Calculus of Variations and Geometric Measure Theory

A. Braides - E. Voglino - M. Zanardini

Microstructures and anti-phase boundaries in long-range lattice systems

created by braidesa on 10 May 2024
modified on 15 May 2024


Submitted Paper

Inserted: 10 may 2024
Last Updated: 15 may 2024

Year: 2024

ArXiv: 2405.06542 PDF


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 $M\ge 2$ 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 increases as $M$ diverges.