First Critical Field of Highly Anisotropic Three-Dimensional Superconductors via a Vortex Density Model

Abstract

We analyze a mean field model for 3D anisotropic superconductors with a layered structure, in the presence of a strong magnetic field. The mean field model arises as the Gammalimit of the Lawrence-Doniach energy in certain regimes. A reformulation of the problem based on convex duality allows us to characterize the first critical field H-c1 of the layered superconductor, up to leading order. In previous work, Alama, Bronsard, and Sandier [J. Eur. Math. Soc. (JEMS), 14 (2012), pp. 1825-1857] derived the asymptotic value of H-c1 for configurations satisfying periodic boundary conditions; in that setting, describing minimizers of the Lawrence-Doniach energy reduces to a 2D problem. In this work, we treat the physical case without any periodicity assumptions and are thus led to studying a delicate and essentially 3D nonlocal obstacle problem first derived by Baldo et al. [Comm. Math. Phys., 818 (2013), pp. 131-171] for the isotropic Ginzburg-Landau energy. We obtain a characterization of H-c1 using the special anisotropic structure of the mean field model.