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Eigenframe discontinuities of the Q-tensor model

Zhiyuan Geng, Changyou Wang

TL;DR

This work analyzes defect structures in the Lyuksyutov-constrained Q-tensor model of nematic liquid crystals in a 3D domain. By a blow-up analysis around points with $\beta(Q)=-1$ and eigenframe discontinuity, the authors reduce the problem to harmonic maps into $S^4$ through the leading order term $U$ and classify tangent maps according to the vanishing order $k$ of $U$. They prove the defect set $\mathcal{S}(Q)$ is countably $1$-rectifiable and provide a detailed local description of eigenframe behavior: for $k=1$ defects lie along lines with half-degree winding, while for $k\ge 2$ higher-order profiles emerge, governed by a reduced energy that neglects higher-order corrections. This extends prior ring-disclination results without symmetry assumptions and clarifies eigenframe discontinuities in the full Q-tensor framework, with implications for understanding biaxial core structures.

Abstract

In this paper, we study the defect structure of minimizer of a Landau-de Gennes energy functional in three-dimensional domains, subject to constraint $|Q|=1$. The set of defects is identified by discontinuities in both the eigenframe and the leading eigenvector. Through a blow-up analysis, we prove that the defect set is 1-rectifiable and classify the asymptotic profile of the leading eigenvector near singularities. This generalizes some previous results on the structure of ring disclinations in the $Q$-tensor model.

Eigenframe discontinuities of the Q-tensor model

TL;DR

This work analyzes defect structures in the Lyuksyutov-constrained Q-tensor model of nematic liquid crystals in a 3D domain. By a blow-up analysis around points with and eigenframe discontinuity, the authors reduce the problem to harmonic maps into through the leading order term and classify tangent maps according to the vanishing order of . They prove the defect set is countably -rectifiable and provide a detailed local description of eigenframe behavior: for defects lie along lines with half-degree winding, while for higher-order profiles emerge, governed by a reduced energy that neglects higher-order corrections. This extends prior ring-disclination results without symmetry assumptions and clarifies eigenframe discontinuities in the full Q-tensor framework, with implications for understanding biaxial core structures.

Abstract

In this paper, we study the defect structure of minimizer of a Landau-de Gennes energy functional in three-dimensional domains, subject to constraint . The set of defects is identified by discontinuities in both the eigenframe and the leading eigenvector. Through a blow-up analysis, we prove that the defect set is 1-rectifiable and classify the asymptotic profile of the leading eigenvector near singularities. This generalizes some previous results on the structure of ring disclinations in the -tensor model.
Paper Structure (6 sections, 7 theorems, 74 equations, 1 figure)

This paper contains 6 sections, 7 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical formulation and main results
  3. Related results
  4. Preliminaries
  5. Energy functional for $U$
  6. Classification of local behaviors of $U(\mathbf{x})$

Key Result

Theorem 1.1

Let $Q\in H^1(B_1,\mathbb{S}^4)$ be a minimizer of the energy functional $\mathcal{E}(Q)$ given by ene, which satisfies the boundary condition bdy cond. The defect set $\mathcal{S}(Q)$, where both the eigenframe and the leading eigenvector are discontinuous (see def:defect), is countably 1-rectifiab

Figures (1)

  • Figure 1: Vector field $s(\mathbf{x})\mathbf{n}(\mathbf{x})$ for $k(\mathbf{x}_0)=1$

Theorems & Definitions (13)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.1
  • proof
  • ...and 3 more