Boundary concentration for eigenvalue problems related to the onset of superconductivity

We examine the asymptotic behavior of the eigenvalue mu(h) and correspondin g eigenfunction associated with the variational problem mu(h) = inf(psi epsilon H1(Omega;C))integral(Omega)/(i del + hA)psi/(2) dxd y/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) yi elding the relationship between the critical temperature vs. applied magnet ic 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 Ome ga is not a disc. For interior decay, we demonstrate that the rate is expon ential.