The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model

We study solutions of the Ginzburg--Landau equations describing superconductors in a magnetic field, just below the "second critical field" Hc2. We thus bridge the gap between the situations described in [E. Sandier and S. Serfaty, Rev. Math. Phys., 12 (2000), pp. 1219--1257] and [X. B. Pan, Comm. Math. Phys., 228 (2002), pp. 327--370]. We prove estimates on the energy, among them one by an algebraic trick inspired by the Bogomoln'yi trick for self-duality. We thus show how, for energy-minimizers, superconductivity decreases in average in the bulk of the sample when the applied field increases to Hc2.