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

Abstract

Let $\mathcal{D}=\Omega \setminus \bar{\omega }\subset {\mathbb{R}}^2$ 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 $u:\mathcal{D}\to \mathbb{C}$, and look for minimizers of $E_{\epsilon }$ with prescribed degrees $deg(u,\partial \Omega )=p$, $deg(u,\partial \omega )=q$ on the boundaries of the domain. For large $\epsilon$ 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 $E_{\epsilon }$, for large $\epsilon$, in the case $p\ne q$, $pq>0$ 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 $E_{\infty }(u)=\frac{1}{2}{\int }_{\mathcal{D}}|\nabla u{|}^2$, the existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.

arXiv comment31 pages