Stable periodic configurations in nonlocal sharp interface models

This paper collects results obtained by the author together with Chen Chao-Nien, Choi Yung Sze, Nicola Fusco, Vesa Julin and Massimiliano Morini (in various groupings) in the last years; it is intended to be an introduction to the "geometric" perspective on some physical problems. Equilibrium models based on energy competition between volume and surface terms, in connection with nonlocal effects, got special attention in recent investigations, as their critical points exhibit various patterns with high degree of symmetry. There is interest in both finding the possible equilibrium shapes, and (which is the object of the present works) proving that they actually are (local) isolated minimizers. Particularly the latter has been thoroughly investigated for lamellar configurations in a model with long-range interaction governed by a screened Coulomb kernel. A section with open problems concludes the paper.