IUMJ

Title: Large vorticity stable solutions to the Ginzburg-Landau Equations

Authors: Andres Contreras and Sylvia Serfaty

Issue: Volume 61 (2012), Issue 5, 1737-1763

Abstract:

We construct local minimizers to the Ginzburg-Landau functional of superconductivity whose number of vortices $N$ is prescribed and blows up as the parameter $\epsilon$, inverse of the Ginzburg-Landau parameter $\kappa$, tends to $0$. We treat the case of $N$ as large as $|\log\epsilon|$, and a wide range of intensity of external magnetic field. The vortices of our solutions arrange themselves with uniform density over a subregion of the domain bounded by a "free boundary" determined via an obstacle problem, and asymptotically tend to minimize the "Coulombian renormalized energy" $W$ introduced in [\'E. Sandier and S. Serfaty, \textit{From the Ginzburg-Landau model to vortex lattice problems}, Comm. Math. Phys. \textbf{313} (2012), no. 3, 635--743]. The method, inspired by [S. Serfaty, \textit{Stable configurations in superconductivity: uniqueness, multiplicity, and vortex-nucleation}, Arch. Ration. Mech. Anal. \textbf{149} (1999), no. 4, 329--365], consists in minimizing the energy over a suitable subset of the functional space, and in showing that the minimum is achieved in the interior of the subset. It also relies heavily on refined asymptotic estimates for the Ginzburg-Landau energy obtained in [\'E. Sandier and S. Serfaty, \textit{From the Ginzburg-Landau model to vortex lattice problems}, Comm. Math. Phys. \textbf{313} (2012), no. 3, 635--743].