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Showing harmonicity

  1. Aug 30, 2008 #1
    1. The problem statement, all variables and given/known data

    Suppose u is harmonic on B\{0}, and that u is continuous in the closure of B.
    How would we show that u is in fact harmonic on B?
    (B = unit ball in R^n, n>=2)

    2. Relevant equations

    u is harmonic in Ω if Δu=0 in Ω.

    u is weakly harmonic in Ω if integral over Ω of u(x)(Δψ(x))dx = 0.

    u has the mean value property on Ω if for each x in Ω and r>0 such that the closed ball with centre x and radius r is contained in Ω, we have
    u(x) = 1/ |B(x,r)| . integral over B(x,r) of u(y)dy

    3. The attempt at a solution

    One idea I had was showing u is weakly harmonic on B, and then correcting it at {0} to become harmonic.
    So u is harmonic on B\{0} => u is weakly harmonic on B\{0}
    => integral over B\{0} of u(x)(Δψ(x))dx = 0 for all C^oo functions ψ supported in B\{0}
    Now because {0} has measure 0, then
    integral over B of u(x)(Δψ(x))dx = 0 = integral over B\{0} of u(x)(Δψ(x))dx
    So u is weakly harmonic on B, and so we can correct it at {0} to make it harmonic.
    But then something tells me that's too straightforward, can't be right, can it?

    So then I thought an alternative (and probably safer) way is
    to show u satisfies the mean value property on B, hence harmonic since it is continuous.
    But then I have no idea how to show this...
    Does that involve using that harmonic functions are smooth, or just some tricks at {0}, or...?
  2. jcsd
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