On the homogenization of some linear problems in domains weakly connected by a system of traps

The aim of the paper is to study the asymptotic behaviour of solutions of second-order elliptic and parabolic equations, arising in modelling of flow in cavernous porous media, in a domain (omega)(epsilon)weakly connected by a system of traps P (epsilon), where (epsilon) is the parameter that characterizes the scale of the microstructure. Namely, we consider a strongly perforated domain (omega)(epsilon)(subset of)(omega)a bounded open set of R3 such that (omega)(epsilon)=(omega)1(epsilon) (union)(omega)2(epsilon) (union)P(epsilon)(union)W(epsilon), where (omega)1(epsilon), (omega)2(epsilon) are non-intersecting subdomains strongly connected with respect to (omega), P(epsilon)is a system of traps and meas W(epsilon) (right arrow)0 as (epsilon)(right arrow)0. Without any periodicity assumption, for a large range of perforated media and by means of variational homogenization, we find the homogenized models. The effective coefficients are described in terms of local energy characteristics of the domain (omega)(epsilon)associated with the problem under consideration. The resulting homogenized problem in the parabolic case is a vector model with memory terms. An example is presented to illustrate the methodology.