Boundary Homogenization of a Class of Obstacle Problems

Jingzhi Li; Hongyu Liu; Lan Tang; Jiangwen Wang

doi:10.4208/aam.OA-2022-0001

Boundary Homogenization of a Class of Obstacle Problems

Jingzhi Li
, Hongyu Liu^*
, Lan Tang
, Jiangwen Wang

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

We study the homogenization of a boundary obstacle problem on a C^1,α-domain D for some elliptic equations with uniformly elliptic coefficient matrices γ. For any ∊∊ℝ₊, ∂D=Γ∪Σ, Γ∩Σ=∅ and S_∊⊂Σ with suitable assumptions, we prove that as ∊ tends to zero, the energy minimizer u^∊ of ^\int_D|γ\nablau|²dx, subject to u≥φ on S_ε, up to a subsequence, converges weakly in H¹(D) to ũ, which minimizes the energy functional (Formula presented.) where μ(x) depends on the structure of S_∊ and φ is any given function in C^∞(D). © 2022 Global Science Press. All rights reserved.

Original language	English
Pages (from-to)	240-260
Number of pages	21
Journal	Annals of Applied Mathematics
Volume	38
Issue number	2
DOIs	https://doi.org/10.4208/aam.OA-2022-0001
Publication status	Published - May 2022

Funding

The work of J. Li was partially supported by the NSF of China No. 11971221, Guangdong NSF Major Fund No. 2021ZDZX1001, the Shenzhen Sci-Tech Fund Nos. RCJC20200714114556020, JCYJ20200109115422828, JCYJ20190809150413261, and Guangdong Provincial Key Laboratory of Computational Science and Material Design No. 2019B030301001. The work of H. Liu was supported by the startup fund from City University of Hong Kong and the Hong Kong RGC General Research Fund (projects Nos. 12301420, 12302919 and 12301218). The work of L. Tang was partially supported by NNSFC grants of China (No. 11831009) and the Fundamental Research Funds for the Central Universities (No. CCNU19TS032). The work of J. Wang was partially supported by the Shenzhen Sci-Tech Fund No. JCYJ20180307151603959.

Research Keywords

asymptotic analysis
boundary obstacle
correctors
Homogenization

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title = "Boundary Homogenization of a Class of Obstacle Problems",

abstract = "We study the homogenization of a boundary obstacle problem on a C1,α-domain D for some elliptic equations with uniformly elliptic coefficient matrices γ. For any ∊∊ℝ+, ∂D=Γ∪Σ, Γ∩Σ=∅ and S∊⊂Σ with suitable assumptions, we prove that as ∊ tends to zero, the energy minimizer u∊ of \textbackslash{}intD|γ\textbackslash{}nablau|2dx, subject to u≥φ on Sε, up to a subsequence, converges weakly in H1(D) to ũ, which minimizes the energy functional (Formula presented.) where μ(x) depends on the structure of S∊ and φ is any given function in C∞(D). {\textcopyright} 2022 Global Science Press. All rights reserved.",

keywords = "asymptotic analysis, boundary obstacle, correctors, Homogenization",

author = "Jingzhi Li and Hongyu Liu and Lan Tang and Jiangwen Wang",

year = "2022",

month = may,

doi = "10.4208/aam.OA-2022-0001",

language = "English",

volume = "38",

pages = "240--260",

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TY - JOUR

T1 - Boundary Homogenization of a Class of Obstacle Problems

AU - Li, Jingzhi

AU - Liu, Hongyu

AU - Tang, Lan

AU - Wang, Jiangwen

PY - 2022/5

Y1 - 2022/5

N2 - We study the homogenization of a boundary obstacle problem on a C1,α-domain D for some elliptic equations with uniformly elliptic coefficient matrices γ. For any ∊∊ℝ+, ∂D=Γ∪Σ, Γ∩Σ=∅ and S∊⊂Σ with suitable assumptions, we prove that as ∊ tends to zero, the energy minimizer u∊ of \intD|γ\nablau|2dx, subject to u≥φ on Sε, up to a subsequence, converges weakly in H1(D) to ũ, which minimizes the energy functional (Formula presented.) where μ(x) depends on the structure of S∊ and φ is any given function in C∞(D). © 2022 Global Science Press. All rights reserved.

AB - We study the homogenization of a boundary obstacle problem on a C1,α-domain D for some elliptic equations with uniformly elliptic coefficient matrices γ. For any ∊∊ℝ+, ∂D=Γ∪Σ, Γ∩Σ=∅ and S∊⊂Σ with suitable assumptions, we prove that as ∊ tends to zero, the energy minimizer u∊ of \intD|γ\nablau|2dx, subject to u≥φ on Sε, up to a subsequence, converges weakly in H1(D) to ũ, which minimizes the energy functional (Formula presented.) where μ(x) depends on the structure of S∊ and φ is any given function in C∞(D). © 2022 Global Science Press. All rights reserved.

KW - asymptotic analysis

KW - boundary obstacle

KW - correctors

KW - Homogenization

UR - https://www.scopus.com/pages/publications/105034416548

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-105034416548&origin=recordpage

U2 - 10.4208/aam.OA-2022-0001

DO - 10.4208/aam.OA-2022-0001

M3 - RGC 21 - Publication in refereed journal

SN - 2096-0174

VL - 38

SP - 240

EP - 260

JO - Annals of Applied Mathematics

JF - Annals of Applied Mathematics

IS - 2

ER -

Boundary Homogenization of a Class of Obstacle Problems

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