Uniform profile near the point defect of Landau-de Gennes model

TL;DR

This paper analyzes the detailed structure of minimizers in the Landau-de Gennes model near point defects, showing convergence to a tangent map and approximation by harmonic maps outside small neighborhoods.

Contribution

It establishes the convergence of blow-up profiles of minimizers to tangent maps and relates the Landau-de Gennes minimizers to harmonic maps near defects.

Findings

Blow-up profiles converge to tangent maps in $C^2_{loc}$.

Minimizers are well approximated by harmonic maps outside defect cores.

Tangent maps resemble hedgehog solutions at infinity.

Abstract

For the Landau-de Gennes functional on 3D domains, \begin{equation*} I_{\varepsilon}(Q,\Omega):=\int_{\Omega}\left\{\frac{1}{2}|\nabla Q|^2+\frac{1}{\varepsilon^2}\left( -\frac{a^2}{2}\mathrm{tr}(Q^2)-\frac{b^2}{3}\mathrm{tr}(Q^3)+\frac{c^2}{4}[\mathrm{tr}(Q^2)]^2 \right) \right\}\,dx, \end{equation*} it is well-known that under suitable boundary conditions, the global minimizer $Q_\varepsilon$ converges strongly in $H^1(\Omega)$ to a uniaxial minimizer $Q_*=s_+(n_*\otimes n_*-\frac{1}{3}\mathrm{Id})$ up to some subsequence $\varepsilon_n\rightarrow\infty$ , where $n_*\in H^1(\Omega,\mathbb{S}^2)$ is a minimizing harmonic map. In this paper we further investigate the structure of $Q_{\varepsilon}$ near the core of a point defect $x_0$ which is a singular point of the map $n_*$. The main strategy is to study the blow-up profile of $Q_{\varepsilon_n}(x_n+\varepsilon_n y)$ where $\{x_n\}$ are carefully chosen and converge to $x_0$. We prove that $Q_{\varepsilon_n}(x_n+\varepsilon_n y)$ converges in $C^2_{loc}(\mathbb{R}^n)$ to a tangent map $Q(x)$ which at infinity behaves like a "hedgehog" solution that coincides with the asymptotic profile of $n_*$ near $x_0$. Moreover, such convergence result implies that the minimizer $Q_{\varepsilon_n}$ can be well approximated by the Oseen-Frank minimizer $n_*$ outside the $O(\varepsilon_n)$ neighborhood of the point defect.