I am interested in the following two problems: \begin{equation} \left\{ \begin{array}{ll} - \text{div} A(x)\nabla G(x,y) = \delta(x-y) & \text{in} \ \Omega \\ G=0 & \text{on} \ \partial \Omega \end{array} \right. \end{equation} \begin{equation} \left\{ \begin{array}{ll} - \text{div} A(x)\nabla G(x,y) = \delta(x-y) - \frac{1}{|\Omega|} & \text{in} \ \Omega \\ \left.\frac{\partial G(x,y)}{\partial n_x}\right|_{\partial \Omega}=0 & \text{on} \ \partial \Omega \end{array} \right. \end{equation}
That is, consider the greens function for general elliptic operator with Dirichlet and Neumann boundary conditions.
What are the conditions on $A(x)$ and $\Omega$ that ensure the Greens function exists?
Here $A(x)$ is the usual elliptic matrix which has the bounds $|\zeta|^2\lambda \leq A(x)\zeta \zeta^t \leq \Lambda |\zeta|^2 $ and $\Omega$ is a BOUNDED domain.
Any references would be very helpful. I am mainly interested in the case when $\Omega$ is convex, so how much regularity should I assume on $A(x)$? Can I take any measurable $A$? If yes, please let me know the reference.