In many book I read, problems for electrostatic potential always lead to solving Poisson equation. I saw a problem about a spherical shell carrying some amount of charges uniformly on the surface with density [tex]\rho[/tex], and then someone put a small patch on the sphere. The patch is then made a constant potential [tex]V_0[/tex] on it and everywhere else on the shell has zero potential. I would like to find the potential everywhere inside the spherical shell. In this case, how can I apply Poisson equation to do that?