Abstract. At the base of spontaneous pattern formation is universally believed to be the competition between short range attractive and long range repulsive forces. Though such a phenomenon is observed in experiments and simulations, a rigorous understanding of the mechanisms at its base is still in most physical problems a challenging open problem. The main difficulties are due to the nonlocality of the interactions and, in more than one space dimensions, the symmetry breaking phenomenon (namely the fact that the interactions have a group of symmetries larger than the expected minimizers).

In this talk we will present a new rigorous approach which allows us to prove symmetry breaking for a class of functionals with isotropic interactions. In particular, we show that in a regime in which the competing interactions are of the same order, minimizers are one-dimensional (i.e. stripes/lamellae). This class of functionals includes physical energies related to pattern formation in thin magnetic films and colloidal systems.  This work has been obtained in collaboration with E. Runa.