Uniform Estimates for Dirichlet Problems in Perforated Domains

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This paper studies the Dirichlet problem for Laplace’s equation in a domain Ωε,η perforated with small holes, where ε represents the scale of the minimal distances between holes and η the ratio between the scale of sizes of holes and ε . We establish W1,p estimates for solutions with bounding constants depending explicitly on the small parameters ε and η . We also show that these estimates are either optimal or near optimal.

Original languageEnglish
JournalVietnam Journal of Mathematics
StateAccepted/In press - 2023

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2023, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.


  • Homogenization
  • Laplace’s equation
  • Perforated domain
  • W estimate

ASJC Scopus subject areas

  • Mathematics (all)


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