Speaker
Description
This talk is devoted to the homogenization of elliptic problems posed in microstructured domains. After a general introduction to homogenization, I will present a brush-type geometric setting, consisting of a fixed lower part and a vertically oscillating microstructure in the upper part. I will first consider a prototype elliptic problem of Laplace type, in order to highlight the influence of the fine geometry on the homogenized limit problem. In particular, I will show that the oscillating structure in the upper region leads, in the limit, to a transmission problem coupling a classical diffusion equation in the lower part with an effective degenerate vertical diffusion in the upper part. I will then briefly discuss an extension to a nonlinear monotone framework with a source term in ($L^1$), where the low regularity of the data leads naturally to the notion of renormalized solution. The aim of the talk is to show how microscopic geometric complexity may generate nontrivial effective macroscopic models.