Uniform Estimates for Dirichlet Problems in Perforated Domains

Zhongwei Shen

doi:10.1007/s10013-023-00613-7

Uniform Estimates for Dirichlet Problems in Perforated Domains

Zhongwei Shen

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

This paper studies the Dirichlet problem for Laplace’s equation in a domain Ω_ε_,_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 language	English
Journal	Vietnam Journal of Mathematics
DOIs	https://doi.org/10.1007/s10013-023-00613-7
State	Accepted/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.

Keywords

Homogenization
Laplace’s equation
Perforated domain
W estimate

ASJC Scopus subject areas

Mathematics (all)

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10.1007/s10013-023-00613-7

Cite this

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title = "Uniform Estimates for Dirichlet Problems in Perforated Domains",

abstract = "This paper studies the Dirichlet problem for Laplace{\textquoteright}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.",

keywords = "Homogenization, Laplace{\textquoteright}s equation, Perforated domain, W estimate",

author = "Zhongwei Shen",

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: {\textcopyright} 2023, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.",

year = "2023",

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language = "English",

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T1 - Uniform Estimates for Dirichlet Problems in Perforated Domains

AU - Shen, Zhongwei

N1 - 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.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - Homogenization

KW - Laplace’s equation

KW - Perforated domain

KW - W estimate

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DO - 10.1007/s10013-023-00613-7

M3 - Article

AN - SCOPUS:85159326867

SN - 2305-221X

JO - Vietnam Journal of Mathematics

JF - Vietnam Journal of Mathematics

ER -

Uniform Estimates for Dirichlet Problems in Perforated Domains

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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