Mathematics

We study the effect of long-range interactions in non-convex one-dimensional lattice systems in the simplified yet meaningful assumption that the relevant long-range interactions are between M-neighbours for some M≥2 and are convex. If short-range interactions are non-convex we then have a competition between short-range oscillations and long-range ordering.  In the case of a double-well nearest-neighbour potential, thanks to a recent result by Braides, Causin, Solci and Truskinovsky, we are able to show that such a competition generates M-periodic minimizers whose arrangements are driven by an interfacial energy. Given M, the shape of such minimizers is universal, and independent of the details of the energies, but the number and shapes of such minimizers increase as M diverges.