  In this work we establish solvability and uniqueness for the $D_2$ Dirichlet
problem and the $R_2$ Regularity problem for second order elliptic operators
$L=-{\rm div}(A\nabla\cdot)+b\nabla\cdot$ in bounded Lipschitz domains, where
$b$ is bounded, as well as their adjoint operators $L^t=-{\rm
div}(A^t\nabla\cdot)-{\rm div}(b\,\cdot)$. The methods that we use are
estimates on harmonic measure, and the method of layer potentials.
  The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$
leads us to impose a specific size condition on ${\rm div}b$ in order to obtain
solvability. On the other hand, we show that $R_2$ for $L$ and $D_2$ for $L^t$
are uniquely solvable, assuming only that $A$ is Lipschitz continuous (and not
necessarily symmetric) and $b$ is bounded.
