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Comptes Rendus Mathématique
Volume 340, n° 1
pages 87-92 (janvier 2005)
Doi : 10.1016/j.crma.2004.10.024
Received : 18 October 2004 ;  accepted : 20 October 2004
Homogenization of a Ginzburg-Landau functional
Homogénéisation dʼune fonctionnelle de Ginzburg-Landau.
 

Leonid Berlyand a , Doina Cioranescu b , Dmitry Golovaty c
a Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 
b Université Pierre et Marie Curie (Paris VI), laboratoire dʼanalyse numérique, 4, place Jussieu, 75252 Paris cedex 05, France 
c Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325, USA 

Abstract

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 , we obtain a limiting functional as  . 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. To cite this article: L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

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Résumé

Nous considérons un problème non linéaire dʼhomogénéisation pour une fonctionnelle de Ginzburg-Landau avec un terme correspondant à lʼenergie de surface (positive ou négative) décrivant un milieu cristallin liquide avec des inclusions. On suppose que la distance entre les inclusions est comparable à leur taille. En appliquant la méthode des mesocharactéristiques nous donnons la fonctionnelle limite lorsque   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. Pour citer cet article : L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

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