3 Removed former incorrect "answer" I wrote

Sorry

Well, now I think I don't understand . I think the Wikipedia article is wrong (or rather, misleading).

The harmonic function to be "physically" produced is not the Coulomb potentialelectrostatics, but the charge density inside a conducting medium. Since in equilibrium there can be no electric field, the charge density here is harmonic.

So, impose another physical heuristic:

Impose a charge temperature distribution on at the exterior, and measure (after some time has passed) the charge density temperature in the interior; that gives the solution to the Dirichlet problem. This is probably roughly equivalent to: impose gives a temperature distribution at harmonic function extending the exterior , and measure (after some time has passed) the temperaturein the interior. [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 charge density (or 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.]

Sorry, not an answer, just a request for clarification: now I don't think I understand. I think the physical argumentWikipedia article is wrong (or rather, misleading).

We need

The harmonic function to construct be "physically" produced is not the Coulomb potential, but the charge density inside a conducting medium. Since in equilibriumthere can be no electric field, the charge density is harmonicfunction with .

So, impose a prescribed boundary valuecharge distribution on the exterior, and measure (after some time has passed) the charge density in the interior; that gives the solution to the Dirichlet problem. This is probably roughly equivalent tobe constructed : impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior.

I think this strongly suggests a mathematically rigorous argument: We are naturally led to model the time-dependence of charge density (or 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 potential associated 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 certain multiple of the charge distribution density on the boundary. Which charge distribution, though: the former is a vector, and in terms of eithercase imagine the target charge on the boundary value? to be concentrated in a sub-region; neither the electric field nor the potential will be constant outside that sub-region.]

1

Sorry, not an answer, just a request for clarification: I don't understand the physical argument.

We need to construct a harmonic function with a prescribed boundary value. This is to be constructed as the potential associated to a certain charge distribution on the boundary. Which charge distribution, though, in terms of the target boundary value?