Boundary Behavior of the Ginzburg–Landau Order Parameter in the Surface Superconductivity Regime

We study the 2D Ginzburg--Landau theory for a type-II superconductorin an applied magnetic field varying between the second and thirdcritical value. In this regime the order parameter minimizing the GLenergy is concentrated along the boundary of the sample and is wellapproximated to leading order (in L2 norm) by a simplified 1Dprofile in the direction perpendicular to the boundary. Motivated bya conjecture of Xing-Bin Pan, we address the question of whetherthis approximation can hold uniformly in the boundary region. Weprove that this is indeed the case as a corollary of a refined,second order energy expansion including contributions due to thecurvature of the sample. Local variations of the GL order parameterare controlled by the second order term of this energy expansion,which allows us to prove the desired uniformity of the surfacesuperconductivity layer.