A unified homogenization approach for the Dirichlet problem in Perforated Domains

We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains, and establish a unified proof that works for different regimes of hole-cell ratios, that is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and it yields correctors and error estimates for vanishing hole-cell ratios. For positive volume fraction of holes, the approach is just the standard oscillating test function method; for vanishing volume fraction of holes, we study asymptotic behaviors of a properly rescaled cell problems and use them to build oscillating test functions. Our method reveals how the different regimes are intrinsically connected through the cell problems and the connection with periodic layer potentials.