Comptes Rendus
no. 1
Calculus of Variations
Homogenization of a Ginzburg–Landau functional
[Homogénéisation d'une fonctionnelle de Ginzburg–Landau.]

Présenté par : Philippe G. Ciarlet

Leonid Berlyand1 ; Doina Cioranescu2 ; Dmitry Golovaty3
1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2 Université Pierre et Marie Curie (Paris VI), laboratoire d'analyse numérique, 4, place Jussieu, 75252 Paris cedex 05, France
3 Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325, USA
Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 87-92.

Nous considérons un problème non linéaire d'homogénéisation pour une fonctionnelle de Ginzburg–Landau avec un terme correspondant à l'energie de surface (positive ou négative) décrivant un milieu cristallin liquide avec des inclusions. On suppose que la distance ɛ entre les inclusions est comparable à leur taille. En appliquant la méthode des mesocharactéristiques nous donnons la fonctionnelle limite lorsque ɛ→0 et prouvons que le problème homogénéisé pour des géometries arbitraires (périodiques ou non), est décrit par une fonctionnelle de Ginzburg–Landau anisotrope. Nous donnons des formules pour calculer les caractéristiques effectives des matériaux ainsi obtenus.

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that sizes and distances between inclusions are of the same order ɛ, we obtain a limiting functional as ɛ→0. We generalize the method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We give computational formulas for material characteristics of an effective medium.

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Accepté le :
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DOI : 10.1016/j.crma.2004.10.024
Leonid Berlyand 1 ; Doina Cioranescu 2 ; Dmitry Golovaty 3

1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2 Université Pierre et Marie Curie (Paris VI), laboratoire d'analyse numérique, 4, place Jussieu, 75252 Paris cedex 05, France
3 Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325, USA 
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Leonid Berlyand; Doina Cioranescu; Dmitry Golovaty. Homogenization of a Ginzburg–Landau functional. Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 87-92. doi : 10.1016/j.crma.2004.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.10.024/

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