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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics