  We prove some interpolation inequalities which arise in the analysis of
pattern formation in physics. They are the strong version of some already known
estimates in weak form that are used to give a lower bound of the energy in
many contexts (coarsening and branching in micromagnetics and superconductors).
The main ingredient in the proof of our inequalities is a geometric
construction which was first used by Choksi, Conti, Kohn, and one of the
authors in the study of branching in superconductors.
