Homogenization of singularly perturbed Dirichlet problems in perforated domains

Abstract

We study the singularly perturbed problem —ε^αΔu^ε + u^ε = f (α > 0) with the Dirichlet boundary condition in a perforated domain, in which the holes are distributed periodically with period 2ε throughout a fixed domain Ω. The asymptotic behaviour of u^ε when ε → 0, together with corrector results and error estimates in L²(Ω), are deduced for all sizes of holes. The behaviour of u^ε in is obtained for the cases where the size of holes is of order ε or is of a sufficiently smaller order. When the holes' size is of a sufficiently small order, as expected, u^ε has similar behaviour to that in the case of a non-varying domain.