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.
Bibliographical noteFunding 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.
© 2023, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
- Laplace’s equation
- Perforated domain
- W estimate
ASJC Scopus subject areas
- Mathematics (all)