Stochastic homogenization on perforated domains I -- Extension operators

Abstract

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 (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.Acknowledgement. I thank B. Jahnel for his helpful suggestions on literature. Furthermore, I thank D.R.M. Renger and G. Friesecke for critical questions that pushed my research further. Finally I thank the DFG for funding my reasearch via CRC 1114 Project C05. This is Part I of a series of 3 papers that will be uploaded during the next weeks. It is part of a major revision of the original ArXiv 2001.10373. The original work was corrected, improved and extended significantly. In particular, the theory is now able to also deal with extensions that preserve the norm of the symmetrized gradient.

Stochastic Homogenization on Irregularly Perforated Domains

“…and because δ : ∂P → R ≥0 is continuous [Hei21], we can define for bounded P δ(P ) := min p∈∂P δ(p) .…”

“…The second issue poses an actual problem though. In a recent work [Hei21], one of the authors has shown that in some cases an extension operator U ε,• : W 1,p (Q ε ) → W 1,r (Q), 1 ≤ r < p, can be constructed for some geometries including the Boolean model (strictly speaking this was shown for an extension from the balls to the complement in the percolation case). However, [Hei21] also suggests that the Boolean model for the Poisson point process requires p > 2 for U ε,• to be properly defined for some r > 0.…”