Homogenization of the Boundary Value Problem for the Poisson Equation with Rapidly Oscillating Nonlinear Boundary Conditions: Space Dimension n ≥ 3, Critical Case
Alexander Podolskiy, Т. А. Шапошникова · Doklady Mathematics · 2019
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.