Advertisement

Archive for Rational Mechanics and Analysis

, Volume 221, Issue 3, pp 1511–1584 | Cite as

Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth

  • Mitia Duerinckx
  • Antoine GloriaEmail author
Article

Abstract

We consider the well-trodden ground of the problem of the homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has p-growth from below (with p > d, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals.

Keywords

Dirichlet Boundary Condition Lower Semicontinuous Recovery Sequence Full Probability Bound Lipschitz Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E., Chiadò Piat V., Dal Maso G., Percivale D.: An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18(5), 481–496 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86(2), 125–145 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Alber Ya.I.: James orthogonality and orthogonal decompositions of Banach spaces. J. Math. Anal. Appl. 312(1), 330–342 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Alicandro R., Cicalese M., Gloria A.: Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. 200(3), 881–943 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Aliprantis, Ch.D., Border, K.C.: Infinite dimensional analysis. A Hitchhiker’S Guide, 3rd edn. Springer, Berlin (2006)Google Scholar
  6. 6.
    Anza Hafsa, O., Mandallena, J.-P.: Homogenization of singular integrals in \({w^{1,\infty}}\) (2010). arXiv:0912.5408v2
  7. 7.
    Anza Hafsa O., Mandallena J.-P.: Homogenization of nonconvex integrals with convex growth. J. Math. Pures Appl. (9) 96(2), 167–189 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Attouch, H.: Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, 1984Google Scholar
  9. 9.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4), 337–403 (1976/1977)Google Scholar
  10. 10.
    Billingsley, P.: Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics, deuxi edn. Wiley, New York, 1999. (A Wiley-Interscience Publication)Google Scholar
  11. 11.
    Braides A.: Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 9(1), 313–321 (1985)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Braides, A., Defranceschi, A.: Homogenization of multiple integrals. Oxford Lecture Series in Mathematics and its Applications, Vol. 12. The Clarendon Press, Oxford University Press, New York, 1998Google Scholar
  13. 13.
    Braides A., Garroni A.: Homogenization of periodic nonlinear media with stiff and soft inclusions. Math. Models Methods Appl. Sci. 5(4), 543–564 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Braides A., Maslennikov M., Sigalotti L.: Homogenization by blow-up. Appl. Anal. 87(12), 1341–1356 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Carbone L., Cioranescu D., De Arcangelis R., Gaudiello A.: Homogenization of unbounded functionals and nonlinear elastomers. The general case. Asymptot. Anal. 29(3–4), 221–272 (2002)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Carbone, L., Cioranescu, D., De Arcangelis, R., Gaudiello, A.: Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set. ESAIM Control Optim. Calc. Var. 10(1), 53–83 (2004). (Electronic)Google Scholar
  17. 17.
    Carbone, L., De Arcangelis, R.: Unbounded functionals in the calculus of variations. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 125. Chapman & Hall/CRC, Boca Raton, 2002. (Representation, relaxation, and homogenization)Google Scholar
  18. 18.
    Corbo Esposito A., De Arcangelis R.: The Lavrentieff phenomenon and different processes of homogenization. Commun. Partial Differ. Equ. 17(9–10), 1503–1538 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Dal Maso G., Modica L.: Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368, 28–42 (1986)zbMATHMathSciNetGoogle Scholar
  20. 20.
    De Buhan M., Gloria A., Le Tallec P., Vidrascu M.: Reconstruction of a constitutive law for rubber from in silico experiments using Ogden’s laws. Int. J. Solids Struct. 62, 158–173 (2015)CrossRefGoogle Scholar
  21. 21.
    Ekeland, I., Temam, R.: Convex analysis and variational problems. Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam, 1976Google Scholar
  22. 22.
    Fonseca I.: The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. (9) 67(2), 175–195 (1988)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Fonseca I., Müller S.: Quasi-convex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23(5), 1081–1098 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Gloria A., Le Tallec P., Vidrascu M.: Foundation, analysis, and numerical investigation of a variational network-based model for rubber. Contin. Mech. Thermodyn. 26(1), 1–31 (2014)ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    Gloria, A., Neukamm, S., Otto, F.: A regularity theory for random elliptic operators and homogenization (2015). arXiv:1409.2678
  26. 26.
    Gloria A., Penrose M.D.: Random parking, Euclidean functionals, and rubber elasticity. Commun. Math. Phys. 321(1), 1–31 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994. (Traduit du russe par G. A. Iosif\({\prime}\)yan)Google Scholar
  28. 28.
    Koralov, L.B., Sinai, Y.G.: Theory of probability and random processes. Universitext, deuxi edn. Springer, Berlin, 2007Google Scholar
  29. 29.
    Krengel, U.: Ergodic theorems. de Gruyter Studies in Mathematics, Vol. 6. De Gruyter, Berlin, 1985Google Scholar
  30. 30.
    Marcellini P.: Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. (4) 117, 139–152 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Messaoudi K., Michaille G.: Stochastic homogenization of nonconvex integral functionals. RAIRO Modél. Math. Anal. Numér. 28(3), 329–356 (1994)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Müller S.: Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Ration. Mech. Anal. 99(3), 189–212 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. Random fields, Vol. I, II (Esztergom, 1979). Colloq. Math. Soc. János Bolyai Vol. 27. North-Holland, Amsterdam, 835–873, 1981Google Scholar
  34. 34.
    Tartar, L.: Some remarks on separately convex functions. Microstructure and Phase Transition. IMA Vol. Math. Appl., Vol. 54. Springer, New York, 191–204, 1993Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Université Libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.InriaLilleFrance

Personalised recommendations