  The sigma-convergence concept has been up to now used to derive macroscopic
models in full space dimensions. In this work, we generalize it to thin
heterogeneous domains given rise to phenomena in lower space dimensions. More
precisely, we provide a new approach of the sigma-convergence method that is
suitable for the study of phenomena occurring in thin heterogeneous media. This
is made through a systematic study of the sigma-convergence method for thin
heterogeneous domains. Assuming that the thin heterogeneous layer is made of
microstructures that are distributed inside in a deterministic way including as
special cases the periodic and the almost periodic distributions, we make use
of the concept of algebras with mean value to state and prove the main
compactness results. As an illustration, we upscale a Darcy-Lapwood-Brinkmann
micro-model for thin flow. We prove that, according to the magnitude of the
permeability of the porous domain, we obtain as effective models, the Darcy law
in lower dimensions. The effective models are derived through the solvability
of either the local Darcy-Brinkmann problems or the local Hele-Shaw problems.

Analysis of PDEs