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.
In dimension d>1, the main difficulties are due to the nonlocality of the interactions 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 S. Daneri.