Homogenization of variational inequalities for non-linear diffusion problems in perforated domains

We consider the homogenization of non-linear diffusion problems with various boundary conditions in periodically perforated domains. These problems are stated as variational inequalities defined by non-linear strictly monotone operators of second order with periodic rapidly oscillating coefficients. We establish the relevant convergence of solutions of the problems to solutions of two-scale and macroscale limiting variational inequalities. We give methods for deriving such limiting variational inequalities. In the case of potential operators, we establish relations between the limiting variational inequalities obtained and the two-scale and macroscale constrained minimization problems.