Stochastic homogenization on perforated domains I -- Extension operators

TL;DR

This paper establishes the existence of bounded extension and trace operators for Sobolev functions on complex, randomly perforated domains with minimal geometric assumptions, advancing stochastic homogenization techniques.

Contribution

It introduces new methods for constructing extension operators under weak geometric conditions, allowing for unbounded Lipschitz constants and percolating holes, broadening applicability.

Findings

Extension operators exist under weak geometric assumptions.

Applicable to Boolean and Delaunay pipe models.

Method handles unbounded Lipschitz constants and percolating holes.

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 (\delta,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. 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.