Stochastic Homogenization on Irregularly Perforated Domains

Abstract

We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.

Homogenization of the linearized ionic transport equations in random porous media

“…Due to this reason, in order to make the presentation clear, we preferred to consider random disperse media that admit deterministic estimates of their geometric characteristics, such as the distance between inclusions and their diameters. We strongly believe that, using the approaches developed in recent articles [25][26][27] (see also the earlier pioneer work [23]), one can consider a wider class of random geometries and generalize the results of this work to the case of more general random perforated domains.…”

“…we recall that z 1 and z N are valences that satisfy relations (27). Inserting this particular test function into equation ( 48) and using that…”