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Hop maximum principle

  1. May 14, 2010 #1

    haushofer

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    Science Advisor

    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! :)
     
  2. jcsd
  3. May 14, 2010 #2

    haushofer

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    Science Advisor

    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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