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$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

This work was supported in part by the NNSF of China (No. 11971212)

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  • In this paper, we establish the $ W^{1,p} $ 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 $ W^{1,p} $ estimates hold for $ \frac{3}{2}-\varepsilon<p<3+\varepsilon $ for $ d\geq3 $, and the range for $ p $ is sharp. For elliptic systems, we prove that the $ W^{1,p} $ estimates hold for $ \frac{2d}{d+1}-\varepsilon<p<\frac{2d}{d-1}+\varepsilon $ under the assumption that the Lipschitz constant of the domain is small.

    Mathematics Subject Classification: Primary: 35J25, 35J57; Secondary: 35B45.

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