Skip to main content
SpringerLink
Log in

Homogenization of singular elliptic systems with nonlinear conditions on the interfaces

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

We prove existence and homogenization results for elliptic problems involving a singular lower order term and defined on a domain with interfaces having a nonlinear response. The considered system of equations can model heat or electrical conduction in composite media.

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

Availability of data and material

Not applicable.

Code availability

Not applicable.

References

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

    Article  MathSciNet  Google Scholar 

  2. Allaire, G., Briane, M.: Multi-scale convergence and reiterated homogenization. Proc. R. Soc. Edimburg Sect. A 126(2), 297–342 (1996)

    Article  Google Scholar 

  3. Allaire, G., Damlamian, A., Hornung, U.: Two-scale convergence on periodic surfaces and applications. In: Bourgeat, A.P., Carasso, C., Luckhaus, S., Mikelic, A. (eds.) Mathematical Modelling of Flow through Porous Media, pp. 15–25. World Scientific, Singapore (1995)

    Google Scholar 

  4. Amar, M., Andreucci, D., Bellaveglia, D.: The time-periodic unfolding operator and applications to parabolic homogenization, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28, 663–700 (2017)

    Article  MathSciNet  Google Scholar 

  5. Amar, M., Andreucci, D., Bellaveglia, D.: Homogenization of an alternating Robin–Neumann boundary condition via time-periodic unfolding. Nonlinear Anal. 153, 56–77 (2017)

    Article  MathSciNet  Google Scholar 

  6. Amar, M., Andreucci, D., Bisegna, P., Gianni, R.: Evolution and memory effects in the homogeneization limit for electrical conduction in biological tissues. Math. Models Methods Appl. Sci. 14(9), 1261–1295 (2004)

    Article  MathSciNet  Google Scholar 

  7. Amar, M., Andreucci, D., Bisegna, P., Gianni, R.: On a hierarchy of models for electrical conduction in biological tissues. Math. Methods Appl. Sci. 29(7), 767–787 (2006)

    Article  MathSciNet  Google Scholar 

  8. Amar, M., Andreucci, D., Bisegna, P., Gianni, R.: Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range. Commun. Pure Appl. Anal. 9(5), 1131–1160 (2010)

    Article  MathSciNet  Google Scholar 

  9. Amar, M., Andreucci, D., Bisegna, P., Gianni, R.: A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: the nonlinear case. Differ. Integral Equ. 26(9–10), 885–912 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Amar, M., Andreucci, D., Gianni, R., Timofte, C.: Concentration and homogenization in electrical conduction in heterogeneous media involving the Laplace-Beltrami operator. Calc. Var. 59, 99 (2020) (to appear)

    Article  MathSciNet  Google Scholar 

  11. Amar, M., Chiricotto, M., Giacomelli, L., Riey, G.: Mass-constrained minimization of a one-homogeneous functional arising in strain-gradient plasticity. J. Math. Anal. Appl. 397(1), 381–401 (2013)

    Article  MathSciNet  Google Scholar 

  12. Amar, M., De Bonis, I., Riey, G.: Homogenization of elliptic problems involving interfaces and singular data. Nonlinear Anal. 189, 111562 (2019)

    Article  MathSciNet  Google Scholar 

  13. Amar, M., Gianni, R.: Laplace–Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete Contin. Dyn. Syst. Ser. B 23(4), 1739–1756 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Amar, M., Gianni, R.: Error estimate for a homogenization problem involving the Laplace–Beltrami operator. Math. Mech. Complex Syst. 6(1), 41–59 (2018)

    Article  MathSciNet  Google Scholar 

  15. Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Oxford Lecture Series in Mathematics and Its Applications, vol. 12. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  16. Braides, A., Riey, G., Solci, M.: Homogenization of Penrose tilings. C. R. Math. Acad. Sci. Paris 347(11–12), 697–700 (2009)

    Article  MathSciNet  Google Scholar 

  17. Braides, A., Solci, M.: Interfacial energies on Penrose lattices. Math. Models Methods Appl. Sci. 21(5), 1193–1210 (2011)

    Article  MathSciNet  Google Scholar 

  18. Piat, V.Chiatò, Maso, G.Dal, Defranceschi, A.: G-convergence of monotone operators. Ann. Inst. H. Poincaré. Anal. Non Linéaire 7(3), 123–160 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. De Giorgi, E., Spagnolo, S.: Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 8(4), 391–411 (1973)

    MathSciNet  MATH  Google Scholar 

  22. Donato, P., Giachetti, D.: Existence and homogenization for a singular problem through rough surfaces. SIAM J. Math. Anal. 48(6), 4047–4086 (2016)

    Article  MathSciNet  Google Scholar 

  23. Fulks, W., Maybee, J.S.: A singular nonlinear equation. Osaka Math. J. 12, 1–19 (1960)

    MathSciNet  MATH  Google Scholar 

  24. Gurtin, M.E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Claredon Press, Oxford (1993)

    MATH  Google Scholar 

  25. Hummel, H.K.: Homogenization for heat transfer in polycrystals with interfacial resistance. Appl. Anal. 75(3–4), 403–424 (2000)

    Article  MathSciNet  Google Scholar 

  26. Lene, F., Leguillon, D.: Ètude de l’influence d’un glissement entre les constituants d’un matériau composite sur ses coefficients de comportement effectifs. J. Mécanique 20(3), 509–536 (1981)

    MathSciNet  MATH  Google Scholar 

  27. Lipton, R.: Heat conduction in fine mixtures with interfacial contact resistance. SIAM J. Appl. Math. 58(1), 55–72 (1998)

    Article  MathSciNet  Google Scholar 

  28. 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  Google Scholar 

  29. Spagnolo, S.: Sulla convergenza delle soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 22(3), 571–597 (1968)

    MathSciNet  MATH  Google Scholar 

  30. Tartar, L.: Problémes d’homogénéisation dans les équations aux dérivée partielles. In: Cours Peccot Collège de France, 1977, partiaellement rédigé dans: Murat, F. (ed.), \(H\)-convergence. Séminaire d’Analyse Fonctionelle et Numérique, 1977/78, Université d’Alger (polycopié)

  31. Zhykov, V.V.: Averaging of functional of the calculus of variations and elasticity theory. Izk. Akad. Nauk. SSSR Ser. Mat. 50, 675–710 (1986)

    MathSciNet  Google Scholar 

  32. Zhykov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)

    Google Scholar 

Download references

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza-Università di Roma, Via A. Scarpa 16, 00161, Rome, Italy

    M. Amar

  2. Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036, Rende, CS, Italy

    G. Riey

Authors
  1. M. Amar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Riey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Riey.

Ethics declarations

Conflict of interest

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amar, M., Riey, G. Homogenization of singular elliptic systems with nonlinear conditions on the interfaces. J Elliptic Parabol Equ 6, 633–654 (2020). https://doi.org/10.1007/s41808-020-00075-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41808-020-00075-9

Keywords

Mathematics Subject Classification

Search

Navigation