Homogenization of the Poisson equation in a non-periodically perforated domain

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of perforations is denoted by ε > 0, and proportional to distance between neighbouring perforations. In periodic case, homogenized problem (obtained limit → 0) well understood (see (Rocky Mountain J. Math. 10 (1980) 125–140)). extend these results non-periodic case which defined as localized deformation setting. propose geometric assumptions that make precise this setting, we prove those case: existence corrector, convergence problem, two-scale expansion.