Homogenization and correctors for composite media with coated and highly anisotropic fibers

This article presents the homogenization of a quasilinear elliptic-parabolic problem in an $varepsilon$-periodic medium consisting of a set of highly anisotropic fibers surrounded by coating layers, the whole being embedded in a third material having an order 1 conductivity. The conductivity along the fibers is of order 1 but the conductivities in the transverse directions and in the coatings are scaled by $mu=o(varepsilon)$ and $varepsilon^p$, as $varepsilon o 0$, respectively. The heat flux are quasilinear, monotone functions of the temperature gradient. The heat capacities of the medium components are bounded but may vanish on certain subdomains, so the problem may become degenerate. By using the two-scale convergence method, we can derive the two-scale homogenized systems and prove some corrector-type results depending on the critical value $gamma=lim_{varepsilonsearrow 0}varepsilon^p/mu$.