论文标题
Landau-De Gennes模型的点缺陷附近的均匀轮廓模型
Uniform profile near the point defect of Landau-de Gennes model
论文作者
论文摘要
对于在3D域上起作用的Landau-de Gennes，\ begin {equination*} i _ {\ varepsilon}（q，q，ω）：= \int_Ω\ left \ left \ {\ frac {1} {1} {1} {2} {2} {2} | \ nabla q |^2+\ frac}+\ frac} vareps - \ \ frac {a^2} {2} \ mathrm {tr}（q^2） - \ frac {b^2} {3} {3} \ mathrm {tr}（q^3）+\ frac {c^2} \ end {方程*}众所周知，在适当的边界条件下，全局最小化$ q_ \ varepsilon $在$ h^1（ω）$中以$ h^1（ω）$收敛到单轴最小化$ q _*= s _+（n _*\ otime n _*\ otime n _** - $ \ varepsilon_n \ rightarrow \ infty $，其中$ n _*\ in H^1（ω，\ Mathbb {s}^2）$是最小化的谐波映射。在本文中，我们进一步研究了$ q _ {\ varepsilon} $的结构，附近是点缺陷$ x_0 $的核心，这是地图$ n _*$的单数点。主要策略是研究$ q _ {\ varepsilon_n}的爆炸配置文件（x_n+\ varepsilon_n y）$，其中$ \ {x_n \} $被小心地选择并收敛到$ 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 $x_0$.此外，这种融合结果意味着最小化$ q _ {\ varepsilon_n} $可以由Oseen-Frank Minimizer $ n _*$近似$ O（\ Varepsilon_n）$ nover的oseen-frank Minimizer $ n _*$。
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.
