Homogenization of some quasi-linear elliptic equations with gradient constraints

  • Valeria Chiadò Piat
  • Marco ZoboliEmail author


We prove a homogenization formula for quasi-linear elliptic equations with gradient constraints on a disperse set, within the framework of monotonic operator theory and compensated compactness methods.


Homogenization Elliptic equations Monotonic operators Compensated compactness 



  1. 1.
    Braides, A., Garroni, A.: Homogenization of periodic non-linear media with stiff and soft inclusions. Math. Mod. Methods Appl. Sci. 5, 543–564 (1995)CrossRefzbMATHGoogle Scholar
  2. 2.
    Carbone, L., De Arcangelis, R.: Unbounded Functionals in the Calculus of Variations. Representation, Relaxation and Homogenization. Chapman & Hall/CRC, Boca Raton (2001)zbMATHGoogle Scholar
  3. 3.
    Carbone, L., De Arcangelis, R., De Maio, U.: Homogenization of media with periodically distributed conductors. Asymptotic Anal. 23, 157–194 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cardone, G., Corbo Esposito, A., Yosifian, G.A., Zhikov, V.V.: Homogenization of some problems with gradient constraint. Asymptotic Anal. 38, 201–220 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chiadò Piat, V., Sandrakov, G.V.: Homogenization of some variational inequalities for elasto-plastic torsion problems. Asymptotic Anal. 40, 1–23 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cioranescu, D., Saint Jean Paulin, J.: Homogenization in open sets with holes. J. Math Anal. Appl. 71, 590–607 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Arcangelis, R., Gaudiello, A., Paderni, G.: Some cases of homogenization of linearly coercive gradient constrained variational problems. Math. Mod. Methods Appl. Sci. 6, 901–940 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fusco, N., Moscariello, G.: On the homogenization of quasi-linear divergence structure operators. Ann. Mat. Pura Appl. 146, 1–13 (1986)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hartman, G.J., Stampacchia, G.: On some nonlinear elliptic differential equations. Acta Math 115, 271–310 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)CrossRefGoogle Scholar
  11. 11.
    Kolpakov, A.A., Kolpakov, A.G.: Capacity and Trasport in Contrast Composite Structures. Asymptotic Analysis and Applications. CRC Press, Taylor and Francis Group, Boca Raton (2009)Google Scholar
  12. 12.
    Lanchon, H.: Sur la solution du problème du torsion élastiplastique d’un barre cylindrique de section multiconnexe. C.R. Acad. Sc. Paris 271, 1137–1140 (1970)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars, Paris (1969)zbMATHGoogle Scholar
  14. 14.
    Rauch, J., Taylor, M.: Electrostatic screeningin. J. Math. Phys. 16, 284–288 (1975)CrossRefGoogle Scholar
  15. 15.
    Tartar, L.: Problèmes d’homogénéisation dans les equations aux dérivées partielles. Cours Peccot, Collège de France (1977)Google Scholar
  16. 16.
    Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, vol. IV. Research Notes in Mathematics 39, 136–192 (1979)Google Scholar

Copyright information

© Unione Matematica Italiana 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences “G.L. Lagrange”, Dipartimento di Eccellenza 2018-2022Politecnico di TorinoTorinoItaly

Personalised recommendations