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

Sara Daneri; Alicja Kerschbaum; Eris Runa

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arXiv · 1907.06419

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

Sara Daneri·Alicja Kerschbaum·Eris Runa

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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 \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: $τ$ and $\varepsilon$. The parameter $τ$ 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 $τ< 0$ 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 $τ$ 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~$d >1$.

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Sara Daneri, Alicja Kerschbaum, Eris Runa. 2021-04-19. One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension. https://arxiv.org/abs/1907.06419

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