2021, Heida, Martin

This preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W1,p to W1,r with the strict inequality r

2020, Heida, Martin

We study the existence of uniformly bounded extension and trace operators for W1,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 (δ, M)-regularity, isotropic cone mixing and bounded average connectivity. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. The third is the most tricky part and measures the ''mesoscopic'' connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process. We finally introduce suitable Sobolev spaces on Rd and Ω in order to construct a stochastic two-scale convergence method and apply the resulting theory to the homogenization of a p-Laplace problem on a randomly perforated domain.