The higher order regularity Dirichlet problem for elliptic systems in the upper-half space

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, $1<p<\mathrm{\infty }$, of arbitrary smoothness $\mathrm{\ell }$, is well-posed in the class of functions whose nontangential maximal operator of their derivatives up to, and including, order $\mathrm{\ell }$ 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.