Homogenization of a Ginzburg–Landau model for a nematic liquid crystal with inclusions |
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Institution: | 1. Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA;2. Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, boîte courrier 187, 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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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 inclusions are separated by distances of the same order ? as their size, we find a limiting functional as ? 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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