What is the equation describing the boundary of a 2D charge density?

Question

Consider a charge density $\rho(x,y)$ distributed on the 2D plane. The charge density follows the Poisson equation:

$$\nabla^2 \mathbf{\phi}=\mathbf{ \nabla\cdot E}=-4\pi\rho(x,y),$$ where $\phi(x,y)$ is a potential and $\mathbf{E}$ is the electric field.

To solve the Poisson equation, we need boundary conditions, $f(x,y)=0$, which describes all the points $(x,y)$ where the charge density vanishes.

Usually we start from the Poisson equation and are given some boundary conditions and we then can try to solve the equation in order to derive the charge distribution.

My case is the reverse: Assuming we already know the potential $\phi(x,y)$, how can I determine what the form of $f(x,y)$ is? In other words, what equation does $f(x,y)$ satisfy in terms of the potential $\phi(x,y)$?

Edit: Is it even possible? I tried to play around examples and it is not obvious at all that just knowing $\phi(x,y)$ is enough.

Answer 1

Wikipedia gives:

if $$ \nabla^2u=-f $$

Then:

$$ u=\int_V d^3 r \, G.f + \oint_S d^2 r\, G_n.g $$

Where $g$ is the bondary value and $G$ is the Green function. It would seem that solution may be possible, but may not be unique.