Asymptotic analysis of a ferromagnetic Ising system with “diffuse” interfacial energy

We give an example of a one-dimensional scalar Ising-type energy with long-range interactions not satisfying standard decay conditions and which admits a continuum approximation finite for all functions u in $$BV((0,L),[-1,1])$$ and taking into account the total variation of u. The optimal discrete arrangements show a periodic pattern of interfaces. In this sense, the continuum energy is generated by “diffuse” microscopic interfacial energy. We also show that related minimum problems show boundary and size effects in dependence of L.