Asymptotic Behaviour of the Energy Integral of a Two-Parameter Homogenization Problem with Nonlinear Periodic Robin Boundary Conditions

We consider a nonlinear Robin problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ. The relative size of each periodic perforation is determined by a positive parameter e. Under suitable assumptions, such a problem admits a family of solutions which depends on e and δ. We analyse the behaviour the energy integral of such a family as (e, δ) tends to (0, 0) by an approach that represents an alternative to asymptotic expansions and classical homogenization theory.