Convergence Rates in L 2 for Elliptic Homogenization Problems

Carlos E. Kenig; Fanghua Lin; Zhongwei Shen

doi:10.1007/s00205-011-0469-0

Convergence Rates in L ² for Elliptic Homogenization Problems

Carlos E. Kenig, Fanghua Lin, Zhongwei Shen

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

120 Scopus citations

Abstract

We study rates of convergence of solutions in L ² and H ^1/2 for a family of elliptic systems {L _e{open}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L _e{open}}. Most of our results, which rely on the recently established uniform estimates for the L ² Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867-917, 2011; Commun Pure Appl Math 64:1-44, 2011) are new even for smooth domains.

Original language	English
Pages (from-to)	1009-1036
Number of pages	28
Journal	Archive for Rational Mechanics and Analysis
Volume	203
Issue number	3
DOIs	https://doi.org/10.1007/s00205-011-0469-0
State	Published - Mar 2012

Bibliographical note

Funding Information:
Carlos E. Kenig was supported in part by NSF grant DMS-0968472. Fanghua Lin was supported in part by NSF grant DMS-0700517. Zhongwei Shen was supported in part by NSF grant DMS-0855294.

ASJC Scopus subject areas

Analysis
Mathematics (miscellaneous)
Mechanical Engineering

Access to Document

10.1007/s00205-011-0469-0

Cite this

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title = "Convergence Rates in L 2 for Elliptic Homogenization Problems",

abstract = "We study rates of convergence of solutions in L 2 and H 1/2 for a family of elliptic systems {L e{open}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L e{open}}. Most of our results, which rely on the recently established uniform estimates for the L 2 Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867-917, 2011; Commun Pure Appl Math 64:1-44, 2011) are new even for smooth domains.",

author = "Kenig, {Carlos E.} and Fanghua Lin and Zhongwei Shen",

note = "Funding Information: Carlos E. Kenig was supported in part by NSF grant DMS-0968472. Fanghua Lin was supported in part by NSF grant DMS-0700517. Zhongwei Shen was supported in part by NSF grant DMS-0855294.",

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AU - Lin, Fanghua

AU - Shen, Zhongwei

N1 - Funding Information: Carlos E. Kenig was supported in part by NSF grant DMS-0968472. Fanghua Lin was supported in part by NSF grant DMS-0700517. Zhongwei Shen was supported in part by NSF grant DMS-0855294.

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N2 - We study rates of convergence of solutions in L 2 and H 1/2 for a family of elliptic systems {L e{open}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L e{open}}. Most of our results, which rely on the recently established uniform estimates for the L 2 Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867-917, 2011; Commun Pure Appl Math 64:1-44, 2011) are new even for smooth domains.

AB - We study rates of convergence of solutions in L 2 and H 1/2 for a family of elliptic systems {L e{open}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L e{open}}. Most of our results, which rely on the recently established uniform estimates for the L 2 Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867-917, 2011; Commun Pure Appl Math 64:1-44, 2011) are new even for smooth domains.

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