  In this paper, we use a probabilistic approach to show that there exists a
unique, bounded continuous solution to the Dirichlet boundary value problem for
a general class of second order non-symmetric elliptic operators $L$ with
singular coefficients, which does not necessarily have the maximum principle.
The theory of Dirichlet forms and heat kernel estimates play a crucial role in
our approach. A probabilistic representation of the non-symmetric semigroup
$\{T_t\}_{t\ge 0}$ generated by $L$ is also given.
