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 $g:\partial \Omega \to {\mathbb{S}}^1$ 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 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.