We study the Dirichlet problem for a second order linear elliptic equation in a bounded smooth domain $\Omega$ in $\mathbb{ R}^n$, $n \ge 3$, with the drift $\mathbf{ b} $ belonging to the critical weak space $L^{ n,\infty}(\Omega )$. We decompose the drift $\mathbf{ b} = \mathbf{ b}_1 + \mathbf{ b}_2$ in which $\text{ div} \mathbf{ b}_1 \geq 0$ and $\mathbf{ b}_2$ is small only in a small scale quasi-norm of $L^{ n,\infty}(\Omega )$. Under this new smallness condition, we prove existence, uniqueness, and regularity estimates of weak solutions to the problem and its dual. H\"{ o}lder regularity and derivative estimates of weak solutions to the dual problem are also established. As a result, we prove uniqueness of very weak solutions slightly below the threshold. When $\mathbf{ b}_2 =0$, our results recover those by Kim and Tsai in [SIAM J. Math. Anal. 52 (2020)]. Due to the new small scale quasi-norm, our results are new even when $\mathbf{ b}_1=0$.
Categories: math.AP