| In this paper I investigate divergence-form elliptic partial differential equations on Lipschitz domains in R2 whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates.;I show that for Dirichlet boundary data in Lq for q large enough, solutions exist and are controlled by the Lq-norm of the boundary data.;Similarly, for Neumann boundary data in Lp, or for Dirichlet boundary data whose tangential derivative is in Lp ("regularity" boundary data), for p small enough, I show that solutions exist and have gradients which are controlled by the Lp-norm of the boundary data.;I prove similar results for Neumann or regularity boundary data in H1, and for Dirichlet boundary data in L infinity or BMO. Finally, I show some converses: if the solutions are controlled in some sense, then Dirichlet, Neumann, or regularity boundary data must exist. |
