One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension

In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in [9]. The parameters of the model are denoted by 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. Restricting to a periodic box of size L, with L multiple of the period of the minimal one-dimensional minimizers, in [9] the authors prove that in any dimension d≥1 and for small but positive τ and ε (eventually depending on L), the minimizers are non-constant one-dimensional periodic functions. In this paper we prove that periodicity and one-dimensionality of minimizers occurs also in the zero temperature analogue of the thermodynamic limit, namely as L→+∞.