Harmonic functions - pdes

• Jan 16th 2010, 08:56 AM
SamBourne
Harmonic functions - pdes
Let 'E' = element of
a) Suppose that u 'E' C^2*(D) ∩ C(D ̅) is harmonic inside a bounded and connected region D. Shoe that if u(x) for some point x* 'E' D then u(x) is a constant.

b) Suppose that u 'E' C^2 is harmonic on all of R^n explain why (delta u)/(delta xI) is also harmonic on R^n
• Jan 17th 2010, 05:53 AM
HallsofIvy
Quote:

Originally Posted by SamBourne
Let 'E' = element of
a) Suppose that u 'E' C^2*(D) ∩ C(D ̅) is harmonic inside a bounded and connected region D. Shoe that if u(x) for some point x* 'E' D then u(x) is a constant.

"Show that if u(x) for some point x* 'E' D"- there appears to be something missing here! if u(x) has what for some point?

Quote:

b) Suppose that u 'E' C^2 is harmonic on all of R^n explain why (delta u)/(delta xI) is also harmonic on R^n
And here you have not said what "xI" is!
• Jan 17th 2010, 08:46 AM
Jester
Quote:

Originally Posted by SamBourne
Let 'E' = element of
a) Suppose that u 'E' C^2*(D) ∩ C(D ̅) is harmonic inside a bounded and connected region D. Shoe that if u(x) for some point x* 'E' D then u(x) is a constant.

b) Suppose that u 'E' C^2 is harmonic on all of R^n explain why (delta u)/(delta xI) is also harmonic on R^n

I agree with Halls in the first part - something is missing. For the second part, a function is called Harmonic if it satisfies Laplaces equation, ie.

$
\frac{\partial^2 u}{\partial x_1^2} +
\frac{\partial^2 u}{\partial x_2^2}
+ \cdots
+ \frac{\partial^2 u}{\partial x_i^2} +
\cdots +
\frac{\partial^2 u}{\partial x_n^2}
= 0$

To show that $\frac{\partial u}{\partial x_i}$ is harmonic is to show it satisfies Laplaces equation.
• Jan 18th 2010, 03:24 AM
Rebesques
Quote:

Originally Posted by HallsofIvy
"Show that if u(x) for some point x* 'E' D"- there appears to be something missing here! if u(x) has what for some point?

And here you have not said what "xI" is!

Halls is right. In the first part, you forgot to add "u attains a maximum at some point inside D". In the last part, you could have said "all first order derivatives of u are also harmonic".

Both these follow from basic results on the Laplace equation (the "maxumum principle" and "interior regularity theorem"). Check with your notes.