Superconductivity in the presence of magnetic steps

This thesis investigates the distribution of superconductivity in a Type-II planar, bounded, and smooth superconductor submitted to a piecewise-constant magnetic field with a jump discontinuity along smooth curves---the magnetic edge. This discontinuous case has not been treated before in the mathematics literature, where the considered applied magnetic field is usually assumed to be smooth.

We examine the behavior of the sample in different regimes of the intensity of the applied magnetic field. When the magnetic field is relatively weak, we prove that superconductivity exists all over the sample. Increasing the magnetic field's intensity to higher levels, superconductivity is shown to vanish in the interior of the sample away from the magnetic edge and can nucleate near this edge as well as near the boundary. Such nucleation may not be uniform. Under stronger magnetic fields, superconductivity is confined to the vicinity of the intersection of the magnetic edge with the boundary, when such an intersection exists, before being completely destroyed at a certain stage of the field's intensity. The results show a behaviour of the sample that, according to the intensity-regime, may differ from or resemble to that in the case of smooth/corner domains submitted to uniform magnetic fields. This highlights the particularity of our discontinuous case.

The study is modeled by the Ginzburg--Landau (GL) theory, and the obtained results are valid for the minimizers of the two-dimensional GL functional with a large GL parameter and with a field's intensity comparable to this parameter.