On the Ginzburg-Landau Energy of Corners

Abstract: 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. We review such a phenomenon in the framework of the Ginzburg-Landau theory and show how the presence of corners induces geometric terms described through effective models in sector-like regions.  We then focus on the connection between two such effective models for the offset of superconductivity and for surface superconductivity previously introduced by Bonnaillie-Noel, S. Fournais in 2007 and Correggi, Giacomelli in 2021, respectively. We prove that the transition between the two models is continuous with respect to the magnetic field strength and  deduce the existence of a minimizer at the threshold. Furthermore, as a consequence, we disprove a conjecture formulated by Correggi, Giacomelli in 2021 concerning the dependence of the corner energy on the angle close to the threshold.

Joint work with E.L. Giacomelli (LMU, Munich) and A. Kachmar (Chinese University of Hong Kong, Shenzhen).