[9] L. Sigalotti, Asymptotic analysis of periodically-perforated nonlinear media at the critical exponent, in press

[10] L. Sigalotti, Asymptotic analysis of periodically-perforated nonlinear media close to the critical exponent, J. Convex Anal., in press

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