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.

Automated summary

This paper studies the shape optimization problem of liquid crystal droplets and establishes the existence of an optimal shape with two cusps on the boundary. It also investigates the properties of the droplet's boundary and the asymptotic behavior of the optimal shape as the volume approaches extremes.