Vorticity for the Ginzburg-Landau model of superconductors in a magnetic field (Q2752778)

The paper provides a thorough survey of recent approaches to the vorticity problem in the Ginzburg-Landau model of superconduction. It is based on the study of the variational functional NEWLINE\[NEWLINEJ(u,A)= \tfrac{1}{2} \int_\Omega |(\nabla- iA)u|^2+ |\nabla\times A- h_{\text{ex}}|^2+ \tfrac{\kappa^2}{2} (1-|u|^2)^2NEWLINE\]NEWLINE according to the behaviour of the parameter \(h_{\text{ex}}\). The superconductor is a vertical cylinder of bounded and simply connected section \(\Omega\subset \mathbb{R}^2\), the complex-valued function \(u\) describes the local state of the superconductor: the superconducting phase is characterized by \(|u(x)|\sim 1\) whereas \(|u(x)|\sim 0\) means the normal phase. (Moreover, \(A\) is the vector potential of the induced magnetic field.) In the mixed phase (the coexistence of both phases), there are normal regions of size \(1/\kappa\), called vortices. The problem of creation and localization of vortices is referred by using 24 rather advanced papers.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00038].