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, Otto, J. Nonlinear Sci. 14, 119 171 (2004)]. 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 increasing magnetic field.