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 period- ically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter , and the level of anisotropy of the cell is determined by a diagonal matrix with positive diagonal entries. The relative size of each periodic perforation is instead determined by a positive parameter . For a given value ̃ of , we analyze the behavior of the unique solution of the problem as (, , ) tends to (0, 0, ̃ ) by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory.