Homogenization of Randomly Deformed Conductivity Resistant Membranes

TL;DR

This paper investigates the homogenization of a randomly deformed, heterogeneous conductivity medium with membranes, deriving an effective deterministic model and explicitly representing the effective coefficients.

Contribution

It introduces a novel homogenization approach for media with randomly deformed membranes, providing explicit formulas for the effective coefficients and solving an auxiliary problem on an infinite interface domain.

Findings

The random conductivity problem is approximated by a deterministic effective medium.

Explicit representation of the effective coefficients is obtained.

The auxiliary problem on an infinite domain with interfaces is solved successfully.

Abstract

We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris and Lions \cite{BLBL06}. Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that on an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole domain with infinitely many interfaces, in a setting that is neither periodic nor stationary ergodic in the usual sense.