Homogenization of the diffusion equation in domains with the fine-grained boundary with the nonlinear boundary Robin condition

  • Л. О. Хількова Институт химических технологий Восточноукраинского национального университета им. В.Даля
DOI: https://doi.org/10.26565/2221-5646-2016-84-07

Abstract

In this paper we consider the boundary-value problem for the stationary diffusion equation in perforated domains, which are additional of a large number non-overlapping small balls on the surface of which is given the nonlinear Robin condition. We study the asymptotic behavior of the solution of the problem. We derive homogenization equations describing the principal term of the asymptotic of the solutions.

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References

Берлянд Л.В., Гончаренко М.В. Осреднение уравнения диффузии в пористой среде со слабым поглощением // Теория функций, функциональный анализ и их приложения, 1989.-- № 52.-- С. 113-- 122.

Cabarrubias B., Donato P. Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary condition // Applicable Analysis: An International Journal, 2012.-- Vol. 91, №6.-- P. 1111-- 1127.

Chourabi I., Donato P. Homogenization and correctors of a class of elliptic problems in perforated domains // Asymptotic Analysis, 2015.-- № 92, P. 1-- 43.

Chourabi I., Donato P. Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains // Chinese Annals of Mathematics, 2016.-- Vol. 37B, №6.-- P. 833-- 852.

Cioranescu D., Donato P. On Robin problems in perforated domains //Mathematical sciences and applications, 1997.-- № 9.-- P. 123-- 135.

Cioranescu D., Donato P., Zaki R. Asymptotic behaviour of elliptic problems in perforated domains with nonlinear boundary conditions // Asymptotic Analysis, 2007.-- № 53.-- P. 209-- 235.


Conca C., Diaz J., Linan A., Timofte C. Homogenization in chemical reactive floes // Electronic Journal of Differential Equations, 2004.-- № 40.-- P. 1-- 22.

Conca C., Diaz J., Linan A., Timofte C. Homogenization results for chemical reactive flows through porous media // New Trends in Continuum Mechanics, 2005.-- № 6.-- P. 99~-- 107.

Conca C., Diaz J., Timofte C. On the homogenization of a transmission problem arising in chemistry // Romanian Reports in Physics, 2004.-- Vol. 5, №4.-- P. 613-- 622.

Goncharenko M. The asymptotic behaviour of the third boundary-value problem solutions in domains with fine-grained boundaries // Mathematical sciences and applications, 1997.-- № 9.-- P. 203-- 213.

Mel'nyk T.A., Sivak O.A. Asymptotic analysis of a boundary-value problem with the nonlinean multiphase interactions in a perforated domain // Ukrainian Mathematical Journal, 2009.-- Vol.61, № 4.-- P. 494-- 512.

Mel'nyk T.A., Sivak O.A. Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains // Asymptotic Analysis, 2011.-- № 5.-- P. 79-- 92.

Timofte C. On the homogenization of a climatization problem // Studia Universitatis <>, Mathematica, 2007.-- Vol.LII, № 2.-- P. 117-- 125.

Timofte C., Cotfas N., Pavel G. On the asymptotic behaviour of some elliptic problems in perforated domains // Romanian Reports in Physics, 2012.~-- Vol.64, № 1.-- P. 5-- 14.

Berlyand L.V., Khruslov E.Ya. Competition between the surface and the boundary layer energies in a Ginzburg-Landau model of a liquid crystal composite // Asymptotic Analysis, 2002.-- № 29.-- P. 185~-- 219.

Марченко В.А., Хруслов Е.Я. Усреднённые модели микронеоднородных сред.-- К: Наукова думка, 2005.-- 551 с.

Cabarrubias B., Donato P. Existence and Uniqueness for a Quasilinear Elliptic Problem With Nonlinear Robin Condition // Carpathian Journal of Mathematics, 2011.~-- Vol.27, № 2.-- P. 173-- 184.

Хруслов Е.Я. Асимптотическое поведение решений второй краевой задачи при измельчении границы области // Математический сборник, 1978.~-- Т.106(148), № 4(8)}.-- С. 604-- 621.

Владимиров В.С. Обобщённые функции в математической физике.-- М.: Наука, 1979.-- 320 c.
Published
2016-12-15
Cited
How to Cite
Хількова, Л. О. (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. Mathematics, Applied Mathematics and Mechanics, 84, 93-111. https://doi.org/10.26565/2221-5646-2016-84-07
Issue
Vol. 84 (2016)
Section
Статті

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