We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we prove full regularity up to the boundary for the minimizers. As a consequence, in a relevant range (which we call the Lyuksyutov regime) of parameters of the model we show that even without the norm constraint isotropic melting is anyway avoided in the energy minimizing configurations. Finally, we describe a class of boundary data including radial anchoring which yield in both the previous situations as minimizers smooth configurations whose level sets of the biaxiality carry nontrivial topology. Results in this paper will be largely employed and refined in the next papers of our series. In particular, in [DMP2], we will prove that for smooth minimizers in a restricted class of axially symmetric configurations, the level sets of the biaxiality are generically finite union of tori of revolution.

中文翻译：

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Landau-de Gennes 模型的类环面解。第一部分：柳克休托夫政权

我们在三维域中研究了向列液晶的连续谱 Landau-De Gennes 能量泛函的全局极小值。假设内部光滑和单轴（例如垂直）边界条件和相应的物理相关范数约束（Lyuksyutov 约束），我们证明了最小化器边界的完全正则性。因此，在模型参数的相关范围（我们称之为 Lyuksyutov 制度）中，我们表明即使没有范数约束，在能量最小化配置中也可以避免各向同性熔化。最后，我们描述了一类边界数据，包括径向锚定，其在前两种情况下均作为最小化平滑配置产生，其双轴水平集具有非平凡的拓扑结构。本文中的结果将在我们系列的下一篇论文中大量使用和完善。特别是，在 [DMP2] 中，我们将证明对于有限类轴对称配置中的平滑极小器，双轴性的水平集通常是旋转环面的有限并集。