Homogenization of the Boundary Value Problem for the Poisson Equation with Rapidly Oscillating Nonlinear Boundary Conditions: Space Dimension n ≥ 3, Critical Case

Abstract

The homogenization of the Poisson equation in a bounded domain with rapidly oscillating boundary conditions specified on a part of the domain boundary is studied. A Neumann boundary condition alternates with an ε-periodically distributed nonlinear Robin condition involving the coefficient \({{\varepsilon }^{{ - \beta }}}\), where \(\beta \in \mathbb{R}\). The diameter of the boundary portions with a nonlinear Robin condition is of order \(O({{\varepsilon }^{\alpha }}),\)\(\alpha > 1\). A critical relation between the parameters \(\alpha \) and \(\beta \) is considered.