Convergence rates for general elliptic homogenization problems in a bounded Lipschitz domain

The paper extends the results obtained by C. Kenig, F. Lin and Z. Shen in \cite{SZW2} 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 $\epsilon$, and combining some techniques developed by Z. Shen in \cite{SZW12} leads to our main results. In addition, we also obtain sharp $O(\epsilon )$ convergence rates in $L^p$ with $p=2d\mathrm{/}(d-1)$, which were originally established by Z. Shen for elasticity systems in \cite{SZW12}. Also, this work may be regarded as the extension of \cite{TS,TS2} developed by T. Suslina concerned with the bounded Lipschitz domain.