Fundamental Solution of Laplace Equation 2d vs 3d

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Main Question or Discussion Point

When one compares the fundamental solution for Laplace's Equation one might note that in 2 dimensions this solution becomes unbounded as r goes to infinity while in 3 dimensions the solution goes to zero as r goes to infinity.

Now I understand both mathematical derivations so my question is not about that. What I would like to know is can someone give me a good explaination in terms of the possible physics being modeled by this equation that would explain this difference between the 2d and 3d case.

That is I would like a real world explaination as opposed to simply a mathematical explaination.
 

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  • #2
HallsofIvy
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You mean, of course, Laplace's equation with boundary conditions given on a circle or sphere. Laplace's equation is related to "potential theory" and so you might look at the difference between the "gravitational potential" in three dimensions and two.
 
  • #3
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Thanks...I will consider it from that point of view.

Further, I would like to understand the fundamental solution also in terms of
electrostatics and incompressible fluid flow.

It is interesting to note how changing the dimension for certain PDE's will significantly
affect the character of the solution. Another example concerns solutions to the wave equation for which there is a significant difference between even and odd dimensions.

If you or anyone else has more ideas I am all ears.
 

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