Periodic minimizers of the anisotropic Ginzburg–Landau model

We consider the anisotropic Ginzburg–Landau model in a three-dimensional periodic setting, in the London limit as the Ginzburg–Landau parameter $${\kappa=1/{\epsilon}\to\infty}$$ . By means of matching upper and lower bounds on the energy of minimizers, we derive an expression for a limiting energy in the spirit of Γ-convergence. We show that, to highest order as $${\epsilon\to0}$$ , the normalized induced magnetic field approaches a constant vector. We obtain a formula for the lower critical field Hc1 as a function of the orientation of the external field $${h^\epsilon_{ex}}$$ with respect to the principal axes of the anisotropy, and determine the direction of the limiting induced field as a minimizer of a convex geometrical problem.