Competition between the surface and the boundary layer energies in a Ginzburg-Landau model of a liquid crystal composite

We consider a nonlinear homogenization problem for a Ginzburg-Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential. We introduce a small parameter ε such that the average distances between the inclusions are of the order of ε, the inclusions sizes are of the order of ε^α, α > 1, and the coefficient in front of the surface energy term, is of the order of ε^β. We found that the parametric half-plane {(α, β): α > 1, -∞ < β + ∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane. Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems.