Lorentz space estimates for the Ginzburg-Landau energy

In this paper we prove novel lower bounds for the Ginzburg-Landau energy with or without magnetic field. These bounds rely on an improvement of the "vortex-balls construction" estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space L^{2, ∞} we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results.