DIMENSIONAL REDUCTION AND EMERGENCE OF DEFECTS IN THE OSEEN-FRANK MODEL FOR NEMATIC LIQUID CRYSTALS

In this paper we discuss the behavior of the Oseen-Frank model for nematic liquid crystals in the limit of vanishing thickness. More precisely, in a thin slab Omega x (0, h) with Omega subset of R-2 and h > 0 we consider the one-constant approximation of the Oseen-Frank model for nematic liquid crystals. We impose Dirichlet boundary conditions on the lateral boundary and weak anchoring conditions on the top and bottom faces of the cylinder Omega x (0, h). The Dirichlet datum has the form (g, 0), where g: partial derivative Omega -> S-1 has non-zero winding number. Under appropriate conditions on the scaling, in the limit as h -> 0 we obtain a behavior that is similar to the one observed in the asymptotic analysis (see [7]) of the two-dimensional Ginzburg-Landau functional. More precisely, we rigorously prove the emergence of a finite number of defect points in Omega having topological charges that sum to the degree of the boundary datum. Moreover, the position of these points is governed by a Renormalized Energy, as in the seminal results of Bethuel, Brezis and Helein [7].