Exact periodic stripes for a minimizers of a local/non-local interaction functional in general dimension

Abstract

We study the functional considered in~\cite{2011PhRvB..84f4205G,2014CMaPh.tmp..127G,GiuSeirGS} and a continuous version of it, analogous to the one considered in~\cite{GR}. The functionals consist of a perimeter term and a non-local term which are in competition. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one dimensional. Such behaviour for a smaller range of exponents in the discrete setting was already shown in~\cite{GiuSeirGS}.

arXiv commentadded figures; relaxed hypothesis from p> 2d to p>= d+2; added independence of L from τ; rewrote intro

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Cited by in corpus (7)

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• On the Ternary Ohta-Kawasaki Free Energy and Its One-dimensional Global Minimizers
• Asymptotically exact formulas for the stripe domains period in ultrathin ferromagnetic films with out-of-plane anisotropy
• Variational analysis of the $J_1$-$J_2$-$J_3$ model: a non-linear lattice version of the Aviles-Giga functional
• Nucleation and growth of lattice crystals
• Homogenization effects on non-local functionals