On the energy of superconductors in large and small domains

We study the Ginzburg-Landau energy functional for superconductors in an applied magnetic field. We focus on asymptotically large or small domains and establish the asymptotic behavior of the energy as a function of the Ginzburg-Landau parameter, applied magnetic field, and domain size. For a large class of domain sizes, we calculate the critical field strength where vortex nucleation becomes energetically favorable, and describe the vorticity of minimizers. For supercritical magnetic field strengths, we recover the energy of a classical Abrikosov vortex lattice. Our findings generalize several known results of Sandier and Serfaty for domains of fixed size.