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arXiv · 2106.10668

Two dimensional liquid crystal droplet problem with tangential boundary condition

Zhiyuan Geng·Fanghua Lin
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Abstract

This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove the boundary of the droplet is a chord-arc curve with its normal vector field in the VMO space. In fact, the boundary curves of such droplets belong to the so-called Weil-Petersson class. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is also studied.

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Zhiyuan Geng, Fanghua Lin. 2021-06-20. Two dimensional liquid crystal droplet problem with tangential boundary condition. https://doi.org/10.1007/s00205-021-01733-5

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