  For $n\ge 3$ and $0<\epsilon\le 1$, let $\Omega\subset\mathbb R^n$ be a
bounded smooth domain and $u_\epsilon:\Omega \subset\R^n\to \mathbb R^2$ solve
the Ginzburg-Landau equation under the weak anchoring boundary condition:
$$\begin{cases} -\Delta
u_\epsilon=\frac{1}{\epsilon^2}(1-|u_\epsilon|^2)u_\epsilon &\ {\rm{in}}\ \
\Omega, \frac{\partial
u_\epsilon}{\partial\nu}+\lambda_\epsilon(u_\epsilon-g_\epsilon)=0 & \
{\rm{on}}\ \ \partial\Omega, \end{cases} $$ where the anchoring strength
parameter $\lambda_\epsilon=K\epsilon^{-\alpha}$ for some $K>0$ and $\alpha\in
[0,1)$, and $g_\epsilon\in C^2(\partial\Omega, \mathbb S^1)$. Motivated by the
connection with the Landau-De Gennes model of nematic liquid crystals under
weak anchoring conditions, we study the {asymptotic behavior} of $u_\epsilon$
as $\epsilon$ goes to zero under the condition that the total modified
Ginzburg-Landau energy {satisfies} $F_\epsilon(u_\epsilon,\Omega)\le
M|\log\epsilon|$ for some $M>0$.
