Homogenization of A Boundary-Value Problem in A Domain Perforated by Cavities of Arbitrary Shape with A General Nonlinear Boundary Condition On Their Boundaries: The Case of Critical Values of the Parameters

Abstract

A homogenized model is constructed (with rigorous justification) for a boundary-value problem for the Poisson equation in a periodically perforated domain with a nonlinear Robin condition on the boundary of the cavities. This condition contains a parameter depending on the period of the structure and a function σ(x, u) responsible for the nonlinearity. The cavities can have an arbitrary shape and the parameters of the problem have “critical values,” which results in a homogenized problem with a different type of nonlinearity.