Mathematics

Pattern formation is a fascinating phenomenon, which often arises at microscopic scale when two materials are subject to the influence of two competing forces: one short-range attractive and one long-range repulsive. The two materials dispose then themselves to form periodic structures with the form of bubbles, stripes, etc. according to their mutual density and the mutual strength of the competing forces. The most famous model of this kind is due to Ohta and Kawasaki, for diblock copolymers. In this talk we consider the double Yukawa model, used by physicists and chemists to model pattern formation in colloidal suspensions and protein solutions. We prove that, in certain regimes, global minimizers of the interaction functional governing the material are periodic stripes, thus showing for the first time breaking of symmetry in a model where both the short-range and the long-range competing potentials are nonlocal.  This work is in collaboration with Eris Runa.