Nucleation of Instability of the Meissner State of 3-Dimensional Superconductors

This paper concerns a nonlinear partial differential system in a 3-dimensional domain involving the operator curl², which is a simplified model used to examine nucleation of instability of the Meissner state of a superconductor as the applied magnetic field reaches the superheating field. We derive a priori C ^2+α estimates for a weak solution H, the curl of the magnetic potential, and determine the location of the maximal points of |curlH| which correspond to the nucleation of instability of the Meissner state. We show that, if the penetration length is small, the solution exhibits a boundary layer. If the applied magnetic field is homogeneous, |curlH| is maximal around the points on the boundary where the applied field is tangential to the surface.