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Convergence Rates in L 2 for Elliptic Homogenization Problems

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Abstract

We study rates of convergence of solutions in L 2 and H 1/2 for a family of elliptic systems \({\{\mathcal{L}_\varepsilon\}}\) 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 \({\{\mathcal{L}_\varepsilon\}}\) . 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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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Carlos E. Kenig

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Fanghua Lin

  3. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    Zhongwei Shen

Authors
  1. Carlos E. Kenig
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  2. Fanghua Lin
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  3. Zhongwei Shen
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Corresponding author

Correspondence to Fanghua Lin.

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Communicated by The Editors

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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Kenig, C.E., Lin, F. & Shen, Z. Convergence Rates in L 2 for Elliptic Homogenization Problems. Arch Rational Mech Anal 203, 1009–1036 (2012). https://doi.org/10.1007/s00205-011-0469-0

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  • DOI: https://doi.org/10.1007/s00205-011-0469-0

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