One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension

In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains a Modica-Mortola type local term and a nonlocal generalized antiferromagnetic term in competition. The competition between the two terms results in a frustrated system which is believed to lead to the emergence of a wide variety of patterns. The sharp interface limit of our model is considered in \cite{GR} and in \cite{DR}. In the discrete setting it has been previously studied in \cite{GLL, GLS, GS}. The model contains two parameters: $\tau$ and $\varepsilon$. The parameter $\tau$ represents the relative strength of the local term with respect to the nonlocal one, while the parameter $\varepsilon$ describes the transition scale in the Modica-Mortola type term. If $\tau < 0$ one has that the only minimizers of the functional are constant functions with values in $\{0,1\}$. In any dimension $d\geq1$ for small but positive $\tau$ and $\varepsilon$, it is conjectured that the minimizers are non-constant one-dimensional periodic functions. In this paper we are able to prove such a characterization of the minimizers, thus showing also the symmetry breaking in any dimension~$d >1$.