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 Ω × (0, h) with Ω ⊂ R2 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 Ω×(0, h). The Dirichlet datum has the form (g, 0), where g: ∂Ω → S1 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 Ω 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 Hélein [7].