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.

We study pattern formation for a variational model displaying competition between a local term penalizing interfaces and a non-local term favoring oscillations. By means of a Γ−convergence analysis, we show that as the parameter J converges to a critical value J c , the minimizers converge to periodic one-dimensional stripes. A similar analysis has been previously performed by other authors for related discrete systems. In that context, a central point is that each "angle" comes with a strictly positive contribution to the energy. Since this is not anymore the case in the continuous setting, we need to overcome this difficulty by slicing arguments and a rigidity

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In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains Modica-Mortola type local term and a second nonlocal generalized antiferromagnetic term in competition. The competition between the two terms results in a frustrated system which is believed to lead to the emergence of wide variety of pattern formation. The sharp interface limit of our model is considered in [7,8,9] in the discrete and in [13] and [4] in the continuous setting. The model contains two parameter τ, ε. The parameter τ represents the relative strength of the local term with respect to the nonlocal one, while the parameter ε describes the transition scale in the Modica-Mortola type term. If τ < 0 one has that the only minimizers of the functional are constant functions. In any dimension d ≥ 1 for small but positive τ and small ε, it is conjectured that the minimizers are one-dimensional periodic functions. In this paper we are able to prove such a characterization of the minimizers, thus showing also the symmetry breaking in any dimension d > 1.