Lorentz Space Estimates for the Ginzburg-Landau Energy

TL;DR

This paper establishes new lower bounds for the Ginzburg-Landau energy using Lorentz space norms, enhancing vortex analysis and convergence results in superconductivity models.

Contribution

It introduces an improved vortex balls construction that incorporates Lorentz space estimates, providing sharper energy bounds and vortex count estimates.

Findings

Derived novel lower bounds for Ginzburg-Landau energy.

Utilized Lorentz space $L^{2, abla}$ to estimate vortex quantities.

Enhanced convergence results for vortex configurations.

Abstract

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 $L^{2,\infty}$ 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.