Lorentz Space Estimates and Jacobian Convergence for the Ginzburg-Landau Energy with Applied Magnetic Field

TL;DR

This paper advances the understanding of Ginzburg-Landau energy with magnetic fields by providing Lorentz space estimates and demonstrating Jacobian convergence, enhancing analysis of vortex structures in superconductors.

Contribution

It extends previous work by deriving new Lorentz space estimates for the energy and proving convergence of vorticity measures in dual Lorentz space frameworks.

Findings

Vorticity mass is comparable to the Lorentz space norm of the covariant gradient.

Established convergence of gauge-invariant Jacobians in Lorentz space duals.

Provided estimates in regimes of external magnetic field.

Abstract

In this paper we continue the study of Lorentz space estimates for the Ginzburg-Landau energy started in our previous paper, \cite{p1}. We focus on getting estimates for the Ginzburg-Landau energy with external magnetic field $h_{ex}$ in certain interesting regimes of $h_{ex}$. This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass of the configuration $(u,A)$ is comparable to the $L^{2,\infty}$ Lorentz space norm of $\nabla_A u$. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of a function space defined in terms of Lorentz spaces.