Generalized Laplace equation

Question

When considering electric potential $\Phi(\vec{r})$ in the presence of dielectric material described by relative permittivity $\epsilon(\vec{r})$, one has to solve the generalized Laplace equation $$\vec{\nabla} \cdot (\epsilon \vec{\nabla} \Phi) = 0.$$ I am interested in some basic properties of this equation. For example, are Dirichlet and Neumann problems well posed in this case? How many of its properties does this equation inherit from the ordinary Laplace equation?