Geodesic
The Passage from Nonconvex Discrete Systems to Variational Problems in Sobolev Spaces: The One-Dimensional Case
Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 408-427

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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. 
@article{TRSPY_2002_236_a42,
     author = {A. Braides and M. Gelli and M. Sigalotti},
     title = {The {Passage} from {Nonconvex} {Discrete} {Systems} to {Variational} {Problems} in {Sobolev} {Spaces:} {The} {One-Dimensional} {Case}},
     journal = {Informatics and Automation},
     pages = {408--427},
     publisher = {mathdoc},
     volume = {236},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2002_236_a42/}
}
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A. Braides; M. Gelli; M. Sigalotti. The Passage from Nonconvex Discrete Systems to Variational Problems in Sobolev Spaces: The One-Dimensional Case. Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 408-427. http://geodesic.mathdoc.fr/item/TRSPY_2002_236_a42/