One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension
Sara Daneri, Alicja Kerschbaum, Eris Runa · arXiv (Cornell University) · 2019
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~$d >1$.