Boundary Layers in Periodic Homogenization of Neumann Problems

Zhongwei Shen; Jinping Zhuge

doi:10.1002/cpa.21740

Boundary Layers in Periodic Homogenization of Neumann Problems

Zhongwei Shen, Jinping Zhuge

Mathematics

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

13 Scopus citations

Abstract

This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in L² in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher-order convergence rate for Neumann problems with nonoscillating data.

Original language	English
Pages (from-to)	2163-2219
Number of pages	57
Journal	Communications on Pure and Applied Mathematics
Volume	71
Issue number	11
DOIs	https://doi.org/10.1002/cpa.21740
State	Published - Nov 2018

Bibliographical note

Funding Information:
Acknowledgment. Zhongwei Shen is supported in part by National Science Foundation Grant DMS-1600520. Jinping Zhuge is supported in part by National Science Foundation Grant DMS-1161154. Zhongwei Shen thanks David Gérard-Varet for bringing the problem of homogenization of the Neumann problem (1.1) to his attention.

Publisher Copyright:
© 2018 Wiley Periodicals, Inc.

ASJC Scopus subject areas

Mathematics (all)
Applied Mathematics

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title = "Boundary Layers in Periodic Homogenization of Neumann Problems",

abstract = "This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in L2 in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher-order convergence rate for Neumann problems with nonoscillating data.",

author = "Zhongwei Shen and Jinping Zhuge",

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N2 - This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in L2 in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher-order convergence rate for Neumann problems with nonoscillating data.

AB - This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in L2 in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher-order convergence rate for Neumann problems with nonoscillating data.

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Boundary Layers in Periodic Homogenization of Neumann Problems

Abstract

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