TITLE: Can the "physical argument" for the existence of a solution to Dirichlet's problem be made into an actual proof?
QUESTION [15 upvotes]: Caveat: I don't really know anything about PDEs, so this question might not make sense.
In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace equation on bounded domains with nice (smooth) boundary. My sketchy understanding of the history of this problem (gleaned from Wikipedia) is that in the 19th century everybody "knew" that the problem had to have a unique solution, because of physics. Specifically, if I give you a distribution of charge along the boundary, it has to determine an electric potential in the domain, which turns out to be harmonic. But Dirichlet's proof was wrong, and it wasn't until around 1900 that Hilbert found a correct argument for the existence and uniqueness of the solution, given reasonable conditions (the boundary function must be continuous, and the boundary really has to be sufficiently smooth). 
Is the physical heuristic really totally meaningless from a mathematical point of view? Or is there some way to translate it into an actual proof?

REPLY [3 votes]: The electrostatic intuition does lead to a correct mathematical formulation of the Dirichlet problem.
Let's consider an electric charge distribution of two thin layers (one layer is positive and the other is negative) located along a closed surface $S\subset\mathbb R^3$. Assume that $d>0$ is the distance  between charges along the normal $n_p$ to the surface at point $p$. Let $\rho\in C(S,\mathbb R)$ denote the distribution's density. 
A pair of two opposing charges $+Q=\rho/d$ and $-Q=-\rho/d$ creates an electric field. The limit of the field when $d\to 0$ is known as the dipole. For any $x\in \mathbb R^3$, the dipole potential at the point $p\in S$ has the form
$$\frac{\rho}{d}\Phi(x-(p+n_pd/2))-\frac{\rho}{d}\Phi(x-(p-n_pd/2))=\rho\frac{\partial \Phi(x-p)}{\partial n_p} +o(1)\quad{\rm as\ \ } d\to0,\qquad(1)$$
where $\Phi(x)=-(4\pi|x|)^{-1},\quad x\in \mathbb R^3,$
is the fundamental solution of Laplace's equation. (1) gives the dipole potential of a single dipole at the point $p\in S$ and the integral
$$u(x)=\int_{S}\rho(p) \frac{\partial \Phi(x-p)}{\partial n_p} dp,\quad x\in\mathbb R^3,$$
is the potential of the whole distribution.
Now, a simple computation shows that $u(x)$ is a harmonic function when $x$ is not on the surface. It has a jump when $x$ passes through $S$:
$$u_{-}(x_0)=u(x_0)-2\pi \rho(x_0),\quad u_{+}(x_0)=u(x_0)+2\pi \rho(x_0),\quad x_0\in S,\qquad\qquad\qquad\qquad (2)$$
where $u_{-}(x_0)$ ($u_{+}(x_0)$) is the limit from the interior (exterior) of the surface.
Relations (2) are integral equations w.r.t. the unknown density (assuming that the potential on the surface is known). The equations can be solved using the Fredholm approach.  The function $u(x)$ then gives a solution to the Dirichlet problem.
Edit. See a nice little textbook by Arnold where he shows how to make the physical intuition rigorous in this problem.