Analysis and Mathematical Physics

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 larger group of symmetries than that of their minimizers). 

In this talk we consider a general class of isotropic functionals in dimension $d\geq 2$, typical in physical models, in which a surface term favouring pure phases competes with a nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured symmetry breaking and pattern formation for global minimizers, in the shape of domains with flat boundary (e.g. stripes or lamellae). 

Among others, our approach relies on detecting a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries. 

This is a joint work with E. Runa.