I'm following Griffith's Intro to Electrodynamics, and when discussing Laplace's Equation, it states
Very often we are interested in finding the potential in a region where $\rho = 0$.
So, if you had a sphere with equal but opposite charge densities distributed over the northern and southern hemispheres, would Laplace's equation hold only over the entire sphere? But you couldn't analyze a hemisphere using the same method?