Homogenization of High-Conductivity Periodic Problems: Application to a General Distribution of One-Directional Fibers
Marc Briane · SIAM Journal on Mathematical Analysis · 2003
This article is devoted to the asymptotic study, as $\varepsilon\to 0$, of the Dirichlet problem \[ \left\{\begin{array}{@{}rll} -\,\mbox{div}\left(A_\varepsilon({\textstyle{x\over\varepsilon}}) abla u_\varepsilon\right) & \kern -.5em=f & \mbox{in }\Omega, \\*[.4em] u_\varepsilon & \kern -.5em=0 & \mbox{on }\partial\Omega, \end{array} \right. \] where $\Omega$ is an x 3 -axis bounded open cylinder of ${\mathbb R}^3$, and $A_\varepsilon$ is a positive measurable function which does not depend on the variable x 3 , periodic with respect to the two-dimensional torus Y 2 . The conductivity $A_\varepsilon$ is not uniformly bounded in an open set of small measure $Q_\varepsilon\subset Y 2 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 $A_\varepsilon$: \[ -\,\mbox{div}\left(A_\varepsilon abla V_{k,\varepsilon}\right) =\Lambda_k(\varepsilon)\,A_\varepsilon\,V_{k,\varepsilon}\quad\mbox{in }Y_2,\quad k\in{\mathbb N}, \] where the eigenfunctions $V_{k,\varepsilon}$ are $Y_2$-periodic. On the one hand, under suitable conditions on $Q_\varepsilon$ we prove that nonlocal effects appear through a coupling in the limit problem if and only if the sequence $({\Lambda_1(\varepsilon)\over\varepsilon^2})_{\varepsilon > 0}$ is bounded, where $\Lambda_1(\varepsilon)$ is the first nonzero eigenvalue of the previous spectral problem. On the other hand, when $Q_\varepsilon$ is composed of N smooth connected open subsets of small diameter, we prove that the limit problem is a coupled system of second order linear PDEs whose size is $n\leq N+1$. The number n is equal to the smallest integer such that the sequence $({\Lambda_n(\varepsilon)\over\varepsilon^2})_{\varepsilon > 0}$ tends to $+\infty$ as $\varepsilon$ tends to 0. We illustrate this result by studying the case of N=2 highly conducting cylinders in the period cell of the same radius $r_\varepsilon \ll 1$ and separated by distance $d_\varepsilon>0$.