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 Bonnaillie-Noël and Fournais (2007 Rev. Math. Phys. 19 607-37) and Correggi and Giacomelli (2021 Calc. Var. PDE 60 236), 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 Correggi and Giacomelli (2021 Calc. Var. PDE 60 236) concerning the dependence of the corner energy on the angle close to the threshold.