We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in
\mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm.
Under these conditions, we obtain solvability of the Dirichlet problem for $L$, with data in $\la(\rn)$ (which is defined to be $BMO(\rn)$ when $\alpha=0$ and the space of H\older continuous functions $C^\alpha(\rn)$ when $\alpha \in (0