Boundary conditions for pde

[DivergentSpectrum]

As i understand, the purpose of laplaces/poissons equation is to recast the question from a geometrical one to a differential equation.
im trying to figure out what are the appropriate boundary conditions for poissons equation:
http://www.sciweavers.org/upload/Tex2Img_1418842096/render.png [Broken]
where v is potential and p is the local charge density

Also, what method do i use to solve this equation? I cant remember a thing about pde's but i have some knowledge of ODE's. It appears linear Because V doesnt show up anywhere, and P is a function of x,y,z but I dont really know where to begin with this though.

 

[Simon Bridge]

The boundary conditions will be v(x,y,z) and it's partials for some specific points.
The exact method to solve, and the best boundary conditions for that matter, will depend on the exact form of p(x,y,z).

Note: we'd normally write: $$\nabla^2\phi = \frac{\rho_{free}}{\epsilon_0}$$ ... since this does not assume a specific coordinate system.

 

[DivergentSpectrum]

So if i know V at all values along some closed surface then V is defined everywhere inside right?