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. (C) 2007 Wiley Periodicals, Inc.