  We establish a new theory of regularity for elliptic complex valued second
order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the
coefficients of the matrix $A$ satisfy a natural algebraic condition, a
strengthened version of a condition known in the literature as
$L^p$-dissipativity. Precisely, the regularity result is a reverse H\"older
condition for $L^p$ averages of solutions on interior balls, and serves as a
replacement for the De Giorgi - Nash - Moser regularity of solutions to
real-valued divergence form elliptic operators. In a series of papers, Cialdea
and Maz'ya studied necessary and sufficient conditions for $L^p$-dissipativity
of second order complex coefficient operators and systems. Recently, Carbonaro
and Dragi\v{c}evi\'c introduced a condition they termed $p$-ellipticity, and
showed that it had implications for boundedness of certain bilinear operators
that arise from complex valued second order differential operators. Their
$p$-ellipticity condition is exactly our strengthened version of
$L^p$-dissipativity. The regularity results of the present paper are applied to
solve $L^p$ Dirichlet problems for $\mathcal
L=$div$A(\nabla\cdot)+B\cdot\nabla$ when $A$ and $B$ satisfy a natural and
familiar Carleson measure condition. We show solvability of the $L^p$ Dirichlet
boundary value problem for $p$ in the range where $A$ is $p$-elliptic.
