Laplace equation only applies in limited regions ?

In summary: ReIn summary, Laplace's equation and Poisson's equation are two different equations that have different applications. While Laplace's equation can be used in limited regions, Poisson's equation can be applied in unbounded domains. It is possible for Laplace's equation to apply in unbounded domains, but this may depend on the specific functions and conditions involved. Additionally, it is possible to write mathematical expressions on this forum, such as the Laplacian, by using LaTeX formatting.
  • #1

UAR

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Why does laplace's equation only apply in limited regions, while Poisson's equation can apply in unbounded domains ?
 
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  • #2
Where did you get the impression that Laplace's equation only applies in limited regions?

The function satisfying [itex]\nabla^2 f= 0[/itex], with f= 1 on the unit circle, in the region outside the circle is 1/r.
 
  • #3
HallsofIvy said:
Where did you get the impression that Laplace's equation only applies in limited regions?

The function satisfying [itex]\nabla^2 f= 0[/itex], 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?
 
  • #4
I'm now convinced it must be false. Consider the counter-example:

[tex]f(x) =x[/tex] for [tex]x\in\Re[/tex],

then:

[tex]\nabla^{2}f(x)=0[/tex]
 

1. What is Laplace equation and where does it apply?

The Laplace equation is a second-order partial differential equation that describes the distribution of a scalar quantity in a given region. It applies in regions where there are no sources or sinks of the scalar quantity and the boundary conditions are well-defined.

2. Can Laplace equation apply in an unbounded region?

No, Laplace equation only applies in limited regions that have well-defined boundaries. In an unbounded region, the boundary conditions cannot be specified and thus, the equation cannot be solved.

3. What are some examples of limited regions where Laplace equation applies?

Laplace equation is commonly used in electrostatics, heat transfer, fluid mechanics, and potential theory. Some examples of limited regions where it can be applied are a parallel plate capacitor, a rectangular metal plate with fixed temperature boundaries, and a fluid flow between two parallel plates.

4. Are there any limitations to the applicability of Laplace equation?

Yes, there are limitations to the applicability of Laplace equation. It cannot be used in regions where there are sources or sinks of the scalar quantity, such as in the presence of electric charges or heat sources. It also cannot be used in regions where the boundary conditions are not well-defined or there is a discontinuity in the boundary conditions.

5. How is Laplace equation solved in limited regions?

Laplace equation can be solved using various mathematical methods such as separation of variables, Green's function, and numerical techniques. The specific method used depends on the boundary conditions and the geometry of the region. In some cases, analytical solutions may not be possible and approximations or numerical solutions are used.

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