Uniform W 1,p estimates for systems of linear elasticity in a periodic medium

Jun Geng; Zhongwei Shen; Liang Song

doi:10.1016/j.jfa.2011.11.023

Uniform W ^1,p estimates for systems of linear elasticity in a periodic medium

Jun Geng, Zhongwei Shen, Liang Song

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

28 Scopus citations

Abstract

Let {Lε} be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform W ^1,p estimate {norm of matrix}∇;u _ε{norm of matrix} _p≤C{norm of matrix}f{norm of matrix} _p in a Lipschitz domain Ω in Rn for solutions to the Dirichlet problem: Lε(uε)=div(f) in Ω and u _ε=0 on ∂Ω, where |1p-12|<12n+δ and C, δ>0 are constants independent of ε>0. The ranges are sharp for n=2 or 3.

Original language	English
Pages (from-to)	1742-1758
Number of pages	17
Journal	Journal of Functional Analysis
Volume	262
Issue number	4
DOIs	https://doi.org/10.1016/j.jfa.2011.11.023
State	Published - Feb 15 2012

Bibliographical note

Funding Information:
Jun Geng and Zhongwei Shen were supported in part by NSF Grant DMS-0855294. Liang Song was supported in part by NNSF of China (Grant No. 11001276).

Keywords

Homogenization
Lipschitz domain
System of elasticity

ASJC Scopus subject areas

Analysis

Access to Document

10.1016/j.jfa.2011.11.023

Cite this

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title = "Uniform W 1,p estimates for systems of linear elasticity in a periodic medium",

abstract = "Let {Lε} be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform W 1,p estimate {norm of matrix}∇;u ε{norm of matrix} p≤C{norm of matrix}f{norm of matrix} p in a Lipschitz domain Ω in Rn for solutions to the Dirichlet problem: Lε(uε)=div(f) in Ω and u ε=0 on ∂Ω, where |1p-12|<12n+δ and C, δ>0 are constants independent of ε>0. The ranges are sharp for n=2 or 3.",

keywords = "Homogenization, Lipschitz domain, System of elasticity",

author = "Jun Geng and Zhongwei Shen and Liang Song",

note = "Funding Information: Jun Geng and Zhongwei Shen were supported in part by NSF Grant DMS-0855294. Liang Song was supported in part by NNSF of China (Grant No. 11001276).",

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T1 - Uniform W 1,p estimates for systems of linear elasticity in a periodic medium

AU - Geng, Jun

AU - Shen, Zhongwei

AU - Song, Liang

N1 - Funding Information: Jun Geng and Zhongwei Shen were supported in part by NSF Grant DMS-0855294. Liang Song was supported in part by NNSF of China (Grant No. 11001276).

PY - 2012/2/15

Y1 - 2012/2/15

N2 - Let {Lε} be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform W 1,p estimate {norm of matrix}∇;u ε{norm of matrix} p≤C{norm of matrix}f{norm of matrix} p in a Lipschitz domain Ω in Rn for solutions to the Dirichlet problem: Lε(uε)=div(f) in Ω and u ε=0 on ∂Ω, where |1p-12|<12n+δ and C, δ>0 are constants independent of ε>0. The ranges are sharp for n=2 or 3.

AB - Let {Lε} be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform W 1,p estimate {norm of matrix}∇;u ε{norm of matrix} p≤C{norm of matrix}f{norm of matrix} p in a Lipschitz domain Ω in Rn for solutions to the Dirichlet problem: Lε(uε)=div(f) in Ω and u ε=0 on ∂Ω, where |1p-12|<12n+δ and C, δ>0 are constants independent of ε>0. The ranges are sharp for n=2 or 3.

KW - Homogenization

KW - Lipschitz domain

KW - System of elasticity

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DO - 10.1016/j.jfa.2011.11.023

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

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