The Ginzburg-Landau functional with vanishing magnetic field
We study the infimum of the Ginzburg-Landau functional in a two dimensional simply connected domain and with an external magnetic field allowed to vanish along a smooth curve. We discuss, when the Ginzburg-Landau parameter is large, energy asymptotics in function of the strength of the external field. Compared with the known results when the external magnetic field does not vanish, we show in some regime a concentration of the energy near the zero set of the external magnetic field and we analyze the decay of the order parameter. These results, which complete a study of X.B. Pan and K.H. Kwek, have been obtained by K. Attar in his PHD thesis (2015) and by Helffer-Kachmar (2015-2016).