Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains

Abstract

We consider quasilinear and linear boundary-value problems for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of order 𝒪(ε). The holes are divided into three ε-periodical sets depending on boundary conditions on their surfaces. On the boundaries of holes from one set we have the homogeneous Dirichlet conditions. On the boundaries of holes from the other sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For the solution to the quasilinear problem we find the leading terms of the asymptotics and prove the corresponding asymptotic estimates that show influence of the perturbation parameters. In the linear case we construct and justify the complete asymptotic expansion for the solution using two-scale asymptotic expansion method.