The distribution of 3D superconductivity near the second critical field

TL;DR

This paper analyzes the distribution of superconductivity in three-dimensional samples near the second critical magnetic field, providing a new formula for the order parameter's $L^2$-norm in this regime.

Contribution

It derives a formula for the $L^2$-norm distribution of the order parameter in 3D near the second critical field, extending previous 2D and $L^4$-norm results.

Findings

Established a formula for the $L^2$-norm distribution of the order parameter.

Valid in the regime of large Ginzburg-Landau parameter and magnetic field close to the second critical field.

Extends earlier 2D and $L^4$-norm results to 3D and $L^2$-norm context.

Abstract

We study the minimizers of the Ginzburg-Landau energy functional with a constant magnetic field in a three dimensional bounded domain. The functional depends on two positive parameters, the Ginzburg-Landau parameter and the intensity of the applied magnetic field, and acts on complex valued functions and vector fields. We establish a formula for the distribution of the $L^2$-norm of the minimizing complex valued function (order parameter). The formula is valid in the regime where the Ginzburg-Landau parameter is large and the applied magnetic field is close to the second critical field---the threshold value corresponding to the transition from the superconducting to the normal phase in the bulk of the sample. Earlier results are valid in $2D$ domains and for the $L^4$-norm in $3D$ domains.