Abstract

We prove that under some structure and decay hypotheses, limits of discrete systems with multi-neighbourhood interactions give rise to local energies with bulk and surface part, and that the limit energy densities can be recovered through a scaling and superposition procedure.

Author information

Contact details are reproduced from the original publication and may be historical.

Maria Stella Gelli

S.I.S.S.A., via Beirut 4, 34013 Trieste, Italy

gelli@sissa.it

Keywords

  • Gamma-limit
  • thin films
  • homogenization

Mathematics Subject Classification

35E99, 49J45; 35M10, 73C50, 73M25

References

  1. R. Alicandro, M. Focardi, M. S. Gelli: Finite-difference approximation of energies in fracture mechanics, Annali della Scuola Normale Superiore, serie IV, 29 (2000) 671–709.Google Scholar
  2. L. Ambrosio: A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. 3-B (1989) 857–881.Google Scholar
  3. L. Ambrosio: Existence theory for a new class of variational problems, Arch. Rational Mech. Anal. 111 (1990) 291–322.https://doi.org/10.1007/BF00376024Google Scholar
  4. L. Ambrosio, A. Braides: Energies in SBV and variational models in fracture mechanics, in: Homogenization and Applications to Material Sciences, D. Cioranescu, A. Damlamian, P. Donato (eds.), GAKUTO, Gakkōtosho, Tokio, Japan (1997) 1–22.Google Scholar
  5. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  6. A. Braides: Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics, Springer-Verlag, Berlin (1998).https://doi.org/10.1007/BFb0097344Google Scholar
  7. A. Braides: Non-local variational limits of discrete systems, Comm. Contemporary Math. 2 (2000) 285–297.https://doi.org/10.1142/S021919970000013XGoogle Scholar
  8. A. Braides: Γ-Convergence for Beginners, Oxford University Press, Oxford, to appear.Google Scholar
  9. A. Braides, V. Chiadò Piat: Integral representation results for functionals defined on SBV (Ω; Rm), J. Math. Pures Appl. 75 (1996) 595–626.Google Scholar
  10. A. Braides, G. Dal Maso, A. Garroni: Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case, Arch Rational Mech. Anal. 146 (1999) 23–58.https://doi.org/10.1007/s002050050135Google Scholar
  11. A. Braides, A. Defranceschi: Homogenization of Multiple Integral, Oxford University Press, Oxford (1998).https://doi.org/10.1093/oso/9780198502463.001.0001Google Scholar
  12. A. Braides, M. S. Gelli: Continuum limits of discrete systems without convexity hypotheses, Math. Mech. Solids 6 (2002).https://doi.org/10.1177/1081286502007001229Google Scholar
  13. A. Braides, M. S. Gelli: From Discrete to Continuum: a Variational Approach, Lecture Notes, SISSA, Trieste, 2000 (available on the server http://www.sissa.it/fa/pub.html).Google Scholar
  14. A. Chambolle: Finite differences discretizations of the Mumford-Shah functional, Model. Math. Numer. Anal. 33 (1999) 261–288.https://doi.org/10.1051/m2an:1999115Google Scholar
  15. G. Cortesani, R. Toader: A density result in SBV with respect to non-isotropic energies, Nonlinear Analysis 38 (1999) 585–604.https://doi.org/10.1016/S0362-546X(98)00132-1Google Scholar
  16. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  17. E. De Giorgi, L. Ambrosio: Un nuovo funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199–210.Google Scholar
  18. L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992).Google Scholar
  19. H. Federer: Geometric Measure Theory, Springer-Verlag, New York (1969).Google Scholar
  20. M. Gobbino: Finite difference approximation of the Mumford-Shah functional, Comm. Pure Appl. Math 51 (1998) 197–228.https://doi.org/10.1002/(SICI)1097-0312(199802)51:2<197::AID-CPA3>3.0.CO;2-6Google Scholar
  21. M. Gobbino, M. G. Mora: Finite difference approximation of free discontinuity problems, Proc. Royal. Soc. Edinburgh A, 131 (2001) 567–595.https://doi.org/10.1017/S0308210501000257Google Scholar
  22. D. Mumford, J. Shah: Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 17 (1989) 577–685.https://doi.org/10.1002/cpa.3160420503Google Scholar
  23. S. Pagano, R. Paroni: A simple model for phase transitions: from the discrete to the continuum problem, Q. Appl. Math., to appear.Google Scholar
  24. L. Truskinovsky: Fracture as a phase transition, Contemporary Research in the Mechanics and Mathematics of Materials, R. C. Batra, M. F. Beatty (eds.), CIMNE, Barcelona (1996) 322–332.Google Scholar
  25. W. Ziemer: Weakly Differentiable Functions, Springer-Verlag, Berlin (1989).https://doi.org/10.1007/978-1-4612-1015-3Google Scholar

Suggested citation

A. Braides, M. S. Gelli. “Limits of Discrete Systems with Long-Range Interactions.” Journal of Convex Analysis 9 (2002), No. 2, 363–399. https://doi.org/10.68381/jca09022

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.