Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity

Abstract:

We examine the asymptotic behavior of the eigenvalue μ(h) and corresponding eigenfunction associated with the variational problem

in the regime h>>1. Here A is any vector field with curl equal to 1. The problem arises within the Ginzburg–Landau model for superconductivity with the function μ(h) yielding the relationship between the critical temperature vs. applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross-section $\Omega\subset\R^{2}$. We first carry out a rigorous analysis of the associated problem on a half-plane and then rigorously justify some of the formal arguments of [BS], obtaining an expansion for μ while also proving that the first eigenfunction decays to zero somewhere along the sample boundary when Ω is not a disc. For interior decay, we demonstrate that the rate is exponential.