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

Rodiac, Rémy

doi:10.1016/j.anihpc.2018.10.001

Rodiac, Rémy

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 783-809.

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Résumé

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 $- Δ h + h = μ$ and $div (T_{h}) = 0$ in Ω, where $T_{h}$ is a $2 \times 2$ matrix given by ${(T_{h})}_{i j} = 2 \partial_{i} h \partial_{j} h - (| \nabla h |^{2} + h^{2}) δ_{i j}$ 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 Ω is in the support of μ and is such that $| \nabla h | (z_{0}) \neq 0$ then μ is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a $C^{1}$ -curve near $z_{0}$ whereas $μ_{⌊ {| \nabla h | = 0}} = h_{| {| \nabla h | = 0}} L^{2}$ . We also prove that if Ω is smooth bounded and star-shaped and if $h = 0$ on ∂Ω then $h \equiv 0$ 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 $h_{e x}$ in that case.

MR Zbl

DOI : 10.1016/j.anihpc.2018.10.001

Mots-clés : Ginzburg–Landau theory, Vorticity, Inner-variational problem

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     author = {Rodiac, R\'emy},
     title = {Description of limiting vorticities for the magnetic {2D} {Ginzburg{\textendash}Landau} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {783--809},
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     volume = {36},
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Rodiac, Rémy. Description of limiting vorticities for the magnetic 2D Ginzburg–Landau equations. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 783-809. doi : 10.1016/j.anihpc.2018.10.001. http://www.numdam.org/articles/10.1016/j.anihpc.2018.10.001/

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