The Two-Dimensional Liquid Crystal Droplet Problem with a Tangential Boundary Condition

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 that the boundary of the droplet is a chord–arc curve with its normal vector field in the VMO space, and its arc-length parameterization belongs to the Sobolev space H^{3 / 2}. In fact, the boundary curves of such droplets closely resemble the so-called Weil–Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is studied.