Description of limiting vorticities for the magnetic 2D Ginzburg–Landau equations

Abstract

Let $\Omega$ be a bounded open set in ${\mathbb{R}}^2$. The aim of this article is to describe the functions $h$ in $H^1(\mathrm{\Omega })$ and the Radon measures $\mu$ which satisfy $-\mathrm{\Delta }h+h=\mu$ and $\mathrm{div}(T_h)=0$ in $\Omega$, where $T_h$ is a $2\times 2$ matrix given by $(T_h{)}_{ij}=2{\partial }_{i}h{\partial }_{j}h-(|\nabla h{|}^2)+h^2){\delta }_{ij}$ for $i,j=1,2$. These equations arise as equilibrium conditions satisfied by limiting vorticities and limiting induced magnetic fields of solutions of the magnetic Ginzburg–Landau equations. This was shown by Sandier–Serfaty in [32,33]. Let us recall that they obtained that $|\nabla h|$ is continuous in Ω. We prove that if $z_0$ in $\Omega$ is in the support of $\mu$ and is such that $|\nabla h|(z_0)\ne 0$ then $\mu$ is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a ${\mathcal{C}}^1$-curve near $z_0$ whereas ${\mu }_{\lfloor \{|\nabla h|=0\}}=h_{|\{|\nabla h|=0\}}{\mathcal{L}}^2$. We also prove that if $\Omega$ is smooth bounded and star-shaped and if $h=0$ on $\partial \Omega$ then $h\equiv 0$ in $\Omega$. This rules out the possibility of having critical points of the Ginzburg–Landau energy with a number of vortices much larger than the applied magnetic field $h_{ex}$ in that case.