This paper describes the construction of equilibrium configurations for smectic-$A$ liquid crystals subjected to nonuniform physical boundary conditions, with two-dimensional dependence on the director and layer normal, and a nonlinear layer function. Euler-Lagrange equations are constructed that describe key properties of liquid crystals confined between two boundaries exhibiting spatial imperfections. The results of the model are shown to be consistent with previous published findings in simple domains while results are obtained on how the structure of the liquid crystals changes in response to boundary perturbations. Domain sizes are considered representing those currently used in applications while predictions in smaller domains at the limit of current technologies are also made. In particular, it is shown that the curvature along a boundary impacts on the liquid crystal's structure distant from the boundary feature and therefore previously developed mathematical models, that essentially reduced the problem to a single spatial dimension, cannot be used in such circumstances. Consequences for practical applications are briefly discussed.

中文翻译：

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非均匀区域内近晶非液晶的结构：模拟不完美边界的影响

本文介绍了近晶相平衡构型的构造$\mathrm{一种}$液晶受到非均匀物理边界条件的影响，二维依赖于指向矢和层法向，并且具有非线性层函数。构造了Euler-Lagrange方程，该方程描述了局限在表现出空间缺陷的两个边界之间的液晶的关键特性。结果表明，该模型的结果与先前在简单领域中发表的发现是一致的，同时获得了有关液晶结构如何响应边界扰动而变化的结果。域大小被认为代表了当前在应用程序中使用的大小，同时在当前技术的限制下也对较小的域进行了预测。特别地，表明沿边界的曲率影响液晶'。远离边界特征的结构，因此以前开发的数学模型实际上无法将问题简化为单个空间维度，因此无法在这种情况下使用。简要讨论了实际应用的后果。