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 $-neighbors 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-neighbor 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.
Citation: Andrea Braides, Edoardo Voglino, Matteo Zanardini. Microstructures and anti-phase boundaries in long-range lattice systems[J]. Networks and Heterogeneous Media, 2024, 19(3): 992-1012. doi: 10.3934/nhm.2024044
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 $-neighbors 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-neighbor 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.
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Andrea Braides, Edoardo Voglino, Matteo Zanardini. Microstructures and anti-phase boundaries in long-range lattice systems[J]. Networks and Heterogeneous Media, 2024, 19(3): 992-1012. doi: 10.3934/nhm.2024044
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