Confusion over applying the 1st uniqueness theorem to charged regions

[phantomvommand]

1. For regions that contain charge density, does the 1st uniqueness theorem still apply?

2. For regions that contain charge density, does the 'no local extrema' implication of Laplace's equation still apply? I think not, since the relevant equation now is Poisson's equation. Furthermore, considering a charged metal sphere and a spherical boundary surface surrounding it, there is a local maxima at the sphere. Nonetheless, the result that "given a specific charge distribution and specific boundary conditions, there is a specific potential distribution" still holds.

3. What is the main difference between the 1st uniqueness theorem and the 2nd uniqueness theorem? Is it that the 1st uniqueness theorem relates to boundary and charge distribution, while the 2nd uniqueness theorem deals with total charge on conductors?

Thanks for all the help

 

[wrobel]

I know nothing about charges but try to answer:)
If $\Delta u\ge 0$ in a domain $D\subset\mathbb{R}^m$ then
$$\sup_{x\in D} u(x)=\sup_{x\in\partial D}u(x)$$
And yes, Dirichlet boundary value problem for the Poisson equation has a unique solution
This is informal for details see textbooks in PDE