On Minimizers of the Landau-de Gennes Energy Functional on Planar Domains

TL;DR

This paper investigates the behavior of tensor-valued minimizers of the Landau-de Gennes energy in planar domains, showing convergence to a projection map as the elastic constant approaches zero, with implications for nematic liquid crystal modeling.

Contribution

It establishes the convergence of minimizers to a projection-valued map and describes the limiting configuration in the Landau-de Gennes model for nematic liquid crystals.

Findings

Minimizers converge to a projection-valued map as epsilon approaches zero.

The limiting map minimizes the Dirichlet energy away from a single point.

The paper characterizes the structure of the limit in planar domains.

Abstract

We study tensor-valued minimizers of the Landau-de Gennes energy functional on a simply-connected planar domain $\Omega$ with non-contractible boundary data. Here the tensorial field represents the second moment of a local orientational distribution of rod-like molecules of a nematic liquid crystal. Under the assumption that the energy depends on a single parameter---a dimensionless elastic constant $\eps>0$---we establish that, as $\eps\to0$, the minimizers converge to a projection-valued map that minimizes the Dirichlet integral away from a single point in $\Omega$. We also provide a description of the limiting map.