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Exact periodic stripes for a local/nonlocal minimization problem with volume constraint
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Exact periodic stripes for a local/nonlocal minimization problem with volume constraint
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Exact periodic stripes for a local/nonlocal minimization problem with volume constraint
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Paper
Exact periodic stripes for a local/nonlocal minimization problem with volume constraint
2021
Overview
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~gr,dr_arma and previously by~gs_cmp in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in~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 \\(ın(0,1)\\). We show that, in the large volume limit, minimizers are periodic stripes of volume density \\(\\), namely stripes whose one-dimensional slices in the direction orthogonal to their boundary are simple periodic with volume density \\(\\) in each period. Results of this type in the one-dimensional setting, where no symmetry breaking occurs, have been previously obtained in muller1993singular, alberti2001new,ren2003energy,chen2005periodicity,giuliani2009modulated.
Publisher
Cornell University Library, arXiv.org
Subject
