[Analyse asymptotique dans l'étude de milieux perforés au voisinage d'un exposant critique]
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 ; 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 . 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.
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@article{CRMATH_2008__346_5-6_363_0,
author = {Braides, Andrea and Sigalotti, Laura},
title = {Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent},
journal = {Comptes Rendus. Math\'ematique},
pages = {363--367},
publisher = {Elsevier},
volume = {346},
number = {5-6},
year = {2008},
doi = {10.1016/j.crma.2008.01.010},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.01.010/}
}
TY - JOUR AU - Braides, Andrea AU - Sigalotti, Laura TI - Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent JO - Comptes Rendus. Mathématique PY - 2008 SP - 363 EP - 367 VL - 346 IS - 5-6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.01.010/ DO - 10.1016/j.crma.2008.01.010 LA - en ID - CRMATH_2008__346_5-6_363_0 ER -
%0 Journal Article %A Braides, Andrea %A Sigalotti, Laura %T Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent %J Comptes Rendus. Mathématique %D 2008 %P 363-367 %V 346 %N 5-6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.01.010/ %R 10.1016/j.crma.2008.01.010 %G en %F CRMATH_2008__346_5-6_363_0
Braides, Andrea; Sigalotti, Laura. Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent. Comptes Rendus. Mathématique, Tome 346 (2008) no. 5-6, pp. 363-367. doi : 10.1016/j.crma.2008.01.010. http://www.numdam.org/articles/10.1016/j.crma.2008.01.010/
[1] Variational convergence for functionals of Ginzburg–Landau type, Indiana Univ. Math. J., Volume 54 (2005), pp. 1411-1472
[2] 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] Ginzburg–Landau Vortices, Birkhäuser, Boston, 1994
[4] 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] 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] Boundary Value Problems in Domains with Fine-Granulated Boundaries, Naukova Dumka, Kiev, 1974 (in Russian)
[8] 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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