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?



Can the "physical argument" for the existence of a solution to Dirichlet's problem be made into an actual proof?
Well, I don't understand the electrostatics, but here is another physical heuristic:


Impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior. This gives a harmonic function extending the exterior temperature. [What's the electrostatic analogue? Formerly I had written "charge density", 
but now I am not sure if that's right.]


I think this strongly suggests a mathematically rigorous argument: We are naturally led to model the time-dependence of temperature in the interior.
This satisfies a diffusion (or heat) equation, but in words:


"After a time \delta, the new temperature is obtained by averaging the old temperature along a circle of radius \sqrt{\delta}."


This process converges under reasonable conditions, as time goes to infinity, to the solution of the Dirichlet problem. Anyway, we are led to the Brownian-motion proof of the existence, which I personally find rather satisfying. Another personal comment: I think one should always take "physical heuristics" rather seriously. 


[In response to Q.Y.'s comments below, which were responses to previous confused remarks that I made: neither the electric field nor the Columb potential is a multiple of the charge density on the boundary: the former is a vector, and in either
case imagine the charge on the boundary to be concentrated in a sub-region; neither the electric field nor the potential will be constant outside that sub-region.]


