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Landau-de Gennes Energy with Weak Planar Anchoring

Ho Man Tai, Yong Yu

TL;DR

This work analyzes the Landau–de Gennes energy for nematic liquid crystals in a smooth, bounded 3D domain with degenerate weak planar anchoring, focusing on the regime of vanishing elastic constant $L$ below the nematic–isotropic transition. The authors prove the minimizers converge, as $L\to0^+$, to a uniaxial limit $Q^*=s_0(u^*\otimes u^*-\tfrac13 I_3)$ where $u^*$ is a tangential-harmonic map solving a Dirichlet energy problem with boundary tangency, and $u^*$ has only finitely many singularities on $\overline{\Omega}$ including boojums on $\partial\Omega$. A key contribution is the development of boundary-sensitive tools: boundary flattening, a tangent-plane projection, and a novel extension lemma that preserves the tangential boundary constraint and yields precise energy controls, enabling a bubbling analysis near boundary points. The combination of compactness for rescaled maps and boundary partial regularity leads to a rigorous description of boundary singularities and confirms boojums as natural boundary phenomena in this degenerate weak anchoring regime. Overall, the paper extends the partial regularity theory for harmonic maps to the Landau–de Gennes setting with tangential boundary data and provides a detailed boundary defect taxonomy with finite boundary singularities.

Abstract

We study global minimizers of the Landau-de Gennes energy for the nematic liquid crystals in simply connected, bounded, smooth domains of dimension 3, subject to a weak planar anchoring. The boundary condition is degenerate. In the regime where the elastic constant tends to zero and the temperature is below the nematic-isotropic transition threshold, the bulk and surface energy enforce the energy minimizers to be uniaxial, with director fields lying in the tangent plane to the boundary in the sense of trace. We establish that the singular set of the limiting minimizer within the closure of the domain is a finite set by studying the associated harmonic map with a strong tangential boundary constraint. The structure of the singularities located on the boundary is also investigated.

Landau-de Gennes Energy with Weak Planar Anchoring

TL;DR

This work analyzes the Landau–de Gennes energy for nematic liquid crystals in a smooth, bounded 3D domain with degenerate weak planar anchoring, focusing on the regime of vanishing elastic constant below the nematic–isotropic transition. The authors prove the minimizers converge, as , to a uniaxial limit where is a tangential-harmonic map solving a Dirichlet energy problem with boundary tangency, and has only finitely many singularities on including boojums on . A key contribution is the development of boundary-sensitive tools: boundary flattening, a tangent-plane projection, and a novel extension lemma that preserves the tangential boundary constraint and yields precise energy controls, enabling a bubbling analysis near boundary points. The combination of compactness for rescaled maps and boundary partial regularity leads to a rigorous description of boundary singularities and confirms boojums as natural boundary phenomena in this degenerate weak anchoring regime. Overall, the paper extends the partial regularity theory for harmonic maps to the Landau–de Gennes setting with tangential boundary data and provides a detailed boundary defect taxonomy with finite boundary singularities.

Abstract

We study global minimizers of the Landau-de Gennes energy for the nematic liquid crystals in simply connected, bounded, smooth domains of dimension 3, subject to a weak planar anchoring. The boundary condition is degenerate. In the regime where the elastic constant tends to zero and the temperature is below the nematic-isotropic transition threshold, the bulk and surface energy enforce the energy minimizers to be uniaxial, with director fields lying in the tangent plane to the boundary in the sense of trace. We establish that the singular set of the limiting minimizer within the closure of the domain is a finite set by studying the associated harmonic map with a strong tangential boundary constraint. The structure of the singularities located on the boundary is also investigated.
Paper Structure (11 sections, 21 theorems, 278 equations, 1 figure)

This paper contains 11 sections, 21 theorems, 278 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivations and literature review
  3. Model setting and main results
  4. Notations
  5. Minimizers of TEXT-energy
  6. Strong $H^1$-limit of TEXT
  7. Boundary flattening map and the associated projection
  8. Extension lemma
  9. Bubbling analysis at boundary singularities
  10. Compactness of rescaled maps
  11. Energy monotonicity and Partial regularity

Key Result

Theorem 1.1

Suppose $b$, $c$, $s_1$ and $s_2$ are given positive material constants. The constant $a\in \mathbb{R}$ satisfies $b^2>27ac$. Then the following hold:

Figures (1)

  • Figure 1: Extension in Part 1B

Theorems & Definitions (36)

  • Theorem 1.1
  • Lemma 2.1: Existence of minimizer
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4: Regularity of minimizers
  • proof
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem. f_s']}
  • Proposition 3.2
  • ...and 26 more