Random homogenization and singular perturbations in perforated domains (Q5938864)

This paper deals with the singularly perturbed Dirichlet problem \[ -\varepsilon\Delta u^\varepsilon+u^\varepsilon=f, \] in a randomly perforated domain \(\Omega^\varepsilon\) obtained by perforating holes, whose size is of a positive order of \(\varepsilon\), from a bounded domain \(\Omega\) in \(\mathbb{R}^n\). Assuming \(f\in L^2(\Omega)\), the author studies the asymptotic behaviour when \(\varepsilon\to 0\) of the solution \(u^\varepsilon\). To this end the perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Here test functions and the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size \(\varepsilon^p\) with the intensity \(\lambda\cdot \varepsilon^{-r}\) is considered.