Existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees

Abstract

Let $\mathcal{D} =Ω\setminus\barω \subset \mathbb{R}^2$ be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy $E_ε(u)=\frac{1}{2}\int_{\mathcal{D}} |\nabla u|^2 +\frac{1}{4ε^2}\int_{\mathcal{D}} (1-|u|^2)^2$, where $u: \mathcal{D} \rightarrow \mathbb{C}$, and look for minimizers of $E_ε$ with prescribed degrees $deg(u,\partial Ω)=p$, $deg(u,\partial ω)=q$ on the boundaries of the domain. For large $ε$ and for balanced degrees, i.e., $p=q$, we obtain existence of minimizers for {\it thin} domain. We also prove non-existence of minimizers of $E_ε$, for large $ε$, in the case $p\neq q$, $pq>0$ and $\mathcal{D}$ is a circular annulus with large capacity (corresponding to "thin" annulus). Our approach relies on similar results obtained for the Dirichlet energy $E_\infty(u)=\frac{1}{2}\int_{\mathcal{D}}|\nabla u|^2$, the existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.