Applied Mathematics & Optimization

, Volume 72, Issue 3, pp 523–547 | Cite as

Homogenization of Functionals with Linear Growth in the Context of \(\mathcal A\)-quasiconvexity

  • José Matias
  • Marco MorandottiEmail author
  • Pedro M. Santos


This work deals with the homogenization of functionals with linear growth in the context of \(\mathcal A\)-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the coupling between homogenization and the \(\mathcal A\)-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects.


\(\mathcal A\)-quasiconvexity Homogenization Representation of integral functionals Concentration effects 

Mathematics Subject Classification

Primary 35B27 Secondary 49J40 49K20 35E99 



Partial support for this research was provided by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Carnegie Mellon Portugal Program under Grant FCT-UTA_CMU/MAT/0005/2009 “Thin Structures, Homogenization, and Multiphase Problems”. The authors warmly thank the Centro de Análise Matemática, Geometria e Sistemas Dinâmicos (CAMGSD) at the Departamento de Matemática of the Instituto Superior Técnico, Universidade de Lisboa, where the research was carried out.


  1. 1.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenization. Proc. R. Soc. Edinb. Sect. A 126(2), 297–342 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Dal, G.: Maso: on the relaxation in \(BV(\Omega; \mathbb{R}^{m})\) of quasi convex integrals. J. Funct. Anal. 109, 76–97 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  5. 5.
    Baía, M., Chermisi, M., Matias, J., Santos, P.M.: Lower semicontinuity and relaxation of signed functionals with linear growth in the context of \({\cal A}\)-quasiconvexity. Calc. Var. Partial Differ. Equ. 47, 465–498 (2013)zbMATHCrossRefGoogle Scholar
  6. 6.
    Braides, A.: Homogenization of some almost periodic functional. Rend. Accad. Naz. Sci. XL 103, 313–322 (1985)MathSciNetGoogle Scholar
  7. 7.
    Braides, A.: \(\Gamma \)-convergence for beginners. In: Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford (2002).
  8. 8.
    Braides, A., Defranceschi, A., Vitali, E.: Homogenization of free discontinuity problems. Arch. Rat. Mech. Anal. 135, 297–356 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Braides, A., Fonseca, I., Leoni, G.: \({\cal A}\)-quasiconvexity: relaxation and homogenization. ESAIM Control Optim. Calc. Var. 5, 539–577 (2000). (electronic)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cioranescu, D., Damlamian, A., De Arcangelis, R.: Homogenization of quasiconvex integrals via the periodic unfolding method. SIAM J. Math. Anal. 37(5), 1435–1453 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335(1), 99–104 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization SIAM. J. Math. Anal. 40(4), 585–1620 (2008)MathSciNetGoogle Scholar
  13. 13.
    Dacorogna, B.: Weak Continuity and Weak Lower Semicontinuity of Non-linear Functionals. Lecture Notes in Mathematics. Springer, Berlin (1982)Google Scholar
  14. 14.
    Dal Maso, G.: An introduction to \(\Gamma \)-convergence. In: Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston Inc, Boston, MA (1993).
  15. 15.
    De Arcangelis, R., Gargiulo, G.: Homogenization of integral functionals with linear growth defined on vector-valued functions. NoDEA 2, 371–416 (1995)zbMATHCrossRefGoogle Scholar
  16. 16.
    De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8, 842–850 (1975)Google Scholar
  17. 17.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)Google Scholar
  18. 18.
    Fonseca, I., Krömer, S.: Multiple integrals under differential constraints: two-scale convergence and homogenization. Indiana Univ. Math. J. 59, 427–458 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\) Spaces. In: Springer Monographs in Mathematics. Springer, New York (2007)Google Scholar
  20. 20.
    Fonseca, I., Müller, S.: Relaxation of quasiconvex functionals in \(BV(\Omega; \mathbb{R}^{p})\) for integrands \(f(x, u, \nabla u)\). Arch. Rat. Mech. Anal. 123, 1–49 (1993)zbMATHCrossRefGoogle Scholar
  21. 21.
    Fonseca, I., Müller, S.: \({\cal A}\)-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30, 1355–1390 (1999). (electronic)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kreisbeck, C., Rindler, F.: Thin-film limits of functionals on \({\cal A}\)-free vector fields. Indiana Univ. Math. J., (to appear). arXiv:1105.3848
  23. 23.
    Kristensen, J., Rindler, F.: Relaxation of signed integrals in BV. Calc. Var. Partial Differ. Equ. 37(1–2), 29–62 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2(1), 33–81 (2002)MathSciNetGoogle Scholar
  25. 25.
    Murat, F.: Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothese de rang constante. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8(1), 69–102 (1981)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Müller, S.: Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal. 99, 189–212 (1987)zbMATHCrossRefGoogle Scholar
  27. 27.
    Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Preiss, D.: Geometry of measures in \(\mathbb{R}^{N}\): distributions, rectifiability and densities. Ann. Math. 125, 537–643 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Rindler, R.: Lower semicontinuity and Young measures in BV without Alberti’s rank-one theorem. Adv. Calc. Var. 5(2), 127–159 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. IV (ed. R. Knops), Pitman. Res. Notes Math. 39, 136–212 (1979)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • José Matias
    • 1
  • Marco Morandotti
    • 2
    Email author
  • Pedro M. Santos
    • 1
  1. 1.CAMGSD, Departamento de MatemáticaInstituto Superior TécnicoLisboaPortugal
  2. 2.SISSA – International School for Advanced StudiesTriesteItaly

Personalised recommendations