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Limits of Discrete Systems with Long-Range Interactions

In this paper we consider the problem of the description of the continuum limit of discrete systems with long-range interactions defined on a cubic lattice as the lattice parameters tend to 0. Consider a domain Ω in R , which will parameterize the limit continuum region, and the portion Zn of the lattice λnZ of step size λn contained in Ω. Let u be a function defined in Zn. If u : Zn → R then we may interpret u(x) as the displacement of a particle parameterized by x ∈ Zn. The interaction between each pair of points x, y in Zn will be described by an energy Ψn depending on u(x) and u(y), and on the mutual position of the points in the lattice. The total energy of the interactions among points of this discrete system described by u is then given by the sum of these pairwise interactions, which we can write in the form