  We consider the Dirichlet and Neumann problems for second-order linear
elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{
and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz
domain $\Omega$ in $\mathbb{R}^n$ $(n\geq 3)$, 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 [17] and
Fabes-Mendez-Mitrea [12] for the Poisson equation. We also prove existence and
uniqueness of solutions of the Dirichlet problem with boundary data in
$L^{2}(\partial\Omega)$. Our results for the Dirichlet problems hold even for
the case $n=2$.
