# 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 $Ω$ be a bounded open set in $R_{2}$. The aim of this article is to describe the functions $h$ in $H_{1}(Ω)$ and the Radon measures $μ$ which satisfy $−Δh+h=μ$ and $div(T_{h})=0$ in $Ω$, where $T_{h}$ is a $2×2$ matrix given by $(T_{h})_{ij}=2∂_{i}h∂_{j}h−(∣∇h∣_{2})+h_{2})δ_{ij}$ 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 $∣∇h∣$ is continuous in Ω. We prove that if $z_{0}$ in $Ω$ is in the support of $μ$ and is such that $∣∇h∣(z_{0})=0$ then $μ$ is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a $C_{1}$-curve near $z_{0}$ whereas $μ_{⌊{∣∇h∣=0}}=h_{∣{∣∇h∣=0}}L_{2}$. We also prove that if $Ω$ is smooth bounded and star-shaped and if $h=0$ on $∂Ω$ then $h≡0$ in $Ω$. 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