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 $$\Gamma $$ Γ -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.

中文翻译：

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Landau-de Gennes 对 Oseen-Frank 向列液晶理论的修正

我们研究了小弹性常数范围内二维域中向列液晶的 Landau-de Gennes 模型的最小值的渐近行为。在弹性常数的领先阶下，最小能量配置可以用更简单的 Oseen-Frank 理论来描述。使用 $$\Gamma $$ Γ -development 的精炼概念，我们恢复了对 Oseen-Frank 能量的 Landau-de Gennes 校正。我们根据最佳 Oseen-Frank 导向器提供了在这个顺序上最小化 Q 张量的明确表征，并观察了新兴的双轴性。我们应用我们的结果来区分 Oseen-Frank 能量下界饱和的固定拓扑度的共形导向场类中的最佳配置。