2019/07/15 by Sara Daneri, Daneri, Sara, Alicja Kerschbaum +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1907.06419
In this paper we study a diffuse interface generalized antiferromagnetic\nmodel. The functional describing the model contains a Modica-Mortola type local\nterm and a nonlocal generalized antiferromagnetic term in competition. The\ncompetition between the two terms results in a frustrated system which is\nbelieved to lead to the emergence of a wide variety of patterns. The sharp\ninterface limit of our model is considered in citeGR and in citeDR. In\nthe discrete setting it has been previously studied in citeGLL, GLS, GS. The\nmodel contains two parameters: \τ and \ε. The parameter \τ\nrepresents the relative strength of the local term with respect to the nonlocal\none, while the parameter \ε describes the transition scale in the\nModica-Mortola type term. If \τ < 0 one has that the only minimizers of the\nfunctional are constant functions with values in 0,1 . In any dimension\nd\≥1 for small but positive \τ and \ε, it is conjectured\nthat the minimizers are non-constant one-dimensional periodic functions. In\nthis paper we are able to prove such a characterization of the minimizers, thus\nshowing also the symmetry breaking in any dimension~d >1.\n