Elliptic Partial Differential Equations with Complex Coefficients

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.