First Critical Field in the pinned three-dimensional Ginzburg--Landau Model: A matching upper bound

TL;DR

This paper determines the precise leading-order behavior of the first critical magnetic field in a 3D Ginzburg--Landau model with pinning, confirming the sharpness of previous bounds and linking vortex onset to an isoflux problem.

Contribution

It establishes a matching upper bound for the first critical field in a 3D pinned Ginzburg--Landau model, confirming the sharpness of earlier bounds and connecting vortex formation to an isoflux problem.

Findings

Established a matching upper bound for H_{c_1}

Confirmed the sharpness of the lower bound for H_{c_1}

Linked vortex onset to a weighted isoflux problem

Abstract

We continue our study of the first critical field $H_{c_1}$ for extreme type-II superconductors governed by the three-dimensional magnetic Ginzburg--Landau functional with a pinning term $a_\varepsilon$, as introduced in our previous work [arXiv:2507.10915]. Building upon the lower bound for $H_{c_1}$ and the characterization of the Meissner solution, we now establish a matching upper bound for $H_{c_1}$, thereby identifying its leading-order behavior. This result confirms the sharpness of the previously derived lower bound and further elucidates the connection between the onset of vorticity and a weighted variant of the \emph{isoflux problem}. Our argument is prompted by the upper bound construction we developed in [arXiv:2510.14910], based on the Biot--Savart law.