The Passage from Nonconvex Discrete Systems to Variational Problems in Sobolev Spaces: The One-Dimensional Case

We treat the problem of the description of the limits of discrete variational problems with long-range interactions in a one-dimensional setting. Under some polynomial-growth condition on the energy densities, we show that it is possible to define a local limit problem on a Sobolev space described by a homogenization formula. We give examples to show that, if the growth conditions are not uniformly satisfied, then the limit problem may be of a nonlocal form or with multiple densities.