Homogenization of a Ginzburg–Landau functional

Leonid Berlyand; Doina Cioranescu; Dmitry Golovaty

doi:10.1016/j.crma.2004.10.024

Calculus of Variations

Homogenization of a Ginzburg–Landau functional
[Homogénéisation d'une fonctionnelle de Ginzburg–Landau.]

Présenté par : Philippe G. Ciarlet

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

Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 87-92.

Abstract (VO)
Résumé

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 $ɛ \to 0$ . 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.

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 $ɛ \to 0$ 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.

Reçu le : 2004-10-18
Accepté le : 2004-10-20
Publié le : 2004-12-01

DOI : 10.1016/j.crma.2004.10.024

Affiliations des auteurs :

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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     author = {Leonid Berlyand and Doina Cioranescu and Dmitry Golovaty},
     title = {Homogenization of a {Ginzburg{\textendash}Landau} functional},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {87--92},
     publisher = {Elsevier},
     volume = {340},
     number = {1},
     year = {2005},
     doi = {10.1016/j.crma.2004.10.024},
     language = {en},
}

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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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