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

Martin Heida; Martin Heida

doi:10.3934/nhm.2023062

Networks and Heterogeneous Media

2023, Volume 18, Issue 4: 1410-1433. doi: 10.3934/nhm.2023062

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Research article

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

Martin Heida ^,

Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr, Berlin, Germany

Received: 25 July 2022 Revised: 17 April 2023 Accepted: 27 April 2023 Published: 06 June 2023

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 ${\mathscr{R}}$ .

Keywords:

Citation: Martin Heida. Stochastic homogenization on perforated domains III–General estimates for stationary ergodic random connected Lipschitz domains[J]. Networks and Heterogeneous Media, 2023, 18(4): 1410-1433. doi: 10.3934/nhm.2023062

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Abstract

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