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

## TL;DR

This paper proves that for a class of local/non-local interaction functionals in any dimension, the global minimizers are exactly periodic stripes, revealing a one-dimensional pattern formation despite the functional's higher symmetry.

## Contribution

It demonstrates, for the first time, that minimizers of certain symmetric local/non-local functionals are one-dimensional stripes, even when the functional itself has higher permutation symmetry.

## Key findings

- Minimizers are exact periodic stripes in both continuous and discrete models.
- Minimizers exhibit one-dimensional pattern formation despite higher symmetry.
- First example of such behavior with local/nonlocal competition in a general dimension.

## Abstract

We study the functional considered in~\cite{2011PhRvB..84f4205G,2014CMaPh.tmp..127G,GiuSeirGS} and a continuous version of it, analogous to the one considered in~\cite{GR}. The functionals consist of a perimeter term and a non-local 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~\cite{GiuSeirGS}.