H\"older regularity and convergence for a non-local model of nematic liquid crystals in the large-domain limit

TL;DR

This paper studies a non-local energy model for nematic liquid crystals, establishing regularity and convergence results in large domains, with techniques similar to those used in Ginzburg-Landau theory.

Contribution

It extends previous large-domain convergence results by proving H"older regularity and stronger convergence away from singularities for minimisers of the model.

Findings

Established H"older bounds for minimisers on bounded domains.

Proved stronger convergence of minimisers away from singularities.

Connected techniques with Ginzburg-Landau and Landau-de Gennes models.

Abstract

We consider a non-local free energy functional, modelling a competition between entropy and pairwise interactions reminiscent of the second order virial expansion, with applications to nematic liquid crystals as a particular case. We build on previous work on understanding the behaviour of such models within the large-domain limit, where minimisers converge to minimisers of a quadratic elastic energy with manifold-valued constraint, analogous to harmonic maps. We extend this work to establish H\"older bounds for (almost-)minimisers on bounded domains, and demonstrate stronger convergence of (almost)-minimisers away from the singular set of the limit solution. The proof techniques bear analogy with recent work of singularly perturbed energy functionals, in particular in the context of the Ginzburg-Landau and Landau-de Gennes models.