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 $\kappa$ limit. For applied magnetic fields weaker than the onset field $H_{C_3}$ but greater than $H_{C_2}$ 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 $H_{C_2}$ and $H_{C_3}$, we prove that the solution indeed converges to a constant but in a much weaker sense.

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