Effective membrane permeability: Estimates and low concentration asymptotics

The paper deals with a mathematical model of a steady-state diffusion process through a periodic membrane. For a wide class of periodic membranes, we define the effective permeability and obtain upper and lower estimates of the effective permeability. For periodic membranes made from two materials with different absorbing properties, we study the asymptotic behavior of the effective permeability when the fraction of one material tends to zero (low concentration asymptotics). When the low fraction material forms homothetically vanishing disperse periodic inclusions in the host material, low concentration approximations are built by the method of matched asymptotic expansions. We also show that our results are consistent with those which can be obtained by a boundary homogenization. Finally, we analyze formulas used in physical, chemical, and biological investigations to describe effective membrane properties.