On linear elliptic equations with drift terms in critical weak spaces

We study the Dirichlet problem for a second order linear elliptic equation in a bounded smooth domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n$, $n\ge 3$, with the drift $\mathbf{b}$ belonging to the critical weak space $L^{n,\mathrm{\infty }}(\mathrm{\Omega })$. We decompose the drift $\mathbf{b}={\mathbf{b}}_1+{\mathbf{b}}_2$ in which $\text{div}{\mathbf{b}}_1\ge 0$ and ${\mathbf{b}}_2$ is small only in a small scale quasi-norm of $L^{n,\mathrm{\infty }}(\mathrm{\Omega })$. Under this new smallness condition, we prove existence, uniqueness, and regularity estimates of weak solutions to the problem and its dual. Hölder 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 ${\mathbf{b}}_1=0$.