We study a two-dimensional variational problem which arises as a thin-film limit of the Landau-de Gennes energy of nematic liquid crystals. We impose an oblique angle condition for the nematic director on the boundary, via boundary penalization (weak anchoring.) We show that for strong anchoring strength (relative to the usual Ginzburg-Landau length scale parameter), defects will occur in the interior, as in the case of strong (Dirichlet) anchoring, but for weaker anchoring strength all defects will occur on the boundary. These defects will each carry a fractional winding number; such boundary defects are known as “boojums”. The boojums will occur in ordered pairs along the boundary; for angle $\alpha \in (0,\frac{\pi }{2})$, they serve to reduce the winding of the phase by steps of 2α and $(2\pi -2\alpha )$ in order to avoid the formation of interior defects. We determine the number and location of the defects via a Renormalized Energy and numerical simulations.

我们研究了二维变分问题，该问题是向列型液晶的Landau-de Gennes能量的薄膜极限。我们通过边界惩罚（弱锚定）在边界上向列向矢施加了一个倾斜角度条件。我们表明，对于强锚定强度（相对于通常的Ginzburg-Landau长度尺度参数），内部会出现缺陷，因为在强（狄利克雷）锚固的情况下，但对于较弱的锚固强度，所有缺陷都会在边界上发生。这些缺陷各自带有分数绕组数。这种边界缺陷被称为“布姆”。布尔值将沿着边界成对排列。角度$\alpha \in （0，\frac{\pi }{2}）$，它们的作用是将相位绕组减小2α和$（2\pi -2\alpha ）$以免形成内部缺陷。我们通过重新归一化能量和数值模拟确定缺陷的数量和位置。