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

### Rémy Rodiac

Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Louvain-la-Neuve, Belgium

## 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 $(T_{h})_{ij} = 2\partial _{i}h\partial _{j}h−(|\mathrm{∇}h|^{2}) + h^{2})\delta _{ij}$ 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 $|\mathrm{∇}h|$ is continuous in Ω. We prove that if $z_{0}$ in $\Omega$ is in the support of $\mu$ and is such that $|\mathrm{∇}h|(z_{0}) \neq 0$ then $\mu$ is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a $\mathscr{C}^{1}$-curve near $z_{0}$ whereas $\mu _{\lfloor \{|\mathrm{\nabla}h| = 0\}} = h_{|\{|\mathrm{\nabla}h| = 0\}}\mathscr{L}^{2}$. 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.

## Cite this article

Rémy Rodiac, Description of limiting vorticities for the magnetic 2D Ginzburg–Landau equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 3, pp. 783–809

DOI 10.1016/J.ANIHPC.2018.10.001