  We construct the fundamental solution or Green function for a divergence form
elliptic system in two dimensions with bounded and measurable coefficients. We
consider the elliptic system in a Lipschitz domain with mixed boundary
conditions. Thus we specify Dirichlet data on part of the boundary and Neumann
data on the remainder of the boundary. We require a corkscrew or non-tangential
accessibility condition on the set where we specify Dirichlet boundary
conditions. Our proof proceeds by defining a variant of the space $BMO(\partial
\Omega)$ that is adapted to the boundary conditions and showing that the
solution exists in this space. We also give a construction of the Green
function with Neumann boundary conditions and the fundamental solution in the
plane.
