Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions

For $n\ge 3$ and $0 0$ 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 $M>0$.