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Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem. (English. Russian original) Zbl 1217.35049

Differ. Equ. 47, No. 1, 78-90 (2011); translation from Differ. Uravn. 47, No. 1, 79-91 (2011).
Summary: We study the homogenization problem for the Poisson equation in a periodically perforated domain with a nonlinear boundary condition for the flux on the cavity boundaries. We show that, under certain relations on the problem scale, the homogenized equations may have different character concerning the nonlinearity. In each case considered, we obtain estimates for the convergence of solutions of the original problem to the solution of the homogenized problem in the corresponding Sobolev spaces.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J65 Nonlinear boundary value problems for linear elliptic equations
35A35 Theoretical approximation in context of PDEs
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