Analysis of a variational model for nematic shells

TL;DR

This paper investigates a variational model for nematic liquid crystal thin films on surfaces, highlighting the impact of extrinsic surface properties, establishing existence results, and analyzing minimizers through theoretical and numerical methods.

Contribution

It introduces a novel extrinsic energy approach for nematic shells, linking minimizer existence to surface topology and providing detailed minimizer characterizations.

Findings

Existence of minimizers depends on surface topology.

Gradient flow of the energy is well-posed via Ginzburg-Landau approximation.

On axisymmetric tori, global and local minimizers are characterized.

Abstract

We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the gradient flow of the energy; iii) in the case of a parametrized axisymmetric torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.