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Harmonic Functions and Gauss' Theorem

  1. May 7, 2007 #1
    1. The problem statement, all variables and given/known data
    Let f be an element of C^2 (R^3=> R) be harmonic in the ball |x|< 1 (i.e. the laplace of f would be 0). Prove that f = 0 inside if it vanishes on the surface |x|=1. What if the dF/dn=0 (partial derivative) on the surface |x|=1 [What is div(f grad f)]?

    Also think of heat flow in the ball. The temperature T= T(t,x) obeys the heat equation dT/dt = laplace of T (partial derivative of the heat equation is equal to the laplace of T). For steady temperatures dT/dt = 0, so T is harmonic. Interpret the results about t in this setting.

    2. Relevant equations

    3. The attempt at a solution
    Very faint idea on how to start, so the following might be completely wrong.

    Proving that f=0 if it vanishes on the surface of the ball at |x|=1 and if the partial derivative of df/dn = 0 on the surface.
    I'm assuming that x in this problem is a vector in R^3.

    From the second half, I figured that d^2T/dt^2 + d^2T/dx^2 = dT / dt.
    I'm assuming that x is a vector in R^3 from the first half of the problem.
    In any case, I assumed that if dT/dt = 0, no heat is coming in or out.
  2. jcsd
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