Exact Periodic Stripes for Minimizers of a Local/Nonlocal Interaction Functional in General Dimension

Abstract

Dedicated to Prof. Stephan Luckhaus on the occasion of his 65th birthday. AbstractWe study the functional considered in [25,26,28] and a continuous version of it, analogous to the one considered in [30]. The functionals consist of a perimeter term and a nonlocal term which are in competition. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one-dimensional. Such behaviour for a smaller range of exponents in the discrete setting was already shown in [28].1) where J is a positive constant, p ≥ d + 2, y ∼ x if x and y are neighbouring points in the lattice, and χ E (x) := 1 if x ∈ E, 0 otherwise. * daneri@math.fau.de † eris.runa@mis.mpg.de arXiv:1702.07334v4 [math.AP] 5 Jun 2018 c (respectively J > J c ), then the global minimizers are trivial, namely either empty or the whole domain. The critical constants J dsc c and J c are (as proven in [25] for the discrete setting and in [30] for the continuous case) J dsc c := y 1 >0, (y 2 ,...,y d )∈Z d−1 y 1 (y 2 1 + · · · + y 2 d ) p/2 and J c :=ˆR d |ζ 1 |K 1 (ζ) dζ. (1.4)When J = J dsc c − τ (resp. J = J c − τ ) with 0 < τ ≤τ for someτ > 0 small enough, it has been conjectured that the minimizers should be periodic unions of stripes of optimal period. In order to define what we mean by unions of stripes, let us fix a canonical basis {e i } d i=1 . A union of stripes in the continuous setting is a [0, L) d -periodic set which is, up to Lebesgue null sets, of the form V ⊥ i +Êe i for some i ∈ {1, . . . , d}, where V ⊥ i is the (d − 1)-dimensional subspace orthogonal to e i andÊ ⊂ R withÊ ∩ [0, L) = ∪ N k=1 (s i , t i ). A union of stripes is periodic if ∃ h > 0, ν ∈ R s.t. E ∩ [0, L) = ∪ N k=0 (2kh + ν, (2k + 1)h + ν). In the following, we will also sometimes call unions of stripes simply stripes.

On the optimality of stripes in a variational model with non-local interactions

“…The extension of this work to the classical perimeter will be the subject of further investigations. Building on the results obtained in this paper, it has been proven recently in [14] that for small enough τ , the minimizers of F τ,L are periodic stripes.…”

A Nonlocal Isoperimetric Problem with Dipolar Repulsion

“…In other regimes, Sternberg and Topaloglu identified stripes as the global minimizers in the case of the two-dimension torus [56], and Morini and Sternberg showed that such patterns also turn out to be minimizers in thin domains [45]. Another class of (anisotropic) nonlocal isoperimetric problems in dimensions n ≥ 2 in which minimizers are one-dimensional and periodic is given by Goldman and Runa [28], and by Daneri and Runa [18].…”

A periodic isoperimetric problem related to the unique games conjecture

“…Any relation of the present work to [9,28] is unclear, since the measures under consideration are different. Different isoperimetric problems exhibiting "crystallization" (or the optimality of sets consisting of parallel stripes) have been studied in, for example [7,12,14,15,33], though these studies have typically focused only on n = 2 or n = 3.…”

“…Different isoperimetric problems exhibiting “crystallization” (or the optimality of sets consisting of parallel stripes) have been studied in, for example , though these studies have typically focused only on n = 2 or n = 3.…”