Harmonic Functions: Why n>2 Condition Imposed

In summary, the function f:\Re^n \rightarrow \Re = |X|^{2-n} is harmonic for n > 2, meaning it satisfies the Laplace equation. The condition n > 2 is imposed because the function cannot exist at 0 and therefore is not harmonic for n = 1. However, for n = 1, the function is still considered harmonic as it satisfies Laplace equation. This concept is important in classical physics, particularly in electromagnetic theory and the potential formulation of the theory. Additionally, harmonic functions must be continuously differentiable twice and satisfy the Laplace equation.
  • #1
praharmitra
311
1
the function [tex]f:\Re^n \rightarrow \Re = |X|^{2-n} [/tex] is harmonic, but only for n > 2.

why is the condition n > 2 imposed. Isn't harmonic for all values of n??
 
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  • #2
Shouldn't the domain be [tex]\mathbb{R}^n\setminus \{ 0 \}[/tex]? Otherwise, said f could not even exist at 0, and hence is clearly not harmonic.
 
  • #3
ya I'm sorry that is the domain...my mistake...

but still, why shoud n>2, why can't n = 1?..

note: n stands for dimension of the space and hence is an integer >=1
 
  • #4
oh, is this classical physics? ;-)

Try what happens if n=1, is the function harmonic then?
 
  • #5
of course it is classical physics. It plays a MAJOR role in electromagnetic theory. and the potential formulation of the theory.

for n = 1, it is harmonic as laplacian(|X|) = 0 right? so it satisfies.

but then, why put the condition n > 2. I have seen it in many books
 
  • #6
well then calculus problems are classic physics problems aswell. Just trying to make sure you get help.

Now harmonic functions are functions which are continuously differentiable twice and satisfy Laplace equation.

So play around with the criterion for twice continuously differentiable.
 

Related to Harmonic Functions: Why n>2 Condition Imposed

What are harmonic functions?

Harmonic functions are mathematical functions that satisfy the Laplace's equation, which is a second-order partial differential equation that describes the behavior of the physical systems. In simpler terms, harmonic functions are smooth and continuous functions that do not have any extreme values within their domain.

Why is the condition n>2 imposed for harmonic functions?

The condition n>2 is imposed because it ensures the uniqueness of the solution to the Laplace's equation. This means that for any given boundary conditions, there can be only one solution that satisfies the equation. If n=2, the solution may not be unique, leading to various solutions that may not accurately represent the physical system being studied.

What happens if the n>2 condition is not met?

If the n>2 condition is not met, the solution to the Laplace's equation may not be unique. This can result in multiple solutions that do not accurately describe the physical system. In some cases, it may also lead to non-physical or unrealistic solutions.

What are some examples of harmonic functions?

Some examples of harmonic functions include the electric potential in electrostatics, velocity potential in fluid mechanics, and temperature distribution in heat transfer problems. They can also be found in various other fields such as mechanics, acoustics, and optics.

What are the practical applications of harmonic functions?

Harmonic functions have various practical applications in different fields of science and engineering. They are used to model and analyze physical systems, such as electric circuits, fluid flow, and heat transfer. They also play a crucial role in solving boundary value problems and optimization problems in mathematics and physics.

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