2021/06/13 by Daneri, Sara, Runa, Eris
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2106.08135
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~\citegr,drarma and previously by~\citegscmp in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in~\citedrarma 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∈(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 \citemuller1993singular, alberti2001new,ren2003energy,chen2005periodicity,giuliani2009modulated.