arXiv 2207.045252022-07-10math.AP

Landau-de Gennes模型点缺陷附近的均匀分布

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

Zhiyuan Geng, Arghir Zarnescu

备注：: 26 pages

摘要：

对于三维区域上的Landau-de Gennes泛函\begin{equation*} I_{\varepsilon}(Q,Ω):=\int_Ω\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*}，在适当的边界条件下，全局极小元$Q_\varepsilon$在$H^1(Ω)$中强收敛到单轴极小元$Q_*=s_+(n_*\otimes n_*-\frac{1}{3}\mathrm{Id})$直到某个子序列$\varepsilon_n\rightarrow\infty$，其中$n_*\in H^1(Ω,\mathbb{S}^2)$是极小化调和映射.在本文中，我们进一步研究了$Q_{\varepsilon}$的结构附近的点缺陷$x_0$是一个奇异点的映射$n_*$的核心。主要策略是研究$Q_{\varepsilon_n}(x_n+\varepsilon_n y)$的爆破剖面，其中$\{x_n\}$被仔细选择并收敛到$x_0$。我们证明了$Q_{\varepsilon_n}(x_n+\varepsilon_n y)$收敛于$C^2_{loc}(\mathbb{R}^n)$的切线映射$Q(x)$在无穷大的行为像一个“刺猬”的解决方案，符合渐近配置文件的$n_*$附近的$x_0$。此外，这样的收敛结果意味着最小值$Q_{\varepsilon_n}$可以很好地由点缺陷的$O(\varepsilon_n)$邻域之外的Oseen-Frank最小值$n_*$来近似。

英文摘要：

For the Landau-de Gennes functional on 3D domains, \begin{equation*} I_{\varepsilon}(Q,Ω):=\int_Ω\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 $H^1(Ω)$ to a uniaxial minimizer $Q_*=s_+(n_*\otimes n_*-\frac{1}{3}\mathrm{Id})$ up to some subsequence $\varepsilon_n\rightarrow\infty$ , where $n_*\in H^1(Ω,\mathbb{S}^2)$ 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.

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英文摘要：

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