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

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 190; no. 1; pp. 181 - 193
• Main Authors: Zubova, M. N., Shaposhnikova, T. A.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.04.2013
• Subjects: Mathematics; Mathematics and Statistics; Asymptotic Behavior; Integral Identity; Limit Behavior; Integral Relation; Poisson Equation
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-013-1253-5

In this paper, the asymptotic behavior of solutions u ε of the Poisson equation in the ε-periodically perforated domain Ω ε ⊂ , n ≥ 3 , with the third nonlinear boundary condition of the form ∂ ν u ε + ε −γ σ( 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 ε α 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.