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 L2,∞ 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.