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Poisson equation

  1. Jan 7, 2009 #1

    1. The problem statement, all variables and given/known data
    Verify, that

    [tex] u(\vec{x}) := - \frac{1}{2 \pi} \int \limits_{\mathbb{R}^2} \log ||\vec{x} - \vec{y} || f(\vec{y}) d \vec{y} [/tex]

    is the general solution of the 2 dimensional Poisson equation:

    [tex] \Delta u = - f [/tex]

    where [tex] f \in C^2_c(\mathbb{R}^2) [/tex] is differentiable twice and has compact support.
    2. Relevant equations

    3. The attempt at a solution

    My attempt would be to swap integral and Laplace operator but I know it's wrong to just do that...
    Can anyone help me please? Thanks!!
  2. jcsd
  3. Feb 1, 2009 #2
    the Laplace operator is applied with respect to x, and the integration is performed over y -> I think you can swap them ( x is only a parameter inside of the integral)
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