Stochastic homogenization on perforated domains III–General estimates for stationary ergodic random connected Lipschitz domains

This is Part III of a series on the existence of uniformly bounded extension operators on randomly perforated domains in the context of homogenization theory. Recalling that randomly perforated domains are typically not John and hence extension is possible only from $W^{1,p}$ to $W^{1,r}$, $r<p$, we will show that the existence of such extension operators can be guaranteed if the weighted expectations of four geometric characterizing parameters are bounded: The local Lipschitz constant $M$, the local inverse Lipschitz radius ${\delta }^{-1}$ resp. ${\rho }^{-1}$, the mesoscopic Voronoi diameter $\mathfrak{d}$ and the local connectivity radius $\mathcal{R}$.