Interpolation inequalities in pattern formation

In this seminar I will present some interpolation inequalities that involves the BV-norm and some negative norms of a function u. These inequalities are the strong version of some already known estimates in weak form, which play a crucial role in the study of pattern formation. The main ingredient in the proof of these estimates is given by a geometric construction, that was first used by Choksi, Conti, Kohn and Otto in the context of branched patterns in superconductors, and which main idea goes back to De Giorgi. This is a joint work with Felix Otto.