Periodic striped configurations in the large volume limit

We show striped pattern formation in the large volume limit for a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension previously considered in [GR19; DR19; DR21a] and in [GLL06; GS16] in the discrete setting. In such a model the relative strength between the short range attractive term favouring pure phases and the long range repulsive term favouring oscillations is modulated by a parameter τ . For τ < 0 minimizers are trivial uniform states. It is conjectured that ∀ d ≥ 2 there exists 0 < τ̄ ≪ 1 such that for all 0 < τ ≤ τ̄ and for all L > 0 minimizers are striped/lamellar patterns. In [DR19] the authors prove the above for L = 2kh∗ τ , where k ∈ N and h∗ τ is the optimal period of stripes for a given 0 < τ ≤ τ̄ . The purpose of this paper is to show the validity of the conjecture for generic L.