Let be a bounded open set in . The aim of this article is to describe the functions in and the Radon measures which satisfy and in , where is a matrix given by for . 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 is continuous in Ω. We prove that if in is in the support of and is such that then is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a -curve near whereas . We also prove that if is smooth bounded and star-shaped and if on then in . 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 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. 783–809DOI 10.1016/J.ANIHPC.2018.10.001