Diffusion in a highly heterogeneous thin domain

Abstract

The paper considers a diffusion equation in a thin domain of thickness ε in $\mathbb{R}^{n}$ whose diffusivity varies periodically with a period cell of size ε^p (p>0) in the cross section. The domain is highly heterogeneous so that the diffusivity is of the scale ε^α (α>0) in a part of the period cell and is of order 1 in the rest. The asymptotic behaviour of the solution when ε→0 is studied for all the values of p and α. In some cases, the limiting equation is in $\mathbb{R}^{n-1}$ as in the previous works on diffusion in thin domains but in some others, an equation in $\mathbb{R}^{n}$ is obtained.