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 $delta$. The relative size of each periodic perforation is instead determined by a positive parameter $epsilon$. Under suitable assumptions, such a problem admits of a family of solutions which depends on $epsilon$ and $delta$. We analyze the behavior the energy integral of such a family as $(epsilon,delta)$ tends to $(0,0 )$ by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory.