Solving Dirichlet problem using Green's identities.

[yungman]

This is to solve Dirichlet problem using Green's identities. The book gave some examples.

My question is: Why the book keep talking $v$ is harmonic(periodic) function. What is the difference whether $v$ is harmonic function or not as long as $v$ has continuous first and second derivatives.?

Green's identity:

$$\int\int_{\Omega}(u\nabla^2v + \nabla u \cdot \nabla v)dx dy = \int_{\Gamma} u\frac{\partial v}{\partial n} ds$$

If we let u=1:

$$\int\int_{\Omega} \nabla^2 v dx dy = \int_{\Gamma} \frac{\partial v}{\partial n} ds$$

For Dirichlet problem, $\nabla^2 v = 0$, Therefore:

$$\int_{\Gamma} \frac{\partial v}{\partial n} ds = 0$$

I have no issue with the math portion. It will be the same even though $v$ is not harmonic as long as $v$ has continuous first and second derivatives.

As long as $\nabla^2 v = 0$, the result is the same. Why the book keep mentioning $v$ being harmonic function in a few example.

 

[Mute]

Harmonic does not mean periodic. "$v$ is harmonic" means that v satisfies the Laplace equation, i.e., $\nabla^2 v = 0$.

See http://en.wikipedia.org/wiki/Harmonic_function

 

[yungman]

Thanks

Alan