Robin's nonlinear problem in domains with a fine-grained random boundary

1Khruslov, EYa., 2Khilkova, LA
1B. I. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv
2Institute of Chemical Technologies of the Volodymyr Dahl East Ukrainian National University, Rubizhne
Dopov. Nac. akad. nauk Ukr. 2017, 9:3-8
https://doi.org/10.15407/dopovidi2017.09.003
Section: Mathematics
Language: Russian
Abstract: 
We consider the boundary-value problem for the stationary diffusion equation in the domain $\Omega^{\varepsilon} = \Omega \setminus \bigcup_{i = 1}^{N^{\varepsilon}}B_{i}^{\varepsilon}$, which is additional to the large number of fine grains $B_{i}^{\varepsilon} (i = 1, . . ., N^{\varepsilon})$. The Robin's nonlinear boundary condition is given on the grain surfaces. We assume that these grains are small balls, which are randomly distribu t ed in a fixed domain $\Omega \in R^{3}$ and have random radii. The distribution function $f^{\varepsilon}(x, r)$ of the centers $x^{i\varepsilon}$ and of the radii $r_{i}^{\varepsilon}$ of the balls depends on the small parameter $\varepsilon > 0$ so that a mean distance between the nearest centers is $O(\varepsilon)$, and the mean radius is $O(\varepsilon^{\alpha})(\alpha > 2)$. It is proved that, as $\varepsilon \rightarrow 0$, the random solution of the problem $u^{\varepsilon}(x)$ converges in probability in the metric of the space $L ^{2}(\Omega)$ to the nonrandom function $u(x)$, for which a homogenized equation is constructed.
Keywords: Homogenization, random distribution, Robin's boundary condition, stationary diffusion
References: 
  1. Marchenko, V. A. & Khruslov, E. Ya. (1974). Boundary-value problems in domains with a fine-grained boundary. Kiev: Naukova Dumka (in Russian).
  2. Shiryaev, A. N. (2004). Probability-1. Moscow: MTsNMO (in Russian).
  3. Bogolyubov, N. N. (1970). Selected works. Vol. 2. Kiev: Naukova Dumka (in Russian).
  4. Gikhman, I. I., Skorokhod, A. V. & Yadrenko, M. I. (1973). Theory of Probability and Mathematical Statistics. Kiev: Vyshcha shkola (in Russian).
  5. Berlyand, L. V. & Khruslov, E. Ya., (2005). Ginzburg—Landau model of a liquid crystal with random inclusions. J. Math. Phys., 46, 095107, 15 p. https://doi.org/10.1063/1.2013127
  6. Marchenko, V. A. & Khruslov, E. Ya. (2005). Homogenized models of micro-inhomogeneous media. Kiev: Naukova Dumka (in Russian).
  7. Khilkova, L. O. (2016). Homogenization of the diffusion equation in domains with the fine-grained boundary with the nonlinear boundary Robin condition. Visnyk of V. N. Karazin Kharkiv National University. Ser. Math., Appl. Math. and Mech., 84, rp. 93-111 (in Russian).