Small surface energy, coarse-graining, and selection of microstructure

We discuss some static and dynamic problems involving a competition between ''bulk'' and ''surface'' energy, In each case the bulk energy prefers microstructure, and the surface energy is a singular perturbation, It is natural to treat the surface energy density as a small parameter. In our static examples - involving branching of twins and magnetic domains - this leads to scaling laws for the energy and the length scale of microstructure, In our dynamic example - involving the fracture of a ferrofluid thin film by an applied magnetic field - it provides a novel approach to coarse-graining.