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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.

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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.

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  • $\begingroup$ Sort of related to en.wikipedia.org/wiki/Inverse_scattering_problem $\endgroup$
    – Cryo
    Nov 20, 2023 at 14:19
  • $\begingroup$ Sorry, I am not sure how it answers the question? First, the notation is different: in your answer $f$ refers to my $\rho$ and $g$ is the boundary value I am looking for. Second, it seems that this is the 3-dimensional case, I am considering the 2d case. Those two points aside, how can I extract $g$ from your formulae? Assuming I know the potential, ($u$ here) I can directly get $f$ and $G$. But getting $g$ is not obvious? $\endgroup$
    – Matt
    Nov 21, 2023 at 11:26
  • $\begingroup$ The notation matches wiki link I gave. You cannot necessarily extract a unique solution, but you can check whether solution matches. This is typical for inverse problems. Often you will use regularisation of some sort to get a unique solution. For example seek solution with minimal norm $\endgroup$
    – Cryo
    Nov 21, 2023 at 14:51

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