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Neumann Problem: Use the divergence theorem to show it has a solution

  1. Sep 9, 2010 #1
    From Partial Differential Equations: An Introduction, by Walter A. Strauss; Chapter 1.5, no.4 (b).
    1. The problem statement, all variables and given/known data
    "Consider the Neumann problem
    (delta) u = f(x,y,z) in D

    [tex]\frac{\partial u}{\partial n}=0[/tex] on bdy D."

    "(b) Use the divergence theorem and the PDE to show that

    [tex]\int\int\int_{D}f(x,y,z)dxdydz = 0[/tex]

    is a necessary condition for the Neumann problem to have a solution."

    2. Relevant equations
    Divergence Theorem:
    [tex]\int\int\int_{D}\nabla\cdot \vec{f} d\vec{x} = \int\int_{S}f\cdot \vec{n}dS [/tex]

    In this case d[tex]\vec{x}[/tex] is a vector component I understand to translate to dxdydz, and I understand the rest of the divergence theorem.

    3. The attempt at a solution
    I wish I could but I don't know where to start; maybe I am rusty, maybe the question is worded strangely; maybe I'm over-thinking it. This is for a pure math PDE class, so even though I get why the condition is necessary from a physics perspective, I can't explain it in a math context, so I'm not sure how or what to apply the divergence theorem on.

    If anyone can lend me a small clue to start me off that would be wonderful. I'm going to be checking this constantly so I will certainly respond.
    Last edited: Sep 9, 2010
  2. jcsd
  3. Sep 9, 2010 #2
    Not to be critical, but check the number you posted. you have posted that you need to do ch 1.2 #4 (b). that is ch 1.5 #4 (b). So either typed it wrong or you are on the wrong problem, b/c I am in a course using the same book. If you are doing the correct problem, then I can say I have no earthly idea where to start.
  4. Sep 9, 2010 #3
    DarthBane, you're right, it is Chapter 1.5, I've edited it. Thanks for the heads up.
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