Hopf's Maximum Principle: Exploring X(r) & f(r)

[haushofer]

Hi,

it's been a while since I've explicitly dealt with differential equations. I have a question concerning "Hopf's maximum principle". The situation is as follows.

Let's say I have a function X(r) for which I have

$$\lim_{r \rightarrow\infty}X(r) = 0$$

This function X(r) satisfies the following condition for some arbitrary function f(r):

$$X(r) = - \Bigl(\frac{\partial f}{\partial r}\Bigr)^2$$

Can I now use Hopf's maximum principle and state that

$$f(r) = 0$$

everywhere? Do things change when I consider X(r) to have compact support? Maybe there is an easy counterexample if this conclusion is false, but any input would be welcome! :)

 

[haushofer]

Mmmm, if I pick [tex]X = \frac{-1}{r^2} [/itex] I get [tex]f(r) =\log{r}[/itex] which is certainly not zero everywhere.