# On the Ginzburg -Landau energy with a magnetic field vanishing along a curve

## TL;DR

This paper analyzes the asymptotic behavior of the Ginzburg-Landau energy in a non-uniform magnetic field that vanishes along a curve, linking it to a one-dimensional functional and revealing surface superconductivity phenomena.

## Contribution

It introduces a new asymptotic analysis of the Ginzburg-Landau energy with a vanishing magnetic field along a curve, extending surface superconductivity results to non-uniform fields.

## Key findings

- Derived an asymptotic formula for the energy in the vanishing magnetic field regime.
- Linked the two-dimensional functional to a one-dimensional model.
- Identified the zero set of the magnetic field as an effective surface for superconductivity.

## Abstract

The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal characteristic function defined by means of a simplified two dimensional Ginzburg-Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog-Helffer and Fournais-Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor's surface.