  The theory of second order complex coefficient operators of the form
$\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the
assumption of $p$-ellipticity. In particular, if the matrix $A$ is
$p$-elliptic, the solutions $u$ to $\mathcal{L}u = 0$ will satisfy a higher
integrability, even though they may not be continuous in the interior.
Moreover, these solutions have the property that $|u|^{p/2-1}u \in
W^{1,2}_{loc}$. These properties of solutions were used by Dindo\v{s}-Pipher to
solve the $L^p$ Dirichlet problem for $p$-elliptic operators whose coefficients
satisfy a further regularity condition, a Carleson measure condition that has
often appeared in the literature in the study of real, elliptic divergence form
operators. This paper contains two main results. First, we establish
solvability of the Regularity boundary value problem for this class of
operators, in the same range as that of the Dirichlet problem. The Regularity
problem, even in the real elliptic setting, is more delicate than the Dirichlet
problem because it requires estimates on derivatives of solutions. Second, the
Regularity results allow us to extend the previously established range of $L^p$
solvability of the Dirichlet problem using a theorem due to Z. Shen for general
bounded sublinear operators.
