A diffuse-interface Landau-de Gennes model for free-boundary problems in the theory of nematic liquid crystals

TL;DR

This paper develops a diffuse-interface Landau-de Gennes model for nematic liquid crystals with free boundaries, proving existence, regularity, and convergence to sharp-interface limits, supported by numerical experiments.

Contribution

It introduces a novel diffuse-interface model for NLC with free boundaries and establishes rigorous mathematical properties and convergence results.

Findings

Existence and regularity of minimizers proven.

Uniform maximum principle and uniqueness for small domains established.

Numerical experiments validate the diffuse-interface approach.

Abstract

We introduce a diffuse-interface Landau-de Gennes free energy for nematic liquid crystals (NLC) systems, with free boundaries, in three dimensions submerged in isotropic liquid, and a phase field is introduced to model the deformable interface. The energy consists of the original Landau-de Gennes free energy, three penalty terms and a volume constraint. We prove the existence and regularity of minimizers for the diffuse-interface energy functional. We also prove a uniform maximum principle of the minimizer under appropriate assumptions, together with a uniqueness result for small domains. Then, we establish a sharp-interface limit where minimizers of the diffuse-interface energy converge to a minimizer of a sharp-interface energy using methods from $\Gamma$-convergence. Finally, we conduct numerical experiments with the diffuse-interface model and the findings are compared with existing works.