Homogenization of an elliptic differential equation in a composite (multiperiodic) medium (Q1819284)

This paper considers homogenization of an elliptic partial differential equation in a media, consisting of two composites, having different periodic structures. A domain \(\Omega \subset {\mathbb{R}}^ n\) is split into two subdomains, \(\Omega_ 1\) and \(\Omega_ 2\), by closed (n-1)- dimensional closed surface \(\gamma\). The author considers the following Dirichlet problem: \[ (1)\quad L_ 1u^{\epsilon}=f_ 1\quad in\quad \Omega_ 1;\quad L_ 2u^{\epsilon}=f_ 2\quad in\quad \Omega_ 2;\quad [u^{\epsilon}]|_{\gamma}=[\partial u^{\epsilon}/\partial \nu]|_{\gamma}=u^{\epsilon}|_{\partial \Omega}=0. \] The solution to the homogenization of (1) is constructed, and a ''closeness'' estimate between the solution \(u^{\epsilon}\) and the homogenized solution is proved.