Journal of Mathematical Sciences

, Volume 224, Issue 5, pp 744–763 | Cite as

Homogenization Estimates in the Riemann–Hilbert Problem for the General Beltrami Equation on the Plane

  • S. E. PastukhovaEmail author

We study homogenization for the Beltrami equation \( {A}_{\varepsilon }{u}_{\varepsilon}\equiv {\partial}_{\overline{z}}{u}_{\varepsilon }+{\mu}^{\varepsilon }{\partial}_z{u}_{\varepsilon }+{\nu}^{\varepsilon}\overline{\partial_z{u}_{\varepsilon }}=f \) with measurable ε-periodic coefficients μ ε and ν ε , where ε is a small parameter. The coefficients of the equation satisfy the uniform ellipticity condition. The equation is considered in a bounded domain Ω of the complex plane with the Riemann–Hilbert condition on the boundary ∂Ω. For the resolvent \( {A}_{\varepsilon}^{-1} \) of this boundary value problem we obtain an approximation in the operator norm of the Sobolev space W 1,2(Ω) with approximation error of order O(\( \sqrt{\varepsilon } \)).


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  1. 1.
    A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam (1978).zbMATHGoogle Scholar
  2. 2.
    N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht (1989).CrossRefzbMATHGoogle Scholar
  3. 3.
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin (1994).Google Scholar
  4. 4.
    V. V. Zhikov and S. E. Pastukhova, “Operator estimates in homogenization theory” Russ. Math. Surv. 71, No. 3, 417–511 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. V. Zhikov and S. E. Pastukhova, “Operator estimates in homogenization of operators with quasiperiodic coefficients” Russian J. Math. Phys. 22, No 2, 264–278 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Univ. Press, Princeton, NJ (2009).zbMATHGoogle Scholar
  7. 7.
    B. V. Bojarski, “Generalized solutions of a system of first order differential equations of elliptic type with discontinuous coefficients” [in Russian], Mat. Sb. 43, No. 4, 451–503 (1957).Google Scholar
  8. 8.
    L. Bers, F. John, and M. Schechter, Partial Differential Equations, Am. Math. Soc., Providence, RI (1979).zbMATHGoogle Scholar
  9. 9.
    F. Giannetti, T. Iwaniec, L. Kovalev, G. Moscariello, and C. Sbordone, “On G-compactness of the Beltrami operators,” In: Nonlinear Homogenization and Its Applications to Composites, Polycrystals and Smart Materials, pp. 107–138, Kluwer, Norwell, MA (2004).Google Scholar
  10. 10.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York etc. (1975).Google Scholar
  11. 11.
    K. Astala, T. Iwaniec, and E. Saksman, “Beltrami operators in the plane,” Duke Math. J. 107, No. 1, 27–56 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    I. N. Vekua, Generalized Analytic Functions, Addison-Wesley Publ., New York etc. (1962).Google Scholar
  13. 13.
    V. V. Zhikov and M. M. Sirazhudinov, “The averaging of a system of Beltrami equations,” Differ. Equations 24, No. 1, 50–56 (1988).MathSciNetzbMATHGoogle Scholar
  14. 14.
    M. M. Sirazhudinov, “G-convergence and homogenization of generalized Beltrami operators,” Sb. Math. 199, No. 5, 755–786 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    S. E. Pastukhova, “Estimates of homogenization for the Beltrami equation,” J. Math. Sci., New York 219, No. 2, 226–235 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    V. A. Solonnikov, “General boundary value problems for Douglis–Nirenberg elliptic systems,” Proc. Steklov Inst. Math. 92, 269–339 (1968).Google Scholar
  17. 17.
    V. V. Zhikov and S. E. Pastukhova, “Homogenization of degenerate elliptic equations,” Sib. Math. J. 49, No. 1, 80–101 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and Há Tiên Ngoan, “Averaging and Gconvergence of differential operators,” Russ. Math. Surv. 34, No. 5, 65-147 (1979).CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Moscow Technological University (MIREA)MoscowRussia

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