Abstract

Motivated by Ball and Majumdar’s modification of Landau-de Gennes model for nematic liquid crystals, we study energy-minimizer Q of a tensor-valued variational obstacle problem in a bounded 3-D domain with prescribed boundary data. The energy functional is designed to blow up as Q approaches the obstacle. Under certain assumptions, especially on blow-up profile of the singular bulk potential, we prove higher interior regularity of Q, and show that the contact set of Q is either empty, or small with characterization of its Hausdorff dimension. We also prove boundary partial regularity of the energy-minimizer.

中文翻译：

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张量值变分障碍问题极小化子的三维正则性

摘要

受Ball和Majumdar对向列型液晶的Landau-de Gennes模型的修改的启发，我们使用规定的边界数据研究有界3-D域中的张量值可变障碍问题的能量最小化器Q。能量功能设计为在Q接近障碍物时爆炸。在一定条件下，尤其是在大宗奇异潜力吹塑机图，我们证明了更高的内部规律性Q，和表明的触点组Q或者为空，或者它的Hausdorff维数的表征小。我们还证明了能量最小化器的边界偏正则性。