One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension - Daneri, Sara; Kerschbaum, Alicja; Runa, Eris (Journal of Functional Analysis, 2022)
In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains a Modica–Mortola type local term and a 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 a wide variety of patterns. The sharp interface limit of our model is considered in [23] and in [10]. In the discrete setting it has been previously studied in [16,17,19]. The model contains two parameters: τ and 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 with values in 0,1. In any dimension d≥1 for small but positive τ and ε, it is conjectured that the minimizers are non-constant 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.
CRIS-Team Scopus Importer:2022-10-14
Keywords: Diffuse interface problem; Nonlocal minimization problem; Pattern formation

Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than d+2 - Kerschbaum, Alicja (Nonlinear Analysis - Theory Methods & Applications, 2022)
We consider a class of continuous generalized antiferromagnetic models with local/nonlocal interactions. The functional consists of an anisotropic perimeter term and a repulsive nonlocal term with a power law kernel. In certain regimes the two terms enter in competition and symmetry breaking with formation of periodic striped patterns is expected to occur. In this paper we extend some recent results of Daneri and Runa to power law kernels within a range of exponents smaller than d+2 and strictly larger than d+1, being d the dimension of the underlying space. In particular, we prove that in a suitable regime minimizers are periodic unions of stripes with a given optimal period. Notice that the exponent d+1 corresponds to an anisotropic version of the model for pattern formation in thin magnetic films.
CRIS-Team Scopus Importer:2021-12-17
Keywords: Antiferromagnetic interaction functional; Nonlocal minimization; Pattern formation