In this thesis, we consider the Dirichlet and Neumann problems for second-order linear elliptic equations:\[
-\triangle u +\Div(u\boldb) =f \quad\text{ and }\quad -\triangle v -\boldb \cdot \nabla v =g
\]
in a bounded Lipschitz domain $\Omega$ in $\m...
In this thesis, we consider the Dirichlet and Neumann problems for second-order linear elliptic equations:\[
-\triangle u +\Div(u\boldb) =f \quad\text{ and }\quad -\triangle v -\boldb \cdot \nabla v =g
\]
in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$ $(n\geq 3)$, where $\boldb:\Omega \rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\boldb \in L^{n}(\Omega)^n$, 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 \cite{MR1331981} and Fabes-Mendez-Mitrea \cite{MR1658089} for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(\partial\Omega)$.