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Uniform Estimates for Dirichlet Problems in Perforated Domains

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Abstract

This paper studies the Dirichlet problem for Laplace’s equation in a domain \(\varOmega _{\varepsilon , \eta }\) 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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Acknowledgements

Supported in part by NSF grants DMS-1856235, DMS-2153585, and by Simons Fellowship. The author thanks the anonymous referees for helpful comments and corrections.

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  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    Zhongwei Shen

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  1. Zhongwei Shen
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Correspondence to Zhongwei Shen.

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Dedicated to my teacher Professor Carlos E. Kenig on the occasion of his 70th birthday.

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Shen, Z. Uniform Estimates for Dirichlet Problems in Perforated Domains. Vietnam J. Math. 51, 845–867 (2023). https://doi.org/10.1007/s10013-023-00613-7

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