Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model

TL;DR

This paper analyzes the asymptotic behavior of minimizers in a 3D Landau-de Gennes model for nematic liquid crystals as the elastic constant approaches zero, revealing line defect structures and convergence to harmonic maps.

Contribution

It establishes the formation and structure of line defects in the small elastic constant limit of the Landau-de Gennes model, with conditions for their occurrence.

Findings

Minimizers converge to harmonic maps away from a finite set of line defects.

Line defects form a finite union of straight segments in the domain.

Conditions are provided for the existence of such defect structures.

Abstract

We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. More precisely, we study the asymptotic behaviour of minimizers as the elastic constant tends to zero, under the assumption that minimizers are uniformly bounded and their energy blows up as the logarithm of the elastic constant. We show that there exists a closed set S of finite length, such that minimizers converge to a locally harmonic map away from S. Moreover, S restricted to the interior of the domain is a locally finite union of straight line segments. We provide sufficient conditions, depending on the domain and the boundary data, under which our main results apply. We also discuss some examples.