  We identify a large class of constant (complex) coefficient, second order
elliptic systems for which the Dirichlet problem in the upper-half space with
data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness
$\ell$, is well-posed in the class of functions whose nontangential maximal
operator of their derivatives up to, and including, order $\ell$ is
$L^p$-integrable. This class includes all scalar, complex coefficient elliptic
operators of second order, as well as the Lam\'e system of elasticity, among
others.
