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

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.