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Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term

Journal of Mathematical Sciences volume 179, pages 446–459 (2011)Cite this article

Abstract

In this paper, we study the asymptotic behavior of solutions u ε of the initial boundary value problem for parabolic equations in domains \( {\Omega_\varepsilon } \subset {\mathbb{R}^n} \), n ≥ 3, perforated periodically by balls with radius of critical size ε α, α = n/(n − 2), and distributed with period ε. On the boundary of the balls a nonlinear third boundary condition is imposed. The weak convergence of the solutions u ε to the solution of an effective equation is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and a corrector result with respect to the energy norm is proved.

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Authors and Affiliations

  1. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    W. Jäger & M. Neuss-Radu

  2. Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119992, GSP-2, Leninskie Gory, Russia

    T. A. Shaposhnikova

Authors
  1. W. Jäger
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  2. M. Neuss-Radu
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  3. T. A. Shaposhnikova
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Corresponding author

Correspondence to W. Jäger.

Additional information

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 28, Part I, pp. 161–181, 2011.

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Jäger, W., Neuss-Radu, M. & Shaposhnikova, T.A. Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term. J Math Sci 179, 446–459 (2011). https://doi.org/10.1007/s10958-011-0603-4

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  • DOI: https://doi.org/10.1007/s10958-011-0603-4

Keywords

  • Weak Solution
  • Nonlinear Boundary
  • Homogenization Problem
  • Nonlinear Boundary Condition
  • Interior Boundary