Correctors for Parabolic Equations in a Heterogeneous Fibered Medium

Sfaxi, Mourad ; Sili, Ali

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 1025-1053

Résumé

We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\Omega$ made of two materials. The first one is located in a set $F_\epsilon$ of cylindrical parallel fibers periodically distributed with a period of size $\epsilon$, and the second one is located in the "matrix" $M_\epsilon = \Omega \setminus F_\epsilon$. The ratio between the conductivity coefficients of the two materials is of order $1/\epsilon^2$. After writing the homogenized problem, we give a corrector result and prove that the solution ue of the starting problem is of the form $u_\epsilon = \tilde{u}_\epsilon + \hat{u}_\epsilon$, where $\tilde{u}_\epsilon$ is a corrector for $u_{\epsilon}$ and $\hat{u}_\epsilon$ is a time boundary layer. In contrast to the known results for parabolic equations, this boundary layer is not concentrated about the time origin $t = 0$, but it remains at least for all $t \in (0, m)$ with some $m > 0$. The proof of the latter is based on the fact that ue does not converge, in general, in $L^{2}(\Omega \times (0, T))$ for the strong topology.