Periodic striped configurations in the large volume limit

We show striped pattern formation in the large volume limit for a class of generalized anti- ferromagnetic local/nonlocal interaction functionals in general dimension previously considered in [20, 10, 12] and in [14, 17] in the discrete setting. In such a model the relative strength be- tween 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 on periodic boxes of size L are striped/lamellar patterns. In [10] we give a partial proof of the above conjecture 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 in its full generality, namely showing pattern formation in the large volume limit on boxes of arbitrary size L.