Stochastic homogenization on perforated domains Ⅰ – Extension Operators

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 $W^{1,p}$-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimally smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from $W^{1,p}$ to $W^{1,r}$ with the strict inequality $r<p$. In particular, we estimate the $L^r$-norm of the extended gradient in terms of the $L^p$-norm of the original gradient. Similar relations hold for the symmetric gradients (for ${\mathbb{R}}^d$-valued functions) and for traces on the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of the same spirit.

Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions: local $(\delta ,M)$-regularity to quantify statistically the local Lipschitz regularity and isotropic cone mixing to quantify the density of the geometry and the mesoscopic properties. These two properties are sufficient to reduce the problem of extension operators to the 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, for which we can explicitly estimate the connectivity terms.