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][1]. 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][2].  The function $u(x)$ then gives a solution to the Dirichlet problem.



**Edit.** See a nice little [textbook][3] by Arnold where he shows how to make the physical intuition rigorous in this problem.
  


  [1]: http://en.wikipedia.org/wiki/Dipole
  [2]: http://en.wikipedia.org/wiki/Fredholm_theory
  [3]: http://books.google.co.uk/books?id=qlNJAYwmfTcC&pg=PR3&dq=Vladimir+I.+Arnold+-+Lectures+on+Partial+Differential+Equations&cd=1#v=onepage&q=Vladimir%2520I.%2520Arnold%2520-%2520Lectures%2520on%2520Partial%2520Differential%2520Equations&f=false