  We establish a Dahlberg-type perturbation theorem for second order divergence
form elliptic operators with complex coefficients. In our previous paper, we
showed the following result: If ${\mathcal L}_0=\mbox{div}
A^0(x)\nabla+B^0(x)\cdot\nabla$ is a $p$-elliptic operator satisfying certain
Carleson condition on $\nabla A$ and $B$ then the $L^p$ Dirichlet problem for
the operator ${\mathcal L}_0$ is solvable in the upper half-space ${\mathbb
R}^n_+$.
  In this paper we prove that the $L^p$ solvability is stable under small
perturbations of ${\mathcal L}_0$. That is if ${\mathcal L}_1$ is another
divergence form elliptic operator with complex coefficients and the
coefficients of the operators ${\mathcal L}_0$ and ${\mathcal L}_1$ are
sufficiently close in the sense of Carleson measures (considering the
differences of coefficients), then the $L^p$ Dirichlet problem for the operator
${\mathcal L}_1$ is solvable for the same value of $p$.
  As a corollary we obtain a new result on $L^p$ solvability of the Dirichlet
problem for operators of the form ${\mathcal L}=\mbox{div}
A(x)\nabla+B(x)\cdot\nabla$ where the matrix $A$ satisfies weaker Carleson
condition than in our earlier paper; in particular the coefficients of $A$ need
no longer be differentiable and instead satisfy a Carleson condition that
controls the oscillation of the matrix $A$ over Whitney boxes. This result in
the real case has been established by Dindo\v{s}, Petermichl and Pipher.
