Abstract

The Γ\Gamma-limit of a family of functionals u↦∫Ωf(xε,xε2,Dsu) dxu\mapsto \int_{\Omega}f\left(\frac{x}{\varepsilon},\frac{x}{\varepsilon^2},D^su\right)\, dx is obtained for s=1,2s=1,2 and when the integrand f=f(x,y,v)f=f(x,y,v) is a continuous function, periodic in xx and yy, and convex with respect to vv. The 33-scale limits of second order derivatives are characterized.

Author information

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

Irene Fonseca

Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

fonseca@andrew.cmu.edu

Elvira Zappale

Dip. di Ingegneria dell'Informazione e Matematica Applicata, Università degli Studi di
Salerno, 84084 Fisciano, Italy

zappale@diima.unisa.it

Keywords

  • Convexity
  • periodicity
  • multiscale limits
  • Γ-convergence
  • A-quasiconvexity
  • measurability selection criterion

Mathematics Subject Classification

35G99, 49J40, 49J45, 74G65

References

  1. G. Allaire: Homogenization and two scale convergence, SIAM J. Math. Anal. 23 (1992) 1482–1518.https://doi.org/10.1137/0523084Google Scholar
  2. G. Allaire, M. Briane: Multiscale convergence and reiterated homogenization, Proc. Royal Soc. Edin. 126 (1996) 297–342.https://doi.org/10.1017/S0308210500022757Google Scholar
  3. Y. Amirat, K. Hamdache, A. Ziani: Homogénéisation non-locale pour des équations dégénérées à coéfficients périodiques, C. R. Acad. Sci. Paris 312 (1991) 963–966.Google Scholar
  4. K. Bhattacharya, I. Fonseca, G. Francfort: An asymptotic study ot the debonding of thin films, Arch. Rat. Mech. Anal. 161 (2002) 205–229.https://doi.org/10.1007/s002050100177Google Scholar
  5. G. Bouchitté, I. Fonseca, G. Leoni, L. Mascarenhas: A global method for relaxation in W 1,p and SBVp, Arch. Rat. Mech. Anal., to appear.Google Scholar
  6. G. Bouchitté, I. Fonseca, L. Mascarenhas: A global method for relaxation, Arch. Rat. Mech. Anal. 145 (1998) 51–98.https://doi.org/10.1007/s002050050124Google Scholar
  7. A. Braides, A. Defranceschi: Homogenization of Multiple Integrals, Clarendon Press, Oxford (1998).https://doi.org/10.1093/oso/9780198502463.001.0001Google Scholar
  8. A. Braides, D. Lukkassen: Reiterated homogenization of integral functionals, Math. Mod. Meth. Appl.Sc. 10 (2000) 47–71.https://doi.org/10.1142/S0218202500000057Google Scholar
  9. A. Braides, I. Fonseca, G. Francfort: 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J. 49 (2000) 1367–1404.https://doi.org/10.1512/iumj.2000.49.1822Google Scholar
  10. A. Braides, I. Fonseca, G. Leoni: A-quasiconvexity: relaxation and homogenization, ESAIM:COCV 5 (2000) 539–577.https://doi.org/10.1051/cocv:2000121Google Scholar
  11. L. Carbone, R. De Arcangelis: Unbounded Functionals in the Calculus of Variations, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 125, Chapman & Hall/CRC (2001).https://doi.org/10.1201/9781420035582Google Scholar
  12. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  13. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin (1989).https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  14. P. Donato: Alcune osservazioni sulla convergenza debole di funzioni non uniformemente oscillanti, Richerche Mat. 32 (1983) 203–219.Google Scholar
  15. I. Fonseca, G. Leoni: Multiscale Γ-convergence: the non convex case, in preparation.Google Scholar
  16. I. Fonseca, S. Müller: Quasi-convex integrands and lower semicontinuity in L1, SIAM J. Math. Anal. 23 (1992) 1081–1098.https://doi.org/10.1137/0523060Google Scholar
  17. I. Fonseca, S. Müller: A-quasiconvexity, lower semicontinuity and Young measures, SIAM J. Math. Anal. 30 (1999) 1355–1390.https://doi.org/10.1137/S0036141098339885Google Scholar
  18. I. Fonseca, S. Müller, P. Pedregal: Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal. 29 (1998) 736–756.https://doi.org/10.1137/S0036141096306534Google Scholar
  19. A. D. Ioffe: On lower semicontinuity of integral functionals I, SIAM J. Cont. Optim. 15 (1977) 521–538.https://doi.org/10.1137/0315035Google Scholar
  20. A. D. Ioffe: On lower semicontinuity of integral functionals II, SIAM J. Cont. Optim. 15 (1977) 991–1000.https://doi.org/10.1137/0315064Google Scholar
  21. J. Kristensen: Finite functionals and Young measures generated by gradients of Sobolev functions. Report 1994-34, Mathematical Institute, Tech. U. of Denmark, Lyngby, Denmark (1994).Google Scholar
  22. P. Marcellini: Periodic solutions and homogenization of non linear variational problems, Ann. Mat. Pura Appl. 117 (1978) 139–152.https://doi.org/10.1007/BF02417888Google Scholar
  23. P. Marcellini: Approximation of quasiconvex functions and semicontinuity of multiple integrals, Manus. Math. 51 (1985) 1–28.https://doi.org/10.1007/BF01168345Google Scholar
  24. L. Mascarenhas, A. M. Toader: Scale convergence in homogenization, Num. Funct. Anal. Optim. 22 (2001) 127–158.https://doi.org/10.1081/NFA-100103791Google Scholar
  25. Morrey, C. B. jun.: Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin (1966).https://doi.org/10.1007/978-3-540-69952-1Google Scholar
  26. F. Murat: Compacité par compensation : condition necessaire et suffisante de continuité faible sous une hypothése de rang constant, Ann. Sc. Norm. Sup. Pisa 8 (1981) 68–102.Google Scholar
  27. G. Nguetsteng: A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989) 608–623.https://doi.org/10.1137/0520043Google Scholar
  28. G. Nguetsteng: Asymptotic Analysis for a stiff variational problem arising in mechanics, SIAM J. Math. Anal. 21 (1990) 1394–1414.https://doi.org/10.1137/0521078Google Scholar
  29. L. Tartar: Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics, R. J. Knops (ed.), Heriott-Watt Symposium, vol. IV, Research Notes in Matemathics 39, Pitman (1979) 136–212.Google Scholar

Suggested citation

I. Fonseca, E. Zappale. “Multiscale Relaxation of Convex Functionals.” Journal of Convex Analysis 10 (2003), No. 2, 325–350. https://doi.org/10.68381/jca1019

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.