  In this paper we investigate elliptic partial differential equations on
Lipschitz domains in the plane whose coefficient matrices have small (but
possibly nonzero) imaginary parts and depend only on one of the two
coordinates. We show that for Dirichlet boundary data in L^p for p large
enough, solutions exist and are controlled by the L^p-norm of the boundary
data. Similarly, for Neumann boundary data in L^q, or for Dirichlet boundary
data whose tangential derivative is in L^q (regularity boundary data), for q
small enough, we show that solutions exist and are controlled by the L^q-norm
of the boundary data. We prove similar results for Neumann or regularity
boundary data in the Hardy space H^1, and for bounded or BMO Dirichlet boundary
data. Finally, we show some converses: if the solutions are controlled in some
sense, then Dirichlet, Neumann, or regularity boundary data must exist.
