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

Abstract

Let [Formula: see text] be a smooth annular type domain. We consider the simplified Ginzburg–Landau energy [Formula: see text], where [Formula: see text], and look for minimizers of [Formula: see text] with prescribed degrees [Formula: see text], [Formula: see text] on the boundaries of the domain. For large [Formula: see text] and for balanced degrees (i.e. [Formula: see text]), we obtain existence of minimizers for domains with large capacity (corresponding to thin annulus). We also prove non-existence of minimizers of [Formula: see text], for large [Formula: see text], if [Formula: see text], [Formula: see text] and if [Formula: see text] is a circular annulus with large capacity. Our approach relies on similar results obtained for the Dirichlet energy [Formula: see text], on a previous existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.