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Vector calculus problem:

  1. May 10, 2012 #1
    1. The problem statement, all variables and given/known data

    ## \gamma_1 ## and ##\gamma_2 ## are both real continuous solutions of ## \nabla^2 \gamma = \gamma ## in ## V## and ##\gamma_1=\gamma_2 ## on the boundary ##\partial V##. We are looking at the function ##g = \gamma_1 - \gamma_2 ##.

    I have proved

    ##\nabla \cdot \left( g \nabla g) \right) = ||\nabla g||^2 + g\nabla^2 g## already. I used this to show that

    ##\int_V ||\nabla g||^2 dV + \int_{\partial V} g^2 dV = 0 ##

    The question is: what does this say about the value of ##g=\gamma_1-\gamma_2 ## in ##V## and are the solutions unique for ##\nabla^2 \gamma = \gamma##?

    Any help appreciated!
     
    Last edited: May 10, 2012
  2. jcsd
  3. May 10, 2012 #2
    If you can show g=constant then the solution is unique, within a constant (for dirichlet conditions it's obviously unique). Your first equation doesn't make sense, I think you mean to put g*laplacian(g). Also, if you managed to get rid of this term, your logic should probably also apply to the g^2 term on the boundary. Elaborate on what you've done at the boundary.
     
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