Homogenization of High-Conductivity Periodic Problems: Application to a General Distribution of One-Directional Fibers

This article is devoted to the asymptotic study, as $\varepsilon\to 0$, of the Dirichlet problem \[ \left\{\begin{array}{@{}rll} -\,\mbox{div}\left(A_\varepsilon({\textstyle{x\over\varepsilon}})\nabla u_\varepsilon\right) & \kern -.5em=f & \mbox{in }\Omega, \\*[.4em] u_\varepsilon & \kern -.5em=0 & \mbox{on }\partial\Omega, \end{array} \right. \] where $\Omega$ is an x3 -axis bounded open cylinder of ${\mathbb R}^3$, and $A_\varepsilon$ is a positive measurable function which does not depend on the variable x3 , periodic with respect to the two-dimensional torus Y2 . The conductivity $A_\varepsilon$ is not uniformly bounded in an open set of small measure $Q_\varepsilon\subset Y2 and is equal to 1 elsewhere.We propose a new approach to solving this high-conductivity homogenization problem. It is based on the study of the asymptotic behavior of the periodic spectral problem weighted by the conductivity function $A_\varepsilon$: \[ -\,\mbox{div}\left(A_\varepsilon\nabla V_{k,\varepsilon}\right) =\Lambda_k(\...