Dimensional reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals

Abstract

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\times (0, h) $ with $ \Omega\subset \mathbb{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\times (0, h) $. The Dirichlet datum has the form $ (g, 0) $, where $ g\colon\partial\Omega\to \mathbb{S}^1 $ has non-zero winding number. Under appropriate conditions on the scaling, in the limit as $ h\to 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 Hélein [7].