Homogenization of a non-periodic material. (English) Zbl 0835.35016

Summary: We study the homogenization of a non-periodic material of \(\mathbb{R}^N\) in the conductivity case. This material consists of spherical balls of radius \(\varepsilon\) (\(\varepsilon\) is a parameter which will converge to \(0\)) and centered at the points \(\theta(k\varepsilon),k\in \mathbb{Z}^N\), where \(\theta\) is a diffeomorphism of \(\mathbb{R}^N\).
We obtain homogenized material whose coefficients depend on \(\nabla \theta\circ \theta^{- 1}\). We prove the result by comparing this material to a material which is periodic in domains of size \(\varepsilon^\gamma\), \(1/2< \gamma< 1\).


35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J20 Variational methods for second-order elliptic equations