  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.
