Pattern formation for a family of models with local/nonlocal interactions

We establish pattern formation for a family of discrete and continuous functionals consisting of a perimeter term and a nonlocal term. In particular, we show that for both the continuous and discrete functionals the global minimizers are exact periodic stripes. One striking feature is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the literature this phenomenon is often referred to as symmetry breaking. 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 [1].