Two dimensional liquid crystal droplet problem with tangential boundary condition

TL;DR

This paper investigates the shape optimization of 2D liquid crystal droplets with tangential boundary conditions, establishing existence, boundary regularity, and asymptotic behavior of optimal shapes under volume constraints.

Contribution

It proves the existence of optimal droplet shapes with cusps, characterizes their boundary regularity, and analyzes their asymptotic behavior for extreme volumes.

Findings

Existence of optimal shapes with two boundary cusps

Boundary is a chord-arc curve with VMO normal vector

Optimal shapes belong to the Weil-Petersson class

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.