Vector Field Models for Nematic Disclinations

TL;DR

This paper investigates a vector field model for nematic disclinations, providing relaxation and compactness results, and offering insights into defect modeling beyond traditional SBV function space approaches.

Contribution

It introduces a relaxed model for nematic disclinations that extends previous SBV-based models by incorporating a second field and proves key mathematical properties.

Findings

Established a relaxation result for the model with fixed parameters

Proved partial compactness results for the vector field model

Compared the model's regularization to Sobolev space approaches

Abstract

In this paper, a model for defects that was introduced in \cite{ZANV} is studied. In the literature, the setting of most models for defects is the function space SBV (special bounded variation functions) (see, e.g., \cite{ContiGarroni, GoldmanSerfaty}). However, this model regularizes the director field to be in a Sobolev space by adding a second field to incorporate the defect. A relaxation result in the case of fixed parameters is proven along with some partial compactness results.