Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: $$-triangle u +operatorname{div}(umathbf{b}) =f quadtext{ and }quad -triangle v -mathbf{b} cdot abla v =g$$ in a bounded Lipschitz domain $Omega$ in $mathbb{R}^n$ $(ngeq 3)$, where $mathbf{b}:Omega rightarrow mathbb{R}^n$ is a given vector field. Under the assumption that $mathbf{b} in L^{n}(Omega)^n$, we first establish existence and uniqueness of solutions in $L_{alpha}^{p}(Omega)$ for the Dirichlet and Neumann problems. Here $L_{alpha}^{p}(Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995, JFA) and Fabes-Mendez-Mitrea (1998, JFA) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(partialOmega)$.