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 ε, describing both nonperiodic homogenization and reduction of dimension. In other words, we give the limit behaviour, when ε tends to zero, of the diffusion equation in a thin domain, with thickness of order ε, when the coefficients of the equation also depend on ε and may present rapid, nonperiodic oscillations, provided they satisfy a suitable compensated compactness condition. We consider two kinds of reduction of dimension: the case of thin plates (3D → 2D) and the case of thin cylinders (3D → 1D). In particular, we give the limit diffusion equation for laminated plates. This is completely explicit and requires no special assumption, except stratification. In the case of thin cylinders, the formulae are less explicit, but we also indicate some simple applications.