Type-I Superconductors in the Limit as the London Penetration Depth Goes to 0

Abstract

This paper provides an explicit formula for the approximate solution of the static London equations. These equations describe the currents and magnetic fields in a Type-I superconductor. We represent the magnetic field as a 2-form and the current as a 1-form, and assume that the superconducting material is contained in a bounded, connected set, $\Omega,$ with smooth boundary. The London penetration depth gives an estimate for the thickness of the layer near $\partial\Omega$ where the current is largely carried. In an earlier paper, we introduced a system of Fredholm integral equations of second kind, on $\partial\Omega,$ for solving the physically relevant scattering problems in this context. In real Type-I superconductors the penetration depth is very small, typically about $100$nm, which often renders the integral equation approach computationally intractable. In this paper we provide an explicit formula for approximate solutions, with essentially optimal error estimates, as the penetration depth tends to zero. Our work makes extensive use of the Hodge decomposition of differential forms on manifolds with boundary, and thus evokes Kohn's work on the tangential Cauchy-Riemann equations.