A GINZBURG-LANDAU-TYPE PROBLEM FOR HIGHLY ANISOTROPIC NEMATIC LIQUID CRYSTALS

We carry out an asymptotic analysis of a variational problem relevant in the studies of nematic liquid crystalline films when one elastic constant dominates over the others, namely, inf E-epsilon(u), where E-epsilon(u) := E 1/2 integral Omega{epsilon vertical bar del u vertical bar(2 )+ 1/epsilon(vertical bar u vertical bar(2)- 1)(2) + L(div u)(2)}dx. Here u : Omega -> R-2 is a vector field, 0 < epsilon << 1 is a small parameter, and L > 0 is a fixed constant, independent of epsilon. We identify a candidate for the Gamma-limit E-0, which is a sum of a bulk term penalizing divergence and an Aviles-Giga-type wall energy involving the cube of the jump in the tangential component of the S-1 -valued nematic director. We establish the lower bound and provide the recovery sequence for this candidate within a restricted class. Then we consider a set of variational problems for E-0 arising from various choices of domain geometry and boundary conditions. We demonstrate that the criticality conditions for E-0 can be expressed as a pair of scalar conservation laws that share characteristics. We use the method of characteristics to analytically construct critical points of E-0 that we observe numerically.