Suppose $\Omega\subset\mathbb{R}^{3}$ is bounded and smooth, $g\in H^{1}(\Omega;\mathbb{S}^{2})$. For $\varepsilon>0$ and $u\in H^{1}(\Omega;\mathbb{R}^{3})$ define the Ginzburg-Landau energy$$E_{\varepsilon}(u)=\int_{\Omega}\dfrac{1}{2}\lvert\nabla u\rvert^{2}+\dfrac{1}{4\varepsilon^{2}}\big{(}1-\lvert u\rvert^{2}\big{)}^{2}\,dx.$$Prove that$$E_{\varepsilon}(u)=\min_{v\in H_{g}^{1}(\Omega;\mathbb{R}^{3})}E_{\varepsilon}(v)$$has a minimizer $u_{\varepsilon}\in H_{g}^{1}(\Omega;\mathbb{R}^{3})$ and $E_{\varepsilon}(u_{\varepsilon})\leq C$, $C>0$ is independent of $\varepsilon$, where$$H_{g}^{1}(\Omega;\mathbb{R}^{3}):=\big{\{}u\in H^{1}(\Omega;\mathbb{R}^{3}):\ \left. u\right| _{\partial\Omega}=\left. g\right| _{\partial\Omega}\big{\}}.$$Then prove there is a sequence $\varepsilon_{k}\rightarrow0$ and $u\in H_{g}^{1}(\Omega;\mathbb{S}^{2})$ such that $u_{\varepsilon_{k}}\rightharpoonup u$ weakly in $H^{1}(\Omega;\mathbb{R}^{3})$ and strongly in $L^{2}(\Omega;\mathbb{R}^{3})$.
I have proved the first one. While for the second, I gauss the $u$ we required is the solution of$$E(u)=\min_{v\in H_{g}^{1}(\Omega;\mathbb{S}^{2})}E(v),$$where$$E(v)=\int_{\Omega}\dfrac{1}{2}\lvert\nabla v\rvert^{2}\,dx.$$According to Evans, it is also a weak solution of$$\begin{cases} \Delta u+\lvert\nabla u\rvert^{2}u=0\ \text{in}\ \Omega\\ \left. u\right| _{\partial\Omega}=\left. g\right| _{\partial\Omega}.\\ \end{cases} $$Then for $\varepsilon_{k}\rightarrow0$, we have $\lvert u_{\varepsilon_{k}}\rvert^{2}\rightarrow1$ and $\lvert\nabla u_{\varepsilon_{k}}\rvert^{2}\rightarrow\lvert\nabla u\rvert^{2}$, i.e. $u_{\varepsilon_{k}}\rightarrow u$ in $H^{1}(\Omega;\mathbb{R}^{3})$ and strongly in $L^{2}(\Omega;\mathbb{R}^{3})$. Meanwhile, since $H^{1}(\Omega;\mathbb{R}^{3})$ is a Hilbert space, there exists a subsequence of $\varepsilon_{k}$ (still denote as $\varepsilon_{k}$) such that $u_{\varepsilon_{k}}\rightharpoonup u$ weakly. But there is a problem that the solution of$$E(u)=\min_{v\in H_{g}^{1}(\Omega;\mathbb{S}^{2})}E(v),$$may not be unique(or are there any method to prove it?). So I wonder if my thoughts make sense. Any other approaches are welcome, too. Appreciate in advance.