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

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 [GR19; DR19b] and previously by [GS16] in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in [DR19b] 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 [Mül93; AM01; RW03; CO05; GLL09].

Periodic striped states in Ising models with dipolar interactions

“…The proofs in [2,14,16] on the striped periodic nature of the global minimizers of 𝑑 ≥ 2 models with polynomial interactions 1/|𝒙 − 𝒚| 𝑝 , 𝑝 ≥ 𝑑 + 2 − 𝜖, additionally require to combine RP with localization estimates into boxes of appropriate size. Further extensions of these ideas have been successfully applied to the proof of periodicity of the global minimizers of: 2D models of in-plane spins with dipolar interactions [7]; 2D models of martensitic phase transitions [13]; effective functionals with diffuse interfaces in the presence of dipolar-like interactions in 𝑑 = 1 [1,8] and 𝑑 ≥ 2 [3]; models with competing interactions in a magnetic field or with mass constraint in 𝑑 = 1 [9] and in 𝑑 ≥ 2 [4].…”

Periodic striped configurations in the large volume limit

“…In [DR21b] the results in [DKR19] are proved to hold in the large volume limit on boxes whose sizes are even multiples of an optimal period. In [DR21a] a characterization of minimizers for the functional (1.1) under the imposition of an arbitrary volume constraint α ∈ (0, 1) was given. In the regime 0 < τ ≪ 1 and when L ≫ 1 is an even multiple of the optimal period h * τ,α of simple periodic stripes with density α, minimizers among all [0, L) d -periodic sets of density α happen to be given by sets of the form…”

Deterministic particle approximation of aggregation-diffusion equations on unbounded domains

“…The Γ-limit of such a functional as β → +∞ (namely with local attractive term) has been characterized in [21]. In suitable regimes, minimizers of the limit functionals have been proved to be given by periodic unions of stripes (i.e., intervals in one dimension, see [21]) with techniques developed in [35,20,21,26,37,22,23,24,25]. For characterization of minimizers with power law attractive-repulsive potentials see [15].…”