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

TL;DR

This paper proves that in a class of local/nonlocal interaction functionals with volume constraints, the global minimizers are periodic stripes with volume density matching the constraint, extending previous results to arbitrary volume constraints.

Contribution

It establishes that minimizers are periodic stripes of prescribed volume density in the large volume limit, generalizing prior work to any volume constraint.

Findings

Minimizers are periodic stripes with volume density equal to the prescribed constraint.

Stripes have a simple periodic structure in the orthogonal slices.

Results hold in the large volume limit for general dimension.

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 $\alpha\in(0,1)$. We show that, in the large volume limit, minimizers are periodic stripes of volume density $\alpha$, namely stripes whose one-dimensional slices in the direction orthogonal to their boundary are simple periodic with volume density $\alpha$ 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}.