Optimal bounds for periodic mixtures of nearest-neighbour ferromagnetic interactions

Abstract

We provide a general framework for the optimal design of surface energies on networks. We prove sharp bounds for the homogenization of discrete systems describing mixtures of ferromagnetic interactions by constructing optimal microgeometries, and we prove a localization principle which allows to reduce to the periodic setting in the general nonperiodic case.

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

“…With this identification the energy becomes a simple nearest-neighbour interaction energy in dimension two, of which we can compute the Γ-limit in the surface scaling. Reinterpreting the limit in dimension one gives the form (11) after some technical arguments. The interest in this example is that the limit is characterized by the non-trivial topology of the graph of the connections i, j with c ε ij = 1, which is the same as that of nearest-neighbours in dimension two.…”

“…In all those cases the limit problem is of the form (4). Applications of this result comprise the description of quasicrystalline structures [8,14] and optimal design problems for networks [11,12]. Moreover, the homogenization result for periodic systems has been recently extended to some types of antiferromagnetic interactions when the limit is instead parameterized on partitions into sets of finite perimeter [9] and can be written as a sum of energies of the form (4).…”

Crystallinity of the homogenized energy density of periodic lattice systems

“…Indeed, example 2.8 shows that for infinite range interactions this is in general not true. In [15,16] it is shown that, as the periodicity T of the interactions tends to +∞, it is possible to approximate any norm as surface energy density satisfying suitable growth conditions. We refer to [2] for a random setting where it is shown that an isotropic energy density (and thus non-crystalline) can be obtained in the limit.…”

Design of lattice surface energies

“…In that case we are mixing two types of connections in a cubic lattice. A simplified description of the two-dimensional setting for nearest neighbour-interactions can be found in [17]. The first step in the homogenization method is to consider all possible ϕ in the periodicbond setting; that is, when i → c i,ξ are periodic, in which case ϕ is independent of x.…”