A Unified Homogenization Approach for the Dirichlet Problem in Perforated Domains

Wenjia Jing Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, People's Republic ofChina

Analysis of PDEs mathscidoc:2206.03014

SIAM Journal on Mathematical Analysis, 52, (2), 1192-1220, 2020.3
We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains and establish a unified proof that works for different regimes of hole-cell ratios, which is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and yields correctors and error estimates for vanishing hole-cell ratios. For a positive volume fraction of holes, the approach is just the standard oscillating test function method; for a vanishing volume fraction of holes, we study asymptotic behaviors of properly rescaled cell problems and use them to build oscillating test functions. Our method reveals how the different regimes are intrinsically connected through the cell problems and the connection with periodic layer potentials.
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@inproceedings{wenjia2020a,
  title={A Unified Homogenization Approach for the Dirichlet Problem in Perforated Domains},
  author={Wenjia Jing},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626165750016269468},
  booktitle={SIAM Journal on Mathematical Analysis},
  volume={52},
  number={2},
  pages={1192-1220},
  year={2020},
}
Wenjia Jing. A Unified Homogenization Approach for the Dirichlet Problem in Perforated Domains. 2020. Vol. 52. In SIAM Journal on Mathematical Analysis. pp.1192-1220. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626165750016269468.
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