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

TL;DR

This paper analyzes the behavior of the Oseen-Frank model for nematic liquid crystals in thin slabs, showing the emergence of defects with topological charges governed by a Renormalized Energy as the thickness vanishes.

Contribution

It provides a rigorous analysis of defect formation in the Oseen-Frank model in the thin limit, connecting it to Ginzburg-Landau asymptotics and topological charge distribution.

Findings

Finite number of defect points with topological charges.

Defect positions governed by a Renormalized Energy.

Behavior similar to Ginzburg-Landau asymptotics.

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 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 H\'elein.