Energy Minimization and Flux Domain Structure in the Intermediate State of a Type-I Superconductor

The intermediate state of a type-Isuperconductor involves a fine-scale mixture of normaland superconducting domains. We take the viewpoint, dueto Landau, that the realizable domain patternsare (local) minima of a nonconvex variationalproblem. We examine the scaling law of the minimumenergy and the qualitative properties of domain patterns achieving that law. Our analysis is restricted to the simplest possible case: a superconducting platein a transverse magnetic field. Our methods include explicit geometric constructions leadingto upper bounds and ansatz-free inequalities leading to lower bounds. The problem is unexpectedly rich when the applied field is near-zero or near-critical.In these regimes there are two small parameters, and the ground state patterns depend on the relation betweenthem.