Uniform W1,p estimates and large-scale regularity for Dirichlet problems in perforated domains

Zhongwei Shen; Jamison Wallace

doi:10.1016/j.jfa.2023.110118

Uniform W^1,p estimates and large-scale regularity for Dirichlet problems in perforated domains

Zhongwei Shen, Jamison Wallace

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

3 Scopus citations

Abstract

In this paper we study the Dirichlet problem for Laplace's equation in a domain ω_ε,η perforated periodically with small holes in R^d, 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 W^1,p estimates for solutions with bounding constants depending explicitly on ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.

Original language	English
Article number	110118
Journal	Journal of Functional Analysis
Volume	285
Issue number	10
DOIs	https://doi.org/10.1016/j.jfa.2023.110118
State	Published - Nov 15 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

Homogenization
Large-scale regularity
Perforated domain
Uniform estimates

ASJC Scopus subject areas

Analysis

Access to Document

10.1016/j.jfa.2023.110118

Cite this

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abstract = "In this paper we study the Dirichlet problem for Laplace's equation in a domain ωε,η perforated periodically with small holes in Rd, 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 ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.",

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AU - Wallace, Jamison

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N2 - In this paper we study the Dirichlet problem for Laplace's equation in a domain ωε,η perforated periodically with small holes in Rd, 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 ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.

AB - In this paper we study the Dirichlet problem for Laplace's equation in a domain ωε,η perforated periodically with small holes in Rd, 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 ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.

KW - Homogenization

KW - Large-scale regularity

KW - Perforated domain

KW - Uniform estimates

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