Upper critical field and location of surface nucleation of superconductivity (Q1868109)

The two-dimensional Landau-Ginzburg functional as a function of the order parameter \(\psi\) and the magnetic vector potential \(A\) as well as the corresponding Euler-Lagrange equations are considered on a domain \(\Omega\). In this article, the authors consider only Neumann boundary conditions. The external magnetic field \({\mathcal H} = (0,0,\sigma)\) is assumed to be a constant orthogonal to the two-dimensional plane considered. The behavior of the minimizers as the Ginzburg-Landau parameter \(\kappa\) becomes large is investigated. The paper has three major results that are motivated by results by \textit{K. Lu} and \textit{X. B. Pan} [J. Math. Phys. 40, No. 6, 2647--2670 (1999; Zbl 0943.35058)]. The first concerns the critical external magnetic field: It is well known that there exists a unique divergence free vector field \(F\) such that \(\nabla \times F= 1\) on \(\Omega\) and \(F\) is orthogonal to the boundary on \(\partial \Omega\). Let \({\mathcal H}_C(\kappa) = \inf\{\sigma>0\mid (0,\sigma F)\) is the only minimizer\} be the critical external magnetic field. The first result gives an asymptotic formula for \({\mathcal H}_C(\kappa)\) for large \(\kappa\). The second and third result give results on the concentration of \(\psi\). In particular, if the domain is non-degenerate, the minimizer \(\psi\) concentrates near the boundary and there in particular at the points of highest curvature.