Undergrad Good positing in electrostatics problem with dielectrics - Poisson problem with conditions at the Dirichlet or Neumann edge

[Hak]

Often in potential calculus problems, the uniqueness theorem of the solution of the Poisson problem with Dirichlet and Neumann boundary conditions is improperly "invoked," without bothering too much about making such an application rigorous, i.e., showing that indeed the problem we are solving does indeed trace back in no uncertain terms to the Poisson problem with Neumann or Dirichlet boundary conditions (for which uniqueness is proved).
How can we try to show that the solution of the electrostatics problem in the presence of dielectric bodies is unique, i.e., that given real charges, set the potential at 0 at infinity, the polarization of the dielectric, the field outside and the field inside are uniquely determined? This would authorize us to search for "solutions of a particular form" that satisfies the fitting conditions, and once such a solution is found, reassure ourselves that it is precisely the solution sought, as the only one that satisfies these conditions.

Suppose we have a surface $S_0$ with zero potential. We have a $\rho$ distribution of real charges around space, and there is a dielectric of any shape (I call $S_d$ the closed surface enclosing the dielectric. Let us further assume that the dielectric is linear, so it is $D=\epsilon E$.

The problem to be solved is as follows:
$$\nabla^2 u = -4\pi\rho$$ (1)
$$u(S_0) = 0$$ (2)
$$\epsilon \partial_n u_{int} = \epsilon_0 \partial_n u_{ext}$$ along $S_d$ (3)

(3) expresses the discontinuity of fields along the dielectric shell (these are boundary conditions).
One of the problems is due to the fact that the uniqueness theorem holds only for "$C^2$ fields" (it is among the assumptions).
The second problem is that (1) and (2) already give a unique solution, so if I found a solution (1)-(2)-(3) this would have to coincide with the unique solution of (1)-(2) (i.e., it is as if the dielectric is not there).
A professor suggested that I add Dirac deltas to the second member of (1) to express the discontinuity given in (3), but I have no idea how to do that (having to be expressed with deltas here is not a discontinuity of $u$, but the normal derivative!)

Do you have any ideas for tracing problem (1)-(2)-(3) back to a Poisson problem with Dirichlet or Neumann edge conditions?

 

[pasmith]

The solution based on u(S_0) = 0 is only valid outside of the closed surface S_d; on the inside a different condition must be found to make the solution of the PDE unique, in the same way that <br /> f&#039;(x) = 0, \quad x \in \mathbb{R} \setminus \{0\} has general solution <br /> f(x) = \begin{cases} C_1 &amp; x &lt; 0 \\ C_2 &amp; x &gt; 0.\end{cases}