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Prove the mean value property of harmonic functions

  1. Mar 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]A \subset \mathbb{R}^{2}[/itex] be an open connected set, and g: A → ℝ a C2 function. Show that if g is harmonic, i.e. [itex]\frac{\partial ^{2} g}{\partial {x_{1}}^{2}} + \frac{\partial ^{2} g}{\partial {x_{2}}^{2}} = 0[/itex], then [itex]g(x) = \frac{1}{2 \pi r} \int_{\partial B_{r}(x)}gds[/itex].


    2. Relevant equations
    We're to use Green's theorem in the form [itex]\int_{\partial A} G \cdot x = \int_{A} (\frac{\partial G_{2}}{\partial {x_{1}}} - \frac{\partial G_{1}}{\partial {x_{2}}})dx[/itex].


    3. The attempt at a solution
    OK, I've been struggling to wrap my head around this one for at least three hours now, and after failing to make any sense of it, looking for help online, as well. I've stumbled upon many proofs, but they all use different versions of Green's theorem, and I just really don't get what's going on.

    All I gathered is that I need to somehow show that the derivative of the RHS is 0, implying it's constant, and then use that by taking the limit r → 0 and argue that due to continuity it is equal to the LHS, i.e. g(x).

    Any help would be greatly appreciated, and I'm sorry I don't have more of an attempt to show, but I have zero idea on how to do this. I tried taking the derivative of the RHS myself, but I can't seem to do that, either...
     
  2. jcsd
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