Surface superconductivity in 3 dimensions

We study the Ginzburg-Landau system for a superconductor occupying a 3-dimensional bounded domain, and improve the estimate of the upper critical field H-C3 obtained by K. Lu and X. Pan in J. Diff. Eqns., 168 (2000), 386-452. We also analyze the behavior of the order parameters. We show that, under an applied magnetic field lying below and not far from H-C3, order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field, and exponentially decay both in the normal and tangential directions away from the sheath in the L-2 sense. As the applied field decreases further but keeps in between and away from H-C2 and H-C3, the superconducting sheath expands but does not cover the entire surface, and superconductivity at the surface portion orthogonal to the applied field is always very weak. This phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by X. Pan in Comm. Math. Phys., 228 (2002), 327-370, where under an axial applied field lying in-between H-C2 and H-C3 the entire surface is in the superconducting state.