The ground state energy of the three dimensional Ginzburg-Landau   functional. Part I: Bulk regime
  We consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the magnetic field is comparable with the second critical field and that the Ginzburg-Landau parameter is large, we determine a sharp asymptotic estimate of the minimizing energy. In particular, this shows how bulk superconductivity decreases in average as the applied magnetic field approaches the second critical field from below. Other estimates are also obtained which allow us to obtain, in a subsequent paper, a fine characterization of the second critical field. The approach relies on a careful analysis of several limiting energies, which is of independent interest. 
S. Fournais, A. Kachmar
math.AP math-ph math.MP