A shape optimization problem for nematic and cholesteric liquid crystal drops

TL;DR

This paper extends the shape optimization framework for nematic and cholesteric liquid crystal drops, allowing for more complex domain geometries and director field discontinuities, and proves the existence of optimal configurations.

Contribution

It introduces a broader class of admissible domains with inner boundaries and establishes existence results for the generalized shape optimization problem.

Findings

Existence of optimal configurations under volume constraints.

Generalization of the shape optimization problem to include inner boundaries.

Mathematical proof of existence for the extended problem.

Abstract

We generalize the shape optimization problem for the existence of stable equilibrium configurations of nematic and cholesteric liquid crystal drops surrounded by an isotropic solution to include a broader family of admissible domains with inner boundaries, allowing discontinuities in the director field across them. Within this setting, we prove the existence of optimal configurations under a volume constraint and show that the minimization problem is a natural generalization of that posed for regular domains.