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}} $.
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
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}} $.
| [1] | D. J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes, New York: Springer, 1988. |
| [2] | R. G. Durán, M. A. Muschietti, The Korn inequality for Jones domains, Electron. J. Differ. Equ., 2004 (2004), 1–10. |
| [3] | N. Guillen, I. Kim, Quasistatic droplets in randomly perforated domains, Arch Ration Mech Anal, 215 (2015), 211–281. |
| [4] |
M. Heida, An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1–30. https://doi.org/10.1097/INF.0b013e3181f1e704 doi: 10.1097/INF.0b013e3181f1e704
|
| [5] | M. Heida, Stochastic homogenization on perforated domains I: Extension operators, arXiv: 2105.10945, [Preprint], (2021) [cited 2023 June 06 ]. Available from: https://arXiv.org/abs/2105.10945 |
| [6] | M. Heida, Stochastic homogenization on perforated domains II–application to nonlinear elasticity models, Z Angew Math Mech, 102 (2022), e202100407. |
| [7] | M. Höpker, Extension Operators for Sobolev Spaces on Periodic Domains, Their Applications, and Homogenization of a Phase Field Model for Phase Transitions in Porous Media, (German), Doctoral Thesis of University Bremen, Bremen, 2016. |
| [8] |
P. W. Jones, Quasiconformal mappings and extendability of functions in sobolev spaces, Acta Math., 147 (1981), 71–88. https://doi.org/10.1007/BF02392869 doi: 10.1007/BF02392869
|
| [9] |
J. Mecke, Stationäre zufällige Maße auf lokalkompakten abelschen Gruppen, Probab Theory Relat, 9 (1967), 36–58. https://doi.org/10.1007/BF00535466 doi: 10.1007/BF00535466
|
| [10] |
A. Piatnitski, M. Ptashnyk, Homogenization of biomechanical models of plant tissues with randomly distributed cells, Nonlinearity, 33 (2020), 5510. https://doi.org/10.1088/1361-6544/ab95ab doi: 10.1088/1361-6544/ab95ab
|
| [11] | A.A. Tempel'man, Ergodic theorems for general dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 26 (1972), 95–132. |
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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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