Magnetic steps on the threshold of the normal state

Superconductivity in the presence of a step magnetic field has been recently the focus of many works. This contribution examines the behavior of a two-dimensional superconducting domain, when superconductivity is lost in the whole domain except near the intersection points of the discontinuity edge and the boundary. The problem involves its own effective energy. We provide local estimates of the minimizers in neighbourhoods of the intersection points. Consequently, we introduce new critical fields marking the loss of superconductivity in the vicinity of these points. The study is modelled by the Ginzburg--Landau theory, and large Ginzburg--Landau parameters are considered.