3D vortex approximation construction and $\varepsilon$-level estimates for the Ginzburg-Landau functional

TL;DR

This paper introduces a new 3D vortex approximation method for the Ginzburg-Landau functional, providing energy bounds and analysis tools at the $\varepsilon$-level, advancing understanding of vortex behavior in superconductors.

Contribution

It develops a quantitative 3D vortex approximation construction coupled with energy bounds at the $\varepsilon$-level, a novel approach for analyzing Ginzburg-Landau minimizers in 3D.

Findings

Provides a vortex approximation construction valid at the $\varepsilon$-level.

Establishes a lower bound for the energy of vortex configurations.

Enables new analysis of minimizers near the first critical field.

Abstract

We provide a quantitative three-dimensional vortex approximation construction for the Ginzburg-Landau functional. This construction gives an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the 2D ones, and valid for the first time at the $\varepsilon$-level. These tools allow for a new approach to analyze the behavior of global minimizers for the Ginzburg-Landau functional below and near the first critical field in 3D, followed in two forthcoming papers. In addition, they allow to obtain an $\varepsilon$-quantitative product estimate for the study of Ginzburg-Landau dynamics.