Skip to main content
SpringerLink
Log in

Homogenization of integral energies under periodically oscillating differential constraints

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

A homogenization result for a family of integral energies

$$\begin{aligned} u_{\varepsilon }\mapsto \int _\Omega f(u_{\varepsilon }(x))\,dx,\quad \varepsilon \rightarrow 0^+, \end{aligned}$$

is presented, where the fields \(u_{\varepsilon }\) are subjected to periodic first order oscillating differential constraints in divergence form. The work is based on the theory of \(\mathscr {A}\)-quasiconvexity with variable coefficients and on two-scale convergence techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acerbi, E., Fusco, N.: Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86(2), 125–145 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston (1984)

    MATH  Google Scholar 

  4. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63(4), 337–403 (1976/77)

  5. Bendsøe, M.P.: Optimization of Structural Topology, Shape, and Material. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  6. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. AMS Chelsea Publishing, Providence (2011). (Corrected reprint of the 1978 original)

  7. Braides, A., Fonseca, I., Leoni, G.: A-quasiconvexity: relaxation and homogenization. ESAIM Control Optim. Calc. Var. 5, 539–577 (2000). (electronic)

  8. Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335(1), 99–104 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cioranescu, D., Donato, P.: An introduction to homogenization. In: Oxford Lecture Series in Mathematics and its Applications, vol. 17. The Clarendon Press, Oxford University Press, New York (1999)

  11. Dacorogna, B.: Weak continuity and weak lower semicontinuity of nonlinear functionals. Lecture Notes in Mathematics, vol. 922. Springer, Berlin (1982)

  12. Dacorogna, B., Fonseca, I.: A-B quasiconvexity and implicit partial differential equations. Calc. Var. Partial Differ. Equ. 14(2), 115–149 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Davoli, E., Fonseca, I.: Homogenization for A-quasiconvexity with variable coefficients (2016). http://cvgmt.sns.it/paper/2904/

  14. Fonseca, I., Krömer, S.: Multiple integrals under differential constraints: two-scale convergence and homogenization. Indiana Univ. Math. J. 59(2), 427–457 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fonseca, I., Leoni, G.: Modern methods in the calculus of variations: \(L^p\) spaces. In: Springer Monographs in Mathematics. Springer, New York (2007)

    MATH  Google Scholar 

  16. Fonseca, I., Müller, S.: \({\cal A}\)-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30(6), 1355–1390 (1999). (electronic)

  17. Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2(1), 35–86 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Marcellini, P.: Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math. 51(1–3), 1–28 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Morrey, C.B. Jr.: Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer, New York (1966)

  20. Murat, F.: Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(1), 69–102 (1981)

  21. Neukamm, S.: Homogenization, linearization and dimension reduction in elasticity with variational methods. PhD thesis, Technische Universität München (2010)

  22. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Santos, Pedro M.: \({\cal A}\)-quasi-convexity with variable coefficients. Proc. R. Soc. Edinburgh Sect. A 134(6), 1219–1237 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Visintin, A.: Some properties of two-scale convergence. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15(2), 93–107 (2004)

  25. Visintin, A.: Towards a two-scale calculus. ESAIM Control Optim. Calc. Var. 12(3), 371–397 (2006). (electronic)

  26. Visintin, A.: Two-scale convergence of some integral functionals. Calc. Var. Partial Differ. Equ. 29(2), 239–265 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where this research was carried out. The research of E. Davoli, and I. Fonseca was funded by the National Science Foundation under Grant No. DMS- 0905778. I. Fonseca was also supported by the National Science Foundation under Grant No. DMS-1411646. E. Davoli and I. Fonseca acknowledge support of the National Science Foundation under the PIRE Grant No. OISE-0967140. E. Davoli is a member of the INDAM-GNAMPA Project 2015 “Critical phenomena in the mechanics of materials: a variational approach”, and was supported by the Austrian FWF project “Global variational methods for nonlinear evolution equations”.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Vienna, Oskar-Morgenstern Platz 1, 1090, Vienna, Austria

    Elisa Davoli

  2. Department of Mathematics, Carnegie Mellon University, Forbes Avenue, Pittsburgh, PA, 15213, USA

    Irene Fonseca

Authors
  1. Elisa Davoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irene Fonseca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irene Fonseca.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davoli, E., Fonseca, I. Homogenization of integral energies under periodically oscillating differential constraints. Calc. Var. 55, 69 (2016). https://doi.org/10.1007/s00526-016-0988-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-016-0988-5

Mathematics Subject Classification

Search

Navigation