On the symmetry breaking and structure of the minimizers for a family of local/nonlocal interaction functionals

In this paper we review and present some results on the one-dimensionality and periodicity of minimizers for two families of local/nonlocal interaction functionals arising in generalized antiferromagnetic models or in models for colloidal suspensions. The local term is given in both cases by the 1-perimeter, penalizing interfaces. The nonlocal term instead favours oscillations: the interactions between the set and its complementary are modulated through a kernel of power law type for generalized antiferromagnetic models and through the Yukawa (or screened Coulomb) kernel in models for colloidal suspensions. Though the functionals are symmetric w.r.t. permutation of coordinates, we show that in suitable regimes the competition between the two terms causes symmetry breaking and global minimizers are periodic stripes, in any space dimension.