Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions

Abstract

For $n\ge 3$ and $0<\hat{I}\mu \le 1$, let $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ be a bounded smooth domain and $u_{\hat{I}}\mu :\hat{I}\text{c◯}\subset {\mathbb{R}}^n\to {\mathbb{R}}^2$ solve the Ginzburg-Landau equation under the weak anchoring boundary condition: $$\{\begin{array}{llc}-\hat{I}”u_{\hat{I}}\mu =\frac{1}{\hat{I}{\mu }^2}(1-|u_{\hat{I}}\mu {|}^2)u_{\hat{I}}\mu & \mathrm{in}\hat{I}\text{c◯},\frac{\partial u_{\hat{I}}\mu }{\partial \hat{I}½}+\hat{I}{»}_{\hat{I}}\mu (u_{\hat{I}}\mu -g_{\hat{I}}\mu )=0& \mathrm{on}\partial \hat{I}\text{c◯},\end{array}$$ where the anchoring strength parameter $\hat{I}{»}_{\hat{I}}\mu =K\hat{I}{\mu }^{-\hat{I}\pm }$ for some $K>0$ and $\hat{I}\pm \in [0,1)$, and $g_{\hat{I}}\mu \in C^2(\partial \hat{I}\text{c◯},{\mathbb{S}}^1)$. Motivated by the connection with the Landau-De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the {asymptotic behavior} of $u_{\hat{I}}\mu$ as $\hat{I}\mu$ goes to zero under the condition that the total modified Ginzburg-Landau energy {satisfies} $F_{\hat{I}}\mu (u_{\hat{I}}\mu ,\hat{I}\text{c◯})\le M|\log \hat{I}\mu |$ for some $M>0$.

38 pages