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arXiv · 2106.08135

Exact periodic stripes for a local/nonlocal minimization problem with volume constraint

Sara Daneri·Eris Runa
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Abstract

We consider a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension, where a short range attractive term of perimeter type competes with a long range repulsive term characterized by a reflection positive power law kernel. Breaking of symmetry with respect to coordinate permutations and pattern formation for functionals in this class have been shown in~\cite{gr,dr_arma} and previously by~\cite{gs_cmp} in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in~\cite{dr_arma} to be given by periodic stripes of volume density $1/2$ in any cube having optimal period size, also in the large volume limit. In this paper we study the minimization problem with arbitrarily prescribed volume constraint $α\in(0,1)$. We show that, in the large volume limit, minimizers are periodic stripes of volume density $α$, namely stripes whose one-dimensional slices in the direction orthogonal to their boundary are simple periodic with volume density $α$ in each period. Results of this type in the one-dimensional setting, where no symmetry breaking occurs, have been previously obtained in \cite{muller1993singular, alberti2001new,ren2003energy,chen2005periodicity,giuliani2009modulated}.

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Sara Daneri, Eris Runa. 2021-06-13. Exact periodic stripes for a local/nonlocal minimization problem with volume constraint. https://arxiv.org/abs/2106.08135

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