Lecturer
Sara Daneri sara.daneri@gssi.it
Timetable and workload
Lectures: 16 hours
Course content
Patterns, meant as periodically repeated regular geometric structures, emerge in several physical systems at small and large scales. For many of them, the driving mechanism is universally believed to be the competition between short-range attractive interactions (favouring pure phases) and long-range repulsive interactions (favouring alternation between different phases). The rigorous study of such a phenomenon aims at understanding how such a mechanism gives rise to such geometric structures.
One of the frameworks in which the problem can be set is the one of calculus of variations and geometric measure theory. In this setting, the goal is to show that global minimizers of energies with competing attractive/repulsive interactions are given by patterns.
In the course, after introducing the basic preliminary tools, we will dive into recent new techniques which were able to rigorously explain for the first time the emergence of striped patterns for a large class of energies. The key novelty consists in relating quantitatively the energy to an intrinsic nonlocal curvature. Further research directions stemming from such an approach will be also discussed.
No preliminary notions outside of the pillar course are required.