Affiliations: Department of Mathematics, Penn State University, University Park, PA 16802, USA E‐mail: [email protected] | Institute of Low Temperature and Engineering, Ukrainian Academy of Science, Lenin Ave., 47, Kharkov 310164, Ukraine E‐mail: [email protected]
Note: [] Corresponding author. The work of L.B. was supported by NSF grant DMS‐9971999.
Abstract: 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 εα, α>1, and the coefficient in front of the surface energy term, is of the order of εβ. 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.
