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

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