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

Let Ω\Omega be a bounded open set in R2\mathbb{R}^{2}. The aim of this article is to describe the functions hh in H1(Ω)H^{1}(\mathrm{\Omega }) and the Radon measures μ\mu which satisfy Δh+h=μ−\mathrm{\Delta }h + h = \mu and div(Th)=0\mathrm{div}(T_{h}) = 0 in Ω\Omega, where ThT_{h} is a 2×22 \times 2 matrix given by (Th)ij=2ihjh(h2)+h2)δij(T_{h})_{ij} = 2\partial _{i}h\partial _{j}h−(|\mathrm{∇}h|^{2}) + h^{2})\delta _{ij} for i,j=1,2i,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|\mathrm{∇}h| is continuous in Ω. We prove that if z0z_{0} in Ω\Omega is in the support of μ\mu and is such that h(z0)0|\mathrm{∇}h|(z_{0}) \neq 0 then μ\mu is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a C1\mathscr{C}^{1}-curve near z0z_{0} whereas μ{h=0}=h{h=0}L2\mu _{\lfloor \{|\mathrm{\nabla}h| = 0\}} = h_{|\{|\mathrm{\nabla}h| = 0\}}\mathscr{L}^{2}. We also prove that if Ω\Omega is smooth bounded and star-shaped and if h=0h = 0 on Ω\partial\Omega then h0h \equiv 0 in Ω\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 hexh_{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