Uniform Estimates for Dirichlet Problems in Perforated Domains

Zhongwei Shen

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arXiv · 2208.12198

Uniform Estimates for Dirichlet Problems in Perforated Domains

Zhongwei Shen

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Abstract

This paper studies the Dirichlet problem for Laplace's equation in a domain $\Omega_{\varepsilon, \eta}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $\eta$ the ratio between the scale of sizes of holes and $\varepsilon$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on the small parameters $\varepsilon$ and $\eta$. We also show that these estimates are either optimal or near optimal.

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Zhongwei Shen. 2022-08-25. Uniform Estimates for Dirichlet Problems in Perforated Domains. https://arxiv.org/abs/2208.12198

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