Abstract
The paper extends the results obtained by Kenig, Lin, and Shen [Arch. Ration. Mech. Anal., 203 (2012), pp. 1009--1036] to more general elliptic homogenization problems in two perspectives: lower order terms in the operator and no smoothness on the coefficients. We do not repeat their arguments. Instead we find the new weighted-type estimates for the smoothing operator at scale $\varepsilon$, and combining some techniques developed by Shen in [preprint, arXiv:1505.00694v1, 2015] leads to our main results. In addition, we also obtain sharp $O(\varepsilon)$ convergence rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Shen for elasticity systems in [preprint, arXiv:1505.00694v1, 2015]. Also, this work may be regarded as the extension of [T. Suslina, Mathematika, 59 (2013), pp. 463--476; T. Suslina SIAM J. Math. Anal., 45 (2013), pp. 3453--3493] developed by Suslina concerned with the bounded Lipschitz domain.
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