%0 Journal Article
%T The Green function for elliptic systems in two dimensions
%A J. L. Taylor
%A S. Kim
%A R. M. Brown
%J Mathematics
%D 2012
%I arXiv
%R 10.1080/03605302.2013.814668
%X 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.
%U http://arxiv.org/abs/1205.1089v1