Geodesic
Integral conditions for convergence of solutions of non-linear Robin's problem in strongly perforated domain
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017) no. 3, pp. 283-313 
E. Ya. Khruslov; L. O. Khilkova; M. V. Goncharenko. Integral conditions for convergence of solutions of non-linear Robin's problem in strongly perforated domain. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017) no. 3, pp. 283-313. http://geodesic.mathdoc.fr/item/JMAG_2017_13_3_a4/
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     author = {E. Ya. Khruslov and L. O. Khilkova and M. V. Goncharenko},
     title = {Integral conditions for convergence of solutions of non-linear {Robin's} problem in strongly perforated domain},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {283--313},
     year = {2017},
     volume = {13},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2017_13_3_a4/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a boundary-value problem for the Poisson equation in a strongly perforated domain $\Omega^\varepsilon =\Omega\setminus F^\varepsilon \subset R^n$ ($n\geqslant 2$) with non-linear Robin's condition on the boundary of the perforating set $F^\varepsilon$. The domain $\Omega^\varepsilon$ depends on the small parameter $\varepsilon>0$ such that the set $F^\varepsilon$ becomes more and more loosened and distributes more densely in the domain $\Omega$ as $\varepsilon\to0$. We study the asymptotic behavior of the solution $u^\varepsilon(x)$ of the problem as $\varepsilon\to0$. A homogenized equation for the main term $u(x)$ of the asymptotics of $u^\varepsilon(x)$ is constructed and the integral conditions for the convergence of $u^\varepsilon(x)$ to $u(x)$ are formulated.

[1] B. Cabarrubias, P. Donato, “Homogenization of a Quasilinear Elliptic Problem with Nonlinear Robin Boundary Condition”, Appl. Anal.: An Intern. J., 91:6 (2012), 1111–1127 | DOI | MR | Zbl

[2] I. Chourabi, P. Donato, “Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains”, Chin. Ann. Math., 37B:6 (2016), 833–852 | DOI | MR | Zbl

[3] D. Cioranescu, P. Donato, “On Robin Problems in Perforated Domains”, Math. Sci. Appl., 9 (1997), 123–135 | MR

[4] D. Cioranescu, P. Donato, R. Zaki, “Asymptotic Behaviour of Elliptic Problems in Perforated Domains with Nonlinear Boundary Conditions”, Asymptot. Anal., 53 (2007), 209–235 | MR | Zbl

[5] C. Conca, J. Diaz, A. Linan, C. Timofte, “Homogenization in Chemical Reactive Floes”, Electron. J. Differential Equations, 40 (2004), 1–22 | DOI | MR

[6] C. Conca, J. Diaz, A. Linan, C. Timofte, “Homogenization Results for Chemical Reactive Flows Through Porous Media”, New trends in continuum mechanics, Theta Ser. Adv. Math., 3, Theta, Bucharest, 2005, 99–107 | MR | Zbl

[7] C. Conca, J. Diaz, C. Timofte, “On the Homogenization of a Transmission Problem Arising in Chemistry”, Romanian Rep. Phys., 56:4 (2004), 613–622 | MR

[8] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, 1998 | MR | Zbl

[9] Ukrainian Math. J., 67:9 (2016), 1349–1366 | DOI | MR | MR | Zbl

[10] M.V. Goncharenko, L.A. Khilkova, “Homogenized Model of Diffusion in a Locally-Periodic Porous Media with Nonlinear Absorption at the Boundary”, Dopov. Nats. Akad. Nauk Ukr., 10:6 (2016), 15–19 (Russian) | DOI | MR

[11] A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis, Fizmatlit, M., 2004 (Russian)

[12] V.A. Marchenko, E.Ya. Khruslov, Homogenization of Partial Differential Equations, Birkhöuser, Boston–Basel–Berlin, 2006 | MR | Zbl

[13] V.G. Maz'ya, Sobolev Spaces, Izdatel'stvo LGU, L., 1985 (Russian) | Zbl

[14] T.A. Mel'nyk, O.A. Sivak, “Asymptotic Analysis of a Boundary-Value Problem with the Nonlinean Multiphase Interactions in a Perforated Domain”, Ukrain. Mat. Zh., 61:4 (2009), 494–512 | MR | Zbl

[15] A.N. Tikhonov, A.A. Samarskiy, Equations of the Mathematical Physics, Nauka, M., 1972 (Russian) | MR

[16] C. Timofte, “On the Homogenization of a Climatization Problem”, Studia Univ. Babes-Bolyai Math, LII:2 (2007), 117–125 | MR | Zbl

[17] C. Timofte, N. Cotfas, G. Pavel, “On the Asymptotic Behaviour of Some Elliptic Problems in Perforated Domains”, Romanian Rep. Phys., 64:1 (2012), 5–14