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

Abstract

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 [23] and in [10]. In the discrete setting it has been previously studied in [16,17,19]. The model contains two parameters: τ and ε. The parameter τ represents the relative strength of the local term with respect to the nonlocal one, while the parameter ε describes the transition scale in the Modica–Mortola type term. If τ<0 one has that the only minimizers of the functional are constant functions with values in {0,1}. In any dimension d≥1 for small but positive τ and ε, 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.