Exact periodic stripes for minimizers of a local/nonlocal interaction functional in general dimension

We study the functional considered in GIULIANI et al. (Phys Rev B 84:064205, 2011, Commun Math Phys 331(1):333–350, 2014) and Giuliani and Seiringer (Commun Math Phys 347:983–1007, 2016) and a continuous version of it, analogous to the one considered in GOLDMAN and RUNA (On the optimality of stripes in a variational model with nonlocal interactions, 2016. arXiv:1611.07228). The functionals consist of a perimeter term and a nonlocal term which are in competition. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one-dimensional. Such behaviour for a smaller range of exponents in the discrete setting was already shown in GIULIANI and SEIRINGER (2016).