We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter ε entering an Allen-Cahn-type energy functional augmented by a twist term. We consider the behavior of the energy as ε tends to zero. We demonstrate existence of the local energy minimizers classified by their overall twist, find the Γ-limit of the relaxed energies and show that it consists of the twist and jump terms. Further, we extend our results to include the situation when the cholesteric pitch vanishes along with ε.

我们考虑与胆甾型液晶模型有关的一维变化问题。我们研究的主要特征是假设向列导向器的扭曲变形比其他变形方式产生更高的能量损失。弹性常数的适当比率然后给出一个小的参数ε，该参数进入由扭转项扩展的Allen-Cahn型能量函数。当ε趋于零时，我们认为能量的行为。我们证明了存在的按局部能量最小化器的整体扭曲分类，找到了弛豫能量的Γ极限，并表明它由扭曲和跳跃项组成。此外，我们将结果扩展到包括胆甾醇沥青随着ε。