# Landau-de Gennes corrections to the Oseen-Frank theory of nematic liquid crystals

## TL;DR

This paper analyzes the Landau-de Gennes model for nematic liquid crystals, deriving corrections to the Oseen-Frank theory in the small elastic constant regime, and characterizes the resulting configurations including biaxiality.

## Contribution

It introduces a refined $ ext{Γ}$-development approach to recover Landau-de Gennes corrections to the Oseen-Frank energy and explicitly characterizes minimizers at this order.

## Key findings

- Landau-de Gennes corrections are explicitly characterized.
- Emergence of biaxiality in minimizers is observed.
- Distinction between optimal configurations based on topological degree.

## Abstract

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of $\Gamma$-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing $Q$-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy.