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

E.Ya. Khruslov^{1}, L.O. Khilkova^{2}, and M.V. Goncharenko^{3}

^{1,3}B. 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

^{2}Institute 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 Ω^{ε} = Ω\F^{ε} ⊂ R^{n} (n ≥ 2) with non-linear
Robin's condition on the boundary of the perforating set F^{ε}. The domain Ω^{ε} depends on the small parameter ε > 0 such that the set F^{ε} becomes more
and more loosened and distributes more densely in the domain Ω as ε→0. We study the asymptotic behavior of the solution u^{ε}(x) of the problem as
ε→0. A homogenized equation for the main term u(x) of the asymptotics of u^{ε}(x) is constructed and the integral conditions for the convergence of u^{ε}(x) to u(x) are formulated.