[Homogénéisation d'une fonctionnelle de Ginzburg–Landau.]
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
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
Accepté le :
Publié le :
Leonid Berlyand 1 ; Doina Cioranescu 2 ; Dmitry Golovaty 3
@article{CRMATH_2005__340_1_87_0, 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}, }
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/
[1] 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] Homogenization of the Ginzburg–Landau functional with a surface energy term, Asymptotic Anal., Volume 21 (1999), pp. 37-59
[3] 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] Ginzburg–Landau Vortices, Birkhäuser, Boston, 1994
[7] On a Robin problem in perforated domains (D. Cioranescu; A. Damlamian; P. Donato, eds.), Homogenization and Applications to Material Science, Gakuto Int. Ser., Math. Sci. Appl., Gakkokotosho, vol. 9, 1997, pp. 123-135
[8] Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal., Volume 113 (1991), pp. 1067-1074
[9] 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] Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1973
[11] Variational Theories for Liquid Crystals, Chapman and Hall, London, 1994
Cité par Sources :
Commentaires - Politique