Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions

TL;DR

This paper investigates the asymptotic behavior of solutions to higher-dimensional Ginzburg-Landau equations with weak anchoring boundary conditions, relevant to nematic liquid crystals, as the parameter epsilon approaches zero.

Contribution

It extends the analysis of Ginzburg-Landau equations to higher dimensions with weak anchoring, connecting to liquid crystal models and exploring energy bounds.

Findings

Asymptotic behavior characterized under energy constraints

Boundary conditions influence vortex formation

Connection established with Landau-De Gennes model

Abstract

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 $M>0$.