Global minimisers of cholesteric liquid crystal systems

TL;DR

This paper analyzes the global minimizers of cholesteric liquid crystal systems within the Oseen-Frank theory, establishing conditions for their uniqueness and stability in specific boundary and material settings.

Contribution

It provides a rigorous mathematical characterization of global minimizers for cholesteric liquid crystals with general elastic constants, including conditions for their global optimality.

Findings

Unique global minimizers found for one-variable functions.

Global minimizers are also minimizers for the full problem when the cholesteric pitch is long.

Analysis extends to stability of the constant state with unfrustrated boundary conditions.

Abstract

In this paper we examine the modelling and minimisation of cholesteric liquid crystals systems within the Oseen-Frank theory. We focus on a cuboid domain with the frustrated boundary conditions ${\bf n}(0,x,y)=(1,0,0)$ and ${\bf n}(1,x,y)=(0,0,1)$. With general elastic constants, we find the unique global minimisers amongst functions of one variable and prove that these are global minimisers of the entire problem if the cholesteric pitch is sufficiently long. Finally we show that our analysis easily translates over the study the global stability of the constant state ${\bf n}(x,y,z) = (0,0,1)$ with unfrustrated boundary conditions.