Pattern formation is ubiquitous in nature. At micro- and mesoscopic scale physical/chemical systems self-assemble into regular structures, characterized by some periodic alternation of different phases. Among them, the most typical patterns consist of bubbles placed on hexagonal lattices or stripes/lamellae (i.e. one-dimensional structures). Such structures are universally believed to arise from the competition between short range attractive forces (favouring pure phases) and long range repulsive forces (favouring alternation between different phases). Although such a phenomenon is observed in experiments and reproduced by numerical simulations, its mathematical understanding is in most cases a challenging and long-standing open problem. In the zero-temperature approximation, the models are of variational type and the physical states are represented by the minimizers of a free energy functional. The main difficulties from the mathematical point of view lie in the symmetry breaking phenomenon (namely the fact in dimensions larger than 1 that the expected minimizers have less symmetries than the interactions contributing to the energy) and in the nonlocality of the interactions.
The aim of this course is to present a new set of ideas/techniques which have been recently developed by the authors and collaborators and which allowed to give a rigorous proof of symmetry breaking and pattern formation in the form of one-dimensional structures in general dimensions for a large class of functionals (both in the sharp interface and in the diffuse interface setting). In order to explain such results we will also introduce some preliminary notions of Geometric Measure Theory, Calculus of Variations and Reflection Positivity (the latter being the main tool to show periodicity of minimizers in one space dimension).
The schedule of the class is available here.