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We study the functional consideredin~\cite{2011PhRvB..84f4205G,2014CMaPh.tmp..127G,GiuSeirGS} and a continuousversion of it, analogous to the one considered in~\cite{GR}. The functionalsconsist of a perimeter term and a non-local term which are in competition. Forboth the continuous and discrete problem, we show that the global minimizersare exact periodic stripes. One striking feature of the functionals is that theminimizers are invariant under a smaller group of symmetries than thefunctional itself. In the continuous setting, to our knowledge this is thefirst example of a model with local/nonlocal terms in competition such that thefunctional is invariant under permutation of coordinates and the minimizersdisplay a pattern formation which is one dimensional. Such behaviour for asmaller range of exponents in the discrete setting was already shownin~\cite{GiuSeirGS}.

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