We consider a boundary-value problem for the equation of stationary diffusion in a porous medium filled with small inclusions in the form of balls with absorbing surfaces. The process of absorption is described by the Robin nonlinear boundary condition. The locations and radii of the inclusions are randomly distributed and described by a set of finite-dimensional distribution functions. We study the asymptotic behavior of solutions to the problem when the number of balls increases and their radii decrease. We deduce a homogenized equation for the main term of the asymptotics and determine sufficient conditions for the distribution functions under which the solutions converge to the solutions of the homogenized problem in probability.
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E. Ya. Khruslov, L. O. Khilkova, and M. V. Goncharenko, “Integral conditions for convergence of solutions of nonlinear Robin’s problem in strongly perforated domains,” J. Math. Phys., Anal., Geom., 13, No. 3, 1–31 (2017).
B. Cabarrubias and P. Donato, “Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary condition,” Appl. Anal., 91, No. 6, 1111–1127 (2012).
D. Cioranescu and P. Donato, “On Robin problems in perforated domains,” Math. Sci. Appl., 9, 123–135 (1997).
D. Cioranescu, P. Donato, and R. Zaki, “The periodic unfolding method in perforated domains,” Port. Math., 63, No. 4, 467–496 (2006).
C. Conca, J. Diaz, A. Linan, and C. Timofte, “Homogenization in chemical reactive flows,” Electron. J. Different. Equat., 40, 1–22 (2004).
C. Conca, J. Diaz, A. Linan, and C. Timofte, “Homogenization results for chemical reactive flows through porous media,” in: New Trends in Continuum Mechanics (2005), pp. 99–107.
J. Diaz, “Two problems in homogenization of porous media,” Extracta Math., 14, 141–155 (1999).
W. Jäger, O. A. Oleinik, and A. S. Shamaev, “On homogenization of solutions of boundary-value problem for the Laplace equation in partially perforated domain with the third boundary type condition on the boundary of cavities,” Trudy Mosk. Mat. Obshch., 58, 187–223 (1997).
T. A. Mel’nyk and O. A. Sivak, “Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain,” Ukr. Mat. Zh., 61, No. 4, 494–512 (2009); English translation: Ukr. Math. J., 61, No. 4, 592–612 (2009).
T. A. Mel’nyk and O. A. Sivak, “Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains,” Asymptot. Anal., 75, 79–92 (2011).
A. Piatnitski and V. Rybalko, “Homogenization of boundary value problems for monotone operators in perforated domains with rapidly oscillating boundary conditions of Fourier type,” J. Math. Sci., 177, No. 1, 109–140 (2011).
C. Timofte, “Homogenization in nonlinear chemical reactive flows,” in: Proc. of the 9th WSEAS International Conference on Applied Mathematics (Istambul, Turkey, May 27–29, 2006), pp. 250–255.
V. A. Marchenko and E.Ya. Khuslov, “Boundary-value problems with fine-grained boundaries,” Mat. Sb., 65, 458–472 (1964).
S. Kaizu, “The Poisson equation with semilinear boundary conditions in domains with many tiny holes,” J. Fac. Sci. Univ. Tokyo. Sect. IA. Math., 36, 43–86 (1989).
M. Goncharenko, “The asymptotic behavior of the third boundary-value problem solutions in domains with fine-grained boundaries,” Math. Sci. Appl., 9, 203–213 (1997).
A. Brillard, D. Gómez, M. Lobo, E. Pérez, and T. A. Shaposhnikova, “Boundary homogenization in perforated domains for adsorption problems with an advection term,” Appl. Anal., 1–17 (2016).
J. Diaz, D. Gómez-Castro, and C. Timofte, “The effectiveness factor of reaction-diffusion equations: homogenization and existence of optimal pellet shapes,” J. Elliptic Parabol. Equat., 2, 119–129 (2016).
J. Diaz, D. Gómez-Castro, T. A. Shaposhnikova, and M. N. Zubova, “The effectiveness factor of reaction-diffusion equations: homogenization and existence of optimal pellet shapes,” Electron. J. Different. Equat., 178, 1–25 (2017).
D. Gómez, E. Pérez, and T. A. Shaposhnikova, “On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems,” Asymptot. Anal., 80, 289–322 (2012).
W. Jäger, M. Neuss-Radu, and T. A. Shaposhnikova, “Homogenization limit for the diffusion equation with nonlinear flux condition on the boundary of very thin holes periodically distributed in a domain, in case of a critical size,” Dokl. Mat., 82, No. 2, 736–740 (2010).
M. E. Pérez, M. N. Zubova, and T. A. Shaposhnikova, “A homogenization problem in a domain perforated by tiny isoperimetric holes with nonlinear Robin-type boundary conditions,” Dokl. Mat., 90, No. 1, 489–494 (2014).
M. N. Zubova and T. A. Shaposhnikova, “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,” Different. Equat., 47, No. 1, 78–90 (2011).
M. N. Zubova and T. A. Shaposhnikova, “Homogenization of the boundary-value problem for the Laplace operator in a perforated domain with a rapidly oscillating nonhomogeneous Robin-type condition on the boundary of holes in the critical case,” Dokl. Mat., 96, No. 1, 344–347 (2017).
L. A. Khilkova, “Averaging of the diffusion equation in domains with fine-grained boundary with Robin-type nonlinear boundary condition,” Visn. Kharkiv. Nats. Univ., Ser. Mat. Prykl. Mat. Mekh., 84, 93–111 (2016).
E. Ya. Khuslov and L. A. Khilkova, “Robin nonlinear problem in domains with fine-grained random boundaries,” Dop. Nats. Akad. Nauk Ukr., No. 9, 3–8 (2017).
A. N. Shiryaev, Probability-1 [in Russian], MTSNMO, Moscow (2004).
N. N. Bogolyubov, Selected Works [in Russian], Vol. 2, Naukova Dumka, Kiev (2004).
I. I. Gikhman, A.V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics [in Russian], Vyshcha Shkola, Kiev (1973).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 5, pp. 692–705, May, 2019.
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Khruslov, E.Y., Khilkova, L.O. Model of Stationary Diffusion with Absorption In Domains With Fine-Grained Random Boundaries. Ukr Math J 71, 792–807 (2019). https://doi.org/10.1007/s11253-019-01677-w
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DOI: https://doi.org/10.1007/s11253-019-01677-w