On the third critical field in Ginzburg-Landau theory

TL;DR

This paper provides precise estimates for the third critical magnetic field in type II superconductors using spectral asymptotics, showing that superconductivity near this field is localized near boundary points with maximal curvature.

Contribution

The authors derive exact estimates for the critical field H_{C_3} and demonstrate the equivalence of local and global definitions, advancing understanding of boundary effects in superconductivity.

Findings

Superconductivity near H_{C_3} is confined to boundary regions with maximal curvature.

Precise size estimates of the superconducting zone near the boundary.

Decay estimates for superconductivity in normal and parallel directions.

Abstract

Using recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, $H_{C_3}$, describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near $H_{C_3}$ only a small part, near the boundary points where the curvature is maximal, of the sample carries superconductivity. We give precise estimates on the size of this zone and decay estimates in both the normal (to the boundary) and parallel variables.