Magnetic Bottles in Connection with Superconductivity

Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrödinger operator with magnetic field have appeared recently. Here we mention the works by Bernoff–Sternberg, Lu–Pan and Del Pino–Felmer–Sternberg. This partially recovers questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significantly the lower bounds and as a by-product we solve a conjecture proposed by Bernoff–Sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.