Hop maximum principle

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haushofer
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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

[tex]
\lim_{r \rightarrow\infty}X(r) = 0
[/tex]

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

[tex]
X(r) = - \Bigl(\frac{\partial f}{\partial r}\Bigr)^2
[/tex]

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

[tex]
f(r) = 0
[/tex]

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! :)
 

Answers and Replies

  • #2
haushofer
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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.
 

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