TY - JOUR

T1 - Homogenization of a Ginzburg-Landau model for a nematic liquid crystal with inclusions

AU - Berlyand, L.

AU - Cioranescu, D.

AU - Golovaty, D.

N1 - Funding Information:
L.B. was partially supported by the NSF grant DMS-0204637. D.G. was partially supported by the NSF grant DMS-0305577. The work on this paper began when L.B. and D.G. visited Mathematical Geophysics Summer School at Stanford University where they were partially supported by the NSF grant DMS-9709320. A part of this paper was written when L.B. and D.G. were visiting the Laboratoire Jacques-Louis Lions at the Université Pierre et Marie Curie. They gratefully acknowledge the hospitality during their visit. The authors are grateful to E.Ya. Khruslov and O.D. Lavrentovich for useful discussions.

PY - 2005/1

Y1 - 2005/1

N2 - 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 inclusions are separated by distances of the same order ε as their size, we find a limiting functional as ε approaches zero. We generalize the variational 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 obtain computational formulas for material characteristics of an effective medium. As a byproduct of our analysis, we show that the limiting functional is a Γ-limit of a sequence of Ginzburg-Landau functionals. Furthermore, we prove that a cross-term corresponding to interactions between the bulk and the surface energy terms does not appear at the leading order in the homogenized limit.

AB - 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 inclusions are separated by distances of the same order ε as their size, we find a limiting functional as ε approaches zero. We generalize the variational 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 obtain computational formulas for material characteristics of an effective medium. As a byproduct of our analysis, we show that the limiting functional is a Γ-limit of a sequence of Ginzburg-Landau functionals. Furthermore, we prove that a cross-term corresponding to interactions between the bulk and the surface energy terms does not appear at the leading order in the homogenized limit.

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U2 - 10.1016/j.matpur.2004.09.013

DO - 10.1016/j.matpur.2004.09.013

M3 - Article

AN - SCOPUS:11844256386

SN - 0021-7824

VL - 84

SP - 97

EP - 136

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

IS - 1

ER -