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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

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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Correspondence to W. Jäger.

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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