Abstract
In this paper, the asymptotic behavior of solutions u ε of the Poisson equation in the ε-periodically perforated domain Ωε ⊂ \( {{\mathbb{R}}^n} \) , n ≥ 3, with the third nonlinear boundary condition of the form ∂ ν u ε + ε−γσ(x, u ε) = ε −γ g(x) on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order εα with α > 1 and any γ. Here, all types of asymptotic behavior of solutions u ε , corresponding to different relations between parameters α and γ, are studied.
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 39, Partial Differential Equations, 2011.
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Zubova, M.N., Shaposhnikova, T.A. Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities. J Math Sci 190, 181–193 (2013). https://doi.org/10.1007/s10958-013-1253-5
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DOI: https://doi.org/10.1007/s10958-013-1253-5