Laplace equation only applies in limited regions ?

[UAR]

Why does laplace's equation only apply in limited regions, while Poisson's equation can apply in unbounded domains ?

 

[HallsofIvy]

Where did you get the impression that Laplace's equation only applies in limited regions?

The function satisfying $\nabla^2 f= 0$, with f= 1 on the unit circle, in the region outside the circle is 1/r.

 

[UAR]

Where did you get the impression that Laplace's equation only applies in limited regions?

The function satisfying $\nabla^2 f= 0$, with f= 1 on the unit circle, in the region outside the circle is 1/r.

Thanks HallsofIvy:

I saw it in a book, but I agree with you that it doesn't seem to be true. However, I think 1/r may be the Green's function of the Laplacian in unbounded 3D domains.

Again, I agree with you that it seems strange, perhaps even untrue that Laplace equation would apply only to limited regions. If it turns out to be true, it may have something to do with the r^l terms in the harmonic functions in spherical polar coordinates. These terms diverge as r goes to infinity.

My above guess is unconvincing; especially since the r^{-(l+1)} terms will converge as r goes to infinity, and the divergent terms' coeffs can be chosen to decrease faster than the terms diverge. Any thoughts ?

BTW how is one able to write math expressions on this forum, such as the laplacian?

 

[UAR]

I'm now convinced it must be false. Consider the counter-example:

$$f(x) =x$$ for $$x\in\Re$$,

then:

$$\nabla^{2}f(x)=0$$