Periodic striped configurations in the large volume limit

Abstract

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 Goldman-Runa and Daneri-Runa and in Giuliani-Lieb-Lebowitz and Giuliani-Seiringer 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 $\tau$. For $\tau <0$ minimizers are trivial uniform states. It is conjectured that $\forall d\ge 2$ there exists $0<\bar{\tau }\ll 1$ such that for all $0<\tau \le \bar{\tau }$ and for all $L>0$ minimizers are striped/lamellar patterns. In Daneri-Runa arXiv:1702.07334 the authors prove the above for $L=2k{h}_{\tau }^{\ast }$, where $k\in \mathbb{N}$ and $h_{\tau }^{\ast }$ is the optimal period of stripes for a given $0<\tau \le \bar{\tau }$. The purpose of this paper is to show the validity of the conjecture for generic $L$.

arXiv admin note: substantial text overlap with arXiv:2106.08135