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Corrector Problem and Homogenization of Nonlinear Elliptic Monotone PDE

  • Jean Louis Woukeng
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 169)

Abstract

In the deterministic homogenization of nonlinear monotone elliptic PDEs, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficient. The obtained results represent an important step towards the numerical implementation of the results from the deterministic homogenization theory beyond the periodic setting.

Keywords

Corrector problem Deterministic homogenization Approximation 

Mathematics Subject Classification (2010).

Primary 35B40 46J10 

References

  1. 1.
    A. Bourgeat, A.L. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 153–165.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Casado Diaz, I. Gayte, The two-scale convergence method applied to generalized Besicovitch spaces. Proc. R. Soc. Lond. A 458 (2002) 2925–2946.MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Gloria, F. Otto, Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc. (to appear), arXiv:1409.0801.Google Scholar
  4. 4.
    W. Jäger, J.L. Woukeng, Approximation of homogenized coefficients and convergence rates in deterministic homogenization, Preprint, 2017.Google Scholar
  5. 5.
    V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
  6. 6.
    G. Nguetseng, M. Sango, J.L. Woukeng, Reiterated ergodic algebras and applications, Commun. Math. Phys 300 (2010) 835–876.MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Sango, N. Svanstedt, J.L. Woukeng, Generalized Besicovitch spaces and application to deterministic homogenization, Nonlin. Anal. TMA 74 (2011) 351–379.MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.L. Woukeng, Deterministic homogenization of non-linear non-monotone degenerate elliptic operators, Adv. Math. 219 (2008) 1608–1631.MathSciNetCrossRefGoogle Scholar
  9. 9.
    J.L. Woukeng, Homogenization of nonlinear degenerate non-monotone elliptic operators in domains perforated with tiny holes, Acta Appl. Math. 112 (2010) 35–68.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J.L. Woukeng, Σ-convergence of nonlinear monotone operators in perforated domains with holes of small size, Appl. Math. 54 (2009) 465–489.MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.L. Woukeng, Introverted algebras with mean value and applications, Nonlinear Anal. TMA 99 (2014) 190–215.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jean Louis Woukeng
    • 1
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of DschangDschangCameroon
  2. 2.Current address: Interdisciplinary Center for Scientific Computing (IWR)University of HeidelbergHeidelbergGermany

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