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

TL;DR

This paper proves that in a large volume limit, the minimizers of a diffuse interface antiferromagnetic model are one-dimensional, extending previous results from finite periodic boxes to infinite domains.

Contribution

It establishes the one-dimensionality of minimizers in the zero temperature limit as the domain size tends to infinity, generalizing prior finite-volume results.

Findings

Minimizers are one-dimensional in the large volume limit.

Periodicity and one-dimensionality hold as domain size approaches infinity.

Results extend finite-volume minimizer properties to infinite domains.

Abstract

In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in Daneri-Kerschbaum-Runa arXiv:1907.06419. 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 Daneri-Kerschbaum-Runa arXiv:1907.06419 the authors prove that in any dimension $d\geq1$ 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 $L\to+\infty$.