From discrete to continuous variational problems: an introduction

where the sum is performed for k ∈ Z and on indices i ∈ Z such that iε ∈ Ω and (i+ k)ε ∈ Ω. If we picture the lattice εZ ∩ Ω as the reference configuration of a set of material points interacting through some forces, and ui represents the displacement of the i-th point, then ψ ε can be thought as the energy density of the interaction of points with distance kε in the reference lattice. Note that the only assumption we make is that ψ ε depends on {ui} through the differences ui+k−ui. It is usually more convenient to make change in the notation and set φε(z) = ε −Nψj ε(jεz); In such a way we write