Homogenization in perforated domains at the critical scale

We describe the asymptotic behaviour of the minimal heterogeneous d-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set Omega subset of R-d, with d >= 2. Two parameters are involved: epsilon, the radius of the ball, and delta, the length scale of the heterogeneity of the medium. We prove that this capacity behaves as C vertical bar log epsilon vertical bar(1-d), where C = C(lambda) is an explicit constant depending on the parameter lambda := lim(epsilon -> 0) vertical bar log delta vertical bar/vertical bar log epsilon vertical bar. We determine the Gamma-limit of oscillating integral functionals subjected to Dirichlet boundary conditions on periodically perforated domains. Our first result is used to study the behaviour of the functionals near the perforations which, in this instance, are balls of radius epsilon. We prove that an additional strange term arises involving C(lambda).

Automated summary

We investigate the asymptotic behavior of the minimal heterogeneous d-capacity of a small set in a fixed bounded open set Omega. We prove that this capacity is related to the parameter lambda and behaves as C |log epsilon|^(1-d), where C is a constant.