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

Abstract

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 \(\tau \) 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. 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 \(d\ge 1\) and for small but positive \(\tau \) and \(\varepsilon \) (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 \(L\rightarrow +\infty \).