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Homework Help: Dirichlet problem

  1. Feb 22, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex] B_R = \{ x \in \mathbb{R}^n: |x| < R \}. [/tex] Calculate the solution of the following Dirichlet problem:

    [tex] -\Delta v = 1 [/tex] in [tex] B_R [/tex]
    [tex] u = 0 [/tex] on [tex] \partial B_R [/tex]

    Calculate the solution of the problem.

    2. Relevant equations

    3. The attempt at a solution

    I know that the solution must be radial for trivial considerations on the invariance of laplacian under orthogonal transformations and the uniqueness of the solution.
    I thought about integrating a Green function for the problem, but what Green function?
    There must be an easier way I'm missing.
     
  2. jcsd
  3. Feb 22, 2010 #2

    Dick

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    There's a lot easier way. You should be able to guess a solution. What's the laplacian of |x|^2?
     
  4. Feb 22, 2010 #3
    Please don't tell me that the solution is simply [tex] - \frac{1}{2n}|x|^2 - R^2 [/tex] 'cause I'm gonna shoot myself in the head.
     
  5. Feb 22, 2010 #4

    Dick

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    You don't have to shoot yourself in the head. It's not right. But it's ALMOST right. Adjust your constant.
     
  6. Feb 22, 2010 #5
    Of course. You don't know how many lines I've written and how much effort I've put into trying to find a Green function to integrate.
    Maybe you can tell me if I'm missing something similarly trivial here

    https://www.physicsforums.com/showthread.php?t=380559 [Broken]

    and finally pull the trigger.
     
    Last edited by a moderator: May 4, 2017
  7. Feb 22, 2010 #6

    Dick

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    Sorry, that one's not so clear to me as the other one. I'll try and give it some thought.
     
    Last edited by a moderator: May 4, 2017
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