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

Let $\mathcal{D}=\mathrm{\Omega }\setminus \bar{\omega }\subset {\mathbb{R}}^2$ be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy $E_\epsilon(u)=\frac{1}{2}\int_{$\mathcal{D}$} |\nabla u|^2 +\frac{1}{4\epsilon^2}\int_{$\mathcal{D}$} (1-|u|^2)^2$,where$u: $\mathcal{D}$ \rightarrow $\mathbb{C}$$,andlookforminimizersof$E_\epsilon$ with prescribed degrees $deg(u,\mathrm{\partial }\mathrm{\Omega })=p$, $deg(u,\mathrm{\partial }\omega )=q$ 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 $p\mathrm{\ne }q$, $pq>0$ 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 $E_{\mathrm{\infty }}(u)=\frac{1}{2}{\int }_{\mathcal{D}}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$, the existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.