Stochastic homogenization on randomly perforated domains

Martin Heida

Published 2020-01-28Version 1

We study the existence of uniformly bounded extension and trace operators for $W^{1,p}$-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions on the geometry which we call local $(\delta,M)$-regularity and isotropic cone mixing. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. In particular we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. Our conditions on the geometry are such that they can partially be check by rigorous mathematics applied to specific geometric models. What is more important is that they are suited to be verified by computer algorithms. This is relevant in order to check whether a given real world medium is proper for rigorous upscaling of a given microscopic problem. A question which we partially leave unanswered is a suitable characterization of connectedness. We provide a tentative solution which leaves space for improvements.