Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No 3, pp. 283-313   https://doi.org/10.15407/mag13.03.283     ( to contents , go back )
https://doi.org/10.15407/mag13.03.283

Integral Conditions for Convergence of Solutions of Non-Linear Robin's Problem in Strongly Perforated Domain

E.Ya. Khruslov1, L.O. Khilkova2, and M.V. Goncharenko3

1,3B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Nauky Ave., Kharkiv 61103, Ukraine
E-mail: khruslov@ilt.kharkov.ua
marusya61@yahoo.co.uk

2Institute of Chemical Technologies of Volodymyr Dahl East Ukrainian National University 31 Volodymyrska Str., Rubizhne 93009, Ukraine
E-mail: LarisaHilkova@gmail.com

Received May 27, 2017

Abstract

We consider a boundary-value problem for the Poisson equation in a strongly perforated domain Ωε = Ω\FεRn (n ≥ 2) with non-linear Robin's condition on the boundary of the perforating set Fε. The domain Ωε depends on the small parameter ε > 0 such that the set Fε becomes more and more loosened and distributes more densely in the domain Ω as ε→0. We study the asymptotic behavior of the solution uε(x) of the problem as ε→0. A homogenized equation for the main term u(x) of the asymptotics of uε(x) is constructed and the integral conditions for the convergence of uε(x) to u(x) are formulated.

Mathematics Subject Classification 2000: 35Q70.
Key words: homogenization, stationary diffusion, non-linear Robin's boundary condition, homogenized equation.

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