Homogenization of randomly deformed conductivity resistant membranes

Wenjia Jing Department of Mathematics, University of Chicago, Illinois, U.S.A.

Analysis of PDEs mathscidoc:2206.03011

Communications in Mathematical Sciences, 14, (5), 1237-1268, 2016.5
We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris, and Lions (2006). Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin-type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that of an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole space with infinitely many interfaces, a setting that is out of the standard stationary ergodic framework.
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@inproceedings{wenjia2016homogenization,
  title={Homogenization of randomly deformed conductivity resistant membranes},
  author={Wenjia Jing},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626163952408726462},
  booktitle={Communications in Mathematical Sciences},
  volume={14},
  number={5},
  pages={1237-1268},
  year={2016},
}
Wenjia Jing. Homogenization of randomly deformed conductivity resistant membranes. 2016. Vol. 14. In Communications in Mathematical Sciences. pp.1237-1268. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626163952408726462.
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