On the Ginzburg-Landau Energy of Corners

It is a well known fact that the geometry of a superconducting sample influences the distribution of the surface superconductivity for strong applied magnetic fields. For instance, the presence of corners induces geometric terms described through effective models in sector-like regions. We study the connection between two effective models for the offset of superconductivity and for surface superconductivity introduced in \cite{BNF} and \cite{CG2}, respectively. We prove that the transition between the two models is continuous with respect to the magnetic field strength, and, as a byproduct, we deduce the existence of a minimizer at the threshold for both effective problems. Furthermore, as a consequence, we disprove a conjecture stated in \cite{CG2} concerning the dependence of the corner energy on the angle close to the threshold.