Averaging of boundary-value problem in domain perforated along (n − 1)-dimensional manifold with nonlinear third type boundary conditions on the boundary of cavities

Abstract

The research considers the asymptotic behavior of solutions u _ɛ of the Poisson equation in a domain ɛ-periodically perforated along manifold γ = ω ∩ {x ₁ = 0} ≠ Ø with a nonlinear third type boundary condition ∂ _v u _ɛ + ɛ^−ασ(x, u _ɛ) = 0 on the boundary of the cavities. It is supposed that the perforations are balls of radius C ₀ɛ^α, C ₀ > 0, α = n − 1 / n − 2, n ≥ 3, periodically distributed along the manifold γ with period ɛ > 0. It has been shown that as ɛ → 0 the microscopic solutions can be approximated by the solution of an effective problem which contains in a transmission conditions a new nonlinear term representing the macroscopic contribution of the processes on the boundary of the microscopic cavities. This effect was first noticed in [1] where the similar problem was investigated for n = 3 and for the case where Ω is a domain periodically perforated over the whole volume. This paper provides a new method for the proof of the convergence of the solutions {u _ɛ} to the solution of the effective problem is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and more accurate results are obtained with respect to the energy norm proved via a corrector term. Note that this approach can be generalized to achieve results for perforations of more complex geometry.