Non-periodic explicit homogenization and reduction of dimension: the linear case

The aim of this paper is to give explicit limit expressions,for diffusion equations involving a small parameter , describingboth nonperiodic homogenization and reduction of dimension.In other words, we give the limit behaviour, when tends tozero, of the diffusion equation in a thin domain, with thicknessof order , when the coefficients of the equation also dependon and may present rapid, nonperiodic oscillations, providedthey satisfy a suitable compensated compactness condition. Weconsider two kinds of reduction of dimension: the case of thinplates (3D 2D) and the case of thin cylinders (3D 1D). Inparticular, we give the limit diffusion equation for laminatedplates. This is completely explicit and requires no specialassumption, except stratification. In the case of thin cylinders,the formulae are less explicit, but we also indicate some simpleapplications.