<p style='text-indent:20px;'>In this paper, we establish the <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M3">\begin{document}$ \frac{3}{2}-\varepsilon<p<3+\varepsilon $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ d\geq3 $\end{document}</tex-math></inline-formula>, and the range for <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is sharp. For elliptic systems, we prove that the <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M7">\begin{document}$ \frac{2d}{d+1}-\varepsilon<p<\frac{2d}{d-1}+\varepsilon $\end{document}</tex-math></inline-formula> under the assumption that the Lipschitz constant of the domain is small.</p>
We consider a family of second-order elliptic operators { L ε } \{\mathcal {L}_\varepsilon \} in divergence form with rapidly oscillating and almost-periodic coefficients in Lipschitz domains. By using the compactness method, we show that the uniform W 1 , p W^{1,p} estimate of second-order elliptic systems holds for 2 n n + 1 − δ > p > 2 n n − 1 + δ \frac {2n}{n+1}-\delta >p>\frac {2n}{n-1}+\delta ; the ranges are sharp for n = 2 n=2 or n = 3 n=3 . In the scalar case we obtain that the W 1 , p W^{1,p} estimate holds for 3 2 − δ > p > 3 + δ \frac {3}{2}-\delta >p>3+\delta if n ⩾ 3 n\geqslant 3 , and 4 3 − δ > p > 4 + δ \frac {4}{3}-\delta >p>4+\delta if n = 2 n=2 ; the ranges of p p are sharp.
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