Diffusion through a composite medium with a semi-periodic structure and high contrasting diffusivity

Abstract

We address the homogenization of the stationary diffusion equation in a composite medium with two components ${\mathit{M}}_{\mathit{\epsilon }}$ and ${\mathit{B}}_{\mathit{\epsilon }}$ having respectively ${\mathit{A}}^{\mathit{\epsilon }}\mathrm{(}\mathit{x}\mathrm{)}$ and ${\mathit{\alpha }}_{\mathit{\epsilon }}{\mathit{A}}^{\mathit{\epsilon }}\mathrm{(}\mathit{x}\mathrm{)}$ as diffusivity coefficients. We assume periodic distribution with size ε of the “holes” ${\mathit{B}}_{\mathit{\epsilon }}$ but no periodicity is assumed on the matrices ${\mathit{A}}^{\mathit{\epsilon }}$ . The high contrast between the two components is characterized by the assumption that the sequence ${\mathit{\alpha }}_{\mathit{\epsilon }}$ decreases towards zero. We study the three regimes corresponding to the limits $\mathit{\alpha }:={lim}_{\mathit{\epsilon }\to 0}\frac{{\mathit{\alpha }}_{\mathit{\epsilon }}}{\mathit{\epsilon }}$ . It is shown in particular that in the case $\mathit{\alpha }=0$ , the inclusions ${\mathit{B}}_{\mathit{\epsilon }}$ behave as holes on the macroscopic diffusion process.