Potential of an infinite charged plate: Poisson's or Laplace's equation?

Question

It is known that the Poisson's equation $\nabla^2\phi = -4\pi\rho$ is valid for a region of space containing charges, and the Laplace equation $\nabla^2\phi = 0$ is valid for a region without charges.

Someone states the Laplace's equation one should to solve when boundaries condition are given and there are no charges in the region, but if charge distribution is given in region - solve Poisson equation.

Suppose we have an infinite charged plate with some charge density over the plate, say $\sigma$ (homogenious or not). In order to find the potential in the whole space, what equation do we solve here, Poisson's or Laplace's?

Answer 1

The problem with a charged plate can be approach from both points of view. Due to the fact that the plate is very thin, it requires using a delta-function charge distribution, which may be not an obvious thing to deal with. This is however doable, see, e.g., my extended answer to this question.

A more classical approach to a charged plate is to describe it using a boundary condition for the electric field (which is essentially obtained using the integral form of the Maxwell equations, whereas the Poisson and the Laplace equations are the differential form) and a rather trivial in this case solution of the Laplace equation.