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Proving Uniqueness

  1. Oct 22, 2008 #1
    I am struggling with proving uniqueness and have multiple HW questions, which basically just have different boundary conditions. Something just isn't clicking for me when doing uniqueness analyses. The approach we were taught is the following:
    del^2(phi) = 0 in a domain
    del(phi) * n = d(phi)/d(n) = f(x,y,z) on the surface (d's in this case are partials and n is the normal unit vector)

    So, del^2(phi1) = 0 and del^2(phi2) = 0
    We can also sub phi1 and phi2 into the second equation.

    Then let K = phi1 - phi2 and d(K)/d(n) = 0

    We then were taught to apply Green's Theorem and solve so that eventually we get:
    The integral over the volume of (del(K) . del(K) dv = integral of K * d(K)/d(n) ds = 0

    However, I know this doesn't hold for every situation (particularly if del^2(phi) is not equal to zero. Something in this process is missing for me.

    In question one we are asked to show unqiueness for the Dirichlet condition (Laplace problem). Problem 2 gives conditions such as:
    del^2(u) = Pu in volume/domain
    P = P(x,y,z)
    B.C.s = d(u)/d(n) = f on surface

    If anyone can provide some information, link for something to read, or reference of something to read that might help it would be greatly appreciated!
  2. jcsd
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