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

Abstract

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, 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.