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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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