Existence and uniqueness of minimizers for axisymmetric nematic films

TL;DR

This paper rigorously proves the existence and uniqueness of minimizers for a variational model of axisymmetric nematic films, providing a complete geometric characterization and numerical insights.

Contribution

It introduces a reduced one-dimensional variational problem for axisymmetric nematic films and establishes rigorous existence and uniqueness results for minimizers.

Findings

Existence and uniqueness of minimizers are proven.

Complete geometric characterization of minimizers is provided.

Numerical simulations illustrate theoretical results.

Abstract

Nematic surfaces are thin liquid films endowed with in-plane orientational order. We study a variational model in which the nematic director is constrained to lie in the tangent space of an axisymmetric surface, and the associated surface energy accounts for both surface tension and elastic nematic contributions. Here we adopt the surface gradient as the differential operator on the surface, we restrict our analysis to revolution surfaces spanning two coaxial rings, and we assume that the nematic director is aligned along parallels. In this setting, the energy functional reduces to a one-dimensional variational problem. We rigorously prove the existence and uniqueness of minimizers and we provide their complete geometric characterization. Finally, we run some numerical simulations.