# First critical field of highly anisotropic three-dimensional superconductors via a vortex density model

## TL;DR

This paper develops a mean field model for highly anisotropic 3D layered superconductors under strong magnetic fields, characterizing the first critical field $H_{c_1}$ without periodic boundary assumptions, using convex duality and non-local obstacle problems.

## Contribution

It extends previous work by analyzing the non-periodic, three-dimensional case of layered superconductors, deriving a characterization of $H_{c_1}$ using convex duality and anisotropic structures.

## Key findings

- Characterization of $H_{c_1}$ in non-periodic 3D layered superconductors.
- Reduction of the problem to a 3D non-local obstacle problem.
- Application of convex duality to analyze the mean field model.

## Abstract

We analyze a mean field model for $3$d anisotropic superconductors with a layered structure, in the presence of a strong magnetic field. The mean field model arises as the $Gamma$-limit of the Lawrence-Doniach energy in certain regimes. A reformulation of the problem based on convex duality allows us to characterize the first critical field $H_{c_1}$ of the layered superconductor, up to leading order. In previous work, Alama-Bronsard-Sandier \cite{ABS} have derived the asymptotic value of $H_{c_1}$ for configurations satisfying periodic boundary conditions; in that setting describing minimizers of the Lawrence-Doniach energy reduces to a $2$d problem. In this work, we treat the physical case without any periodicity assumptions, and are thus led to studying a delicate and essentially $3$d non-local obstacle problem first derived by Baldo-Jerrard-Orlandi-Soner \cite{BJOS2} for the isotropic Ginzburg-Landau energy. We obtain a characterization of $H_{c_1}$ using the special anisotropic structure of the mean field model.