Revisions to Can the "physical argument" for the existence of a solution to Dirichlet's problem be made into an actual proof?

Removed former incorrect "answer" I wrote

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edited Oct 29, 2009 at 13:02

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SorryWell, 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 densityhere is harmonic.another physical heuristic:

So, imposeImpose a chargetemperature distribution onat the exterior, and measure (after some time has passed) the charge densitytemperature in the interior; that gives the solution to the Dirichlet probleminterior. This is probably roughly equivalent to: imposeThis gives a temperature distribution atharmonic function extending the exterior, and measure (after some time has passed) the temperature in. [What's the interiorelectrostatic 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, now I think I understand. I think the Wikipedia article is wrong (or rather, misleading).

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

So, impose a charge 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 to: 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 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.]

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.]

added 1423 characters in body; added 1 characters in body

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edited Oct 29, 2009 at 12:52

moonface

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5
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Sorry, not an answer, just a request for clarification:now I don'tthink I understand. I think the physical argumentWikipedia article is wrong (or rather, misleading).

We need to construct aThe harmonic function with a prescribed boundary value. This is to be constructed as"physically" produced is not the Coulomb potential associated to, but the charge density inside a certainconducting medium. Since in equilibrium there can be no electric field, the charge density is harmonic.

So, impose a charge distribution on the boundary. Whichexterior, 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 to: impose a temperature distribution at the exterior, thoughand 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 termswords:

"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 targetDirichlet 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 value?: 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: 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?

Sorry, now I think I understand. I think the Wikipedia article is wrong (or rather, misleading).

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

So, impose a charge 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 to: 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 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.]

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created Oct 29, 2009 at 5:15

moonface

666
5
4

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?

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