Variational reduction for semi-stiff Ginzburg-Landau vortices

Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$. For $\epsilon>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $-\Delta u=\frac{1}{\epsilon^2}(1-|u|^2)u$ in $\Omega$ such that on $\partial \Omega$ u satisfies $|u|=1$ and $u\wedge \partial_\nu u=0$. These boundary conditions are called semi-stiff and are intermediate between the Dirichlet and the homogeneous Neumann boundary conditions. In order to construct such solutions we use a variational reduction method very similar to the one used by del Pino-Kowalczyk-Musso to find solutions of the Ginzburg-Landau equations with Dirichlet and homogeneous Neumann boundary conditions. We obtain the exact same result as the authors of the aforementioned article obtained for the Neumann problem. This is because the renormalized energy for the Neumann problem and for the semi-stiff problem are the same. In particular if $\Omega$ is simply connected a solution with degree one on the boundary always exists and if $\Omega$ is not simply connected then for any $k\geq 1$ a solution with $k$ vortices of degree one exists.