Homogenization of a Ginzburg–Landau functional

Berlyand, Leonid; Cioranescu, Doina; Golovaty, Dmitry

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

Berlyand, Leonid ¹ ; Cioranescu, Doina ² ; Golovaty, Dmitry 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, Tome 340 (2005) no. 1, pp. 87-92.

Résumé
Abstract

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.

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.

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 :

Berlyand, Leonid ¹ ; Cioranescu, Doina ² ; Golovaty, Dmitry ³

@article{CRMATH_2005__340_1_87_0,
     author = {Berlyand, Leonid and Cioranescu, Doina and Golovaty, Dmitry},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2004.10.024/}
}

TY  - JOUR
AU  - Berlyand, Leonid
AU  - Cioranescu, Doina
AU  - Golovaty, Dmitry
TI  - Homogenization of a Ginzburg–Landau functional
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 87
EP  - 92
VL  - 340
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2004.10.024/
DO  - 10.1016/j.crma.2004.10.024
LA  - en
ID  - CRMATH_2005__340_1_87_0
ER  -

%0 Journal Article
%A Berlyand, Leonid
%A Cioranescu, Doina
%A Golovaty, Dmitry
%T Homogenization of a Ginzburg–Landau functional
%J Comptes Rendus. Mathématique
%D 2005
%P 87-92
%V 340
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2004.10.024/
%R 10.1016/j.crma.2004.10.024
%G en
%F CRMATH_2005__340_1_87_0

Berlyand, Leonid; Cioranescu, Doina; Golovaty, Dmitry. Homogenization of a Ginzburg–Landau functional. Comptes Rendus. Mathématique, Tome 340 (2005) no. 1, pp. 87-92. doi : 10.1016/j.crma.2004.10.024. http://www.numdam.org/articles/10.1016/j.crma.2004.10.024/

Bibliographie
Cité par

[1] Berlyand, L. Averaging of elasticity equations in domains with fine-grained boundaries. Part 1, Func. Theory, Funct. Anal. Appl., Volume 39 (1983), pp. 16-25 (in Russian)

[2] Berlyand, L. Homogenization of the Ginzburg–Landau functional with a surface energy term, Asymptotic Anal., Volume 21 (1999), pp. 37-59

[3] Berlyand, L.; Goncharenko, M.V. Averaging of a diffusion equation in a porous medium with weak absorption, Teor. Funktsiı̆ Funktsional. Anal. i Prilozhen., Volume 53 (1989), pp. 113-122 (English translation in J. Soviet Math., 1990, pp. 3428-3435)

[4] L. Berlyand, E.J. Khruslov, Competition between the surface and the boundary energies in a Ginzburg–Landau model of a liquid crystal composite, Asymptotic Anal., in press

[5] L. Berlyand, D. Cioranescu, D. Golovaty, Homogenization of a Ginzburg–Landau model for a nematic liquid crystal with inclusions, J. Math. Pures Appl., in press

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

[7] Cioranescu, D.; Donato, P. On a Robin problem in perforated domains (Cioranescu, D.; Damlamian, A.; Donato, P., eds.), Homogenization and Applications to Material Science, Gakuto Int. Ser., Math. Sci. Appl., Gakkokotosho, vol. 9, 1997, pp. 123-135

[8] Ericksen, J.L. Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal., Volume 113 (1991), pp. 1067-1074

[9] Khruslov, E.J. Asymptotic behavior of the solutions of the second boundary value problem in the case of the refinement of the boundary of the domain, Mat. Sb., Volume 106 (1978), pp. 604-621

[10] Ladyzhenskaya, O.A.; Ural'tseva, N.N. Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1973

[11] Virga, E. Variational Theories for Liquid Crystals, Chapman and Hall, London, 1994

Cité par Sources :