The Distribution of Surface Superconductivity Along the Boundary: On a Conjecture of X. B. Pan

We consider the Ginzburg-Landau model of superconductivity in two dimensions in the large κ limit. For applied magnetic fields weaker than the onset field HC3 but greater than HC2 it is well known that the superconductivity order parameter decays exponentially fast away from the boundary. It has been conjectured by X.B. Pan that this surface superconductivity solution converges pointwise to a constant along the boundary. For applied fields that are in some sense between HC2 and HC3 , we prove that the solution indeed converges to a constant but in a much weaker sense. We also discuss the optimality of our result via heuristic arguments suggesting that uniform convergence to a constant might not be true.