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Uniqueness with Laplace's Equation and Robin Boundary Condn

  1. Mar 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that T(x, y) satisfies Laplace’s equation in a bounded region
    D and that
    ∂T/∂n+ λT = σ(x, y) on ∂D,
    where ∂D is the boundary of D, ∂T/∂n is the outward normal deriva-
    tive of T, σ is a given function, and λ is a constant. Prove that there
    is at most one solution to the problem if λ > 0, and that if λ = 0
    the solution is not unique, but any two solutions differ by an additive
    constant.


    2. Relevant equations
    Txx+Tyy=0


    3. The attempt at a solution
    I have managed to show the additive inverse part for λ=0, but I'm having trouble with the λ>0 part.

    I used W=S-T, where S also satisfies the above conditions, so therefore W is also a solution of Laplace's, with
    ∂W/∂n+λW=0.
    However, I don't know where to go from there.
    Do I use the identity I used in the previous part, which gives me ∫∫RWx2+Wy2dxdy=∫∂RW(∂W/∂n)ds ?
    Cheers
     
  2. jcsd
  3. Apr 4, 2012 #2
    I'm still really stuck on this. Is there any more information people require to help me?
    Thanks.
     
  4. Apr 4, 2012 #3

    Office_Shredder

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    Staff Emeritus
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    Gold Member

    The identity you proved in the previous part is exactly what you need. If you substitute -lambda*W for dW/dn in the right integral, compare the signs of each side of the equality
     
  5. Apr 6, 2012 #4
    Thanks for the tip, but could you possibly explain what you mean by the sign of each side? Is it simply that one side is positive, and the other is negative? In which case, I'm not sure how to proceed. Sorry.
     
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