We consider the Dirichlet and Neumann problems for second-order linear elliptic equations:
\[
−
△
u
+
div
(
u
b
)
=
f
and
−
△
v
−
b
⋅
∇
v
=
g
-\triangle u +\operatorname {div}(u\mathbf {b}) =f \quad \text { and }\quad -\triangle v -\mathbf {b} \cdot \nabla v =g
\]
in a bounded Lipschitz domain
Ω
\Omega
in
R
n
\mathbb {R}^n
(
n
≥
3
)
(n\geq 3)
, where
b
:
Ω
→
R
n
\mathbf {b}:\Omega \rightarrow \mathbb {R}^n
is a given vector field. Under the assumption that
b
∈
L
n
(
Ω
)
n
\mathbf {b} \in L^{n}(\Omega )^n
, we first establish existence and uniqueness of solutions in
L
α
p
(
Ω
)
L_{\alpha }^{p}(\Omega )
for the Dirichlet and Neumann problems. Here
L
α
p
(
Ω
)
L_{\alpha }^{p}(\Omega )
denotes the Sobolev space (or Bessel potential space) with the pair
(
α
,
p
)
(\alpha ,p)
satisfying certain conditions. These results extend the classical works of Jerison-Kenig [J. Funct. Anal. 130 (1995), pp. 161–219] and Fabes-Mendez-Mitrea [J. Funct. Anal. 159 (1998), pp. 323–368] for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in
L
2
(
∂
Ω
)
L^{2}(\partial \Omega )
. Our results for the Dirichlet problems hold even for the case
n
=
2
n=2
.