Ground state energy scaling laws during the onset and destruction of the intermediate state in a type I superconductor

The intermediate state of a type I superconductor is a classical example of energy-driven pattern formation, first studied by Landau in 1937. Three of us recently derived five different rigorous upper bounds for the ground-state energy, corresponding to different microstructural patterns, but only one of them was complemented by a lower bound with the same scaling [Choksi, Kohn, and Otto, J. Nonlinear Sci. 14 (2004), 119-171]. This paper completes the picture by providing matching lower bounds for the remaining four regimes, thereby proving that exactly those five different regimes are traversed with an increasing magnetic field.