Ginzburg–Landau energy with prescribed degrees

We consider the simplified Ginzburg–Landau energy $\frac12\int_\Omega|\nabla u|^2+\frac1{(4\varepsilon)^2}\int_\Omega(1-|u|^2)^2$. Here, $\Omega$ is a domain in $\mathbb R^2$ and $u$ is complex-valued. On $\partial\Omega$, we prescribe $|u|=1$ and the winding numbers of $u$. This is one of the simplest models of critical equation leading to non-scalar bubbles. I will discuss existence/nonexistence results for minimizers/critical points. The talk is based on results of Berlyand, Dos Santos, Farina, Golovaty, Rybalko, and the lecturer.