Geodesic
The Passage from Nonconvex Discrete Systems to Variational Problems in Sobolev Spaces: The One-Dimensional Case
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 408-427 
A. Braides; M. Gelli; M. Sigalotti. The Passage from Nonconvex Discrete Systems to Variational Problems in Sobolev Spaces: The One-Dimensional Case. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 408-427. http://geodesic.mathdoc.fr/item/TM_2002_236_a42/
@article{TM_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 = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {408--427},
     year = {2002},
     volume = {236},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2002_236_a42/}
}
TY  - JOUR
AU  - A. Braides
AU  - M. Gelli
AU  - M. Sigalotti
TI  - The Passage from Nonconvex Discrete Systems to Variational Problems in Sobolev Spaces: The One-Dimensional Case
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2002
SP  - 408
EP  - 427
VL  - 236
UR  - http://geodesic.mathdoc.fr/item/TM_2002_236_a42/
LA  - en
ID  - TM_2002_236_a42
ER  - 
%0 Journal Article
%A A. Braides
%A M. Gelli
%A M. Sigalotti
%T The Passage from Nonconvex Discrete Systems to Variational Problems in Sobolev Spaces: The One-Dimensional Case
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2002
%P 408-427
%V 236
%U http://geodesic.mathdoc.fr/item/TM_2002_236_a42/
%G en
%F TM_2002_236_a42

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] Braides A., “Non-local variational limits of discrete systems”, Commun. Contemp. Math., 2 (2000), 285–297 | MR | Zbl

[2] Braides A., $\Gamma$-convergence for beginners, Oxford Univ. Press, Oxford (to appear) | MR

[3] Braides A., Dal'Maso G., Garroni A., “Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case”, Arch. Ration. Mech. and Anal., 146 (1999), 23–58 | DOI | MR | Zbl

[4] Braides A., Defranceschi A., Homogenization of multiple integrals, Oxford Univ. Press, Oxford, 1998 | MR | Zbl

[5] Braides A., Gelli M. S., “Continuum limits of discrete systems without convexity hypotheses”, Math. Mech. Solids, 6 (2002) | MR

[6] Braides A., Gelli M. S., “Limits of discrete systems with long-range interactions”, J. Convex Anal., 9 (2002) | MR | Zbl

[7] Dal'Maso G., An introduction to $\Gamma$-convergence, Birkhäuser, Boston etc., 1993 | MR

[8] Pagano S., Paroni R., “A simple model for phase transitions: from the discrete to the continuum problem”, Quart. Appl. Math. (to appear) | MR

[9] Truskinovsky L., “Fracture as a phase transition”, Contemporary research in the mechanics and mathematics of materials, eds. R. C. Batra, M. F. Beatty, CIMNE, Barcelona, 1996, 322–332