Two†parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. A functional analytic approach

We consider a Dirichlet 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, and the level of anisotropy of the cell is determined by a diagonal matrix γ with positive diagonal entries. The relative size of each periodic perforation is instead determined by a positive parameter ε. For a given value γ∼ of γ, we analyze the behavior of the unique solution of the problem as (ε,δ,γ) tends to (0,0,γ∼) by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory.