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

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 $\sigma(x, u)$ 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.