Abstract

We examine the asymptotic behavior of the eigenvalue mu(h) and corresponding eigenfunction associated with the variational problem mu(h) = inf(psi epsilon H1(Omega;C))integral(Omega)/(i del + hA)psi/(2) dxdy/integral(Omega)/(psi)/(2) dxdy in the regime h much greater than i. Here A is any vector field arith curl equal to i. The problem arises within the Ginzburg-Landau model for superconductivity with the function mu(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 of 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 mu while also proving that the first eigenfunction decays to zero somewhere along the sample boundary partial derivative Omega when Omega is not a disc. For interior decay, we demonstrate that the rate is exponential.