Persistence of superconductivity in thin shells beyond $H_{c1}$

In Ginzburg-Landau theory, a strong magnetic field is responsible for the breakdown of superconductivity. This work is concerned with the identification of the region where superconductivity persists, in a thin shell superconductor modeled by a compact surface $\mathcal M\subset\mathbb R^3$, as the intensity $h$ of the external magnetic field is raised above $H_{c1}$. Using a mean field reduction approach devised by Sandier and Serfaty as the Ginzburg-Landau parameter $\kappa$ goes to infinity, we are led to studying a two-sided obstacle problem. We show that superconductivity survives in a neighborhood of size $(H_{c1}/h)^{1/3}$ of the zero locus of the normal component $H$ of the field. We also describe intermediate regimes, focusing first on a symmetric model problem. In the general case, we prove that a striking phenomenon we call freezing of the boundary takes place: one component of the superconductivity region is insensitive to small changes in the field.