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Is there someone who could please help me prove this statement?

I know that the uniform limit of a convergent sequence of harmonic functions is continuous. Then my goal is to show this function satisfies the mean value property. That is my question, How to show that?

Thanks!!

Yang
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    Since the convergence is uniform, and the sphere is compact, you can interchange the limit with the integral. The mean value property follows directly. – Daniel Fischer Oct 03 '13 at 17:50
  • Thanks. You applied the mean value property to the sequence of harmonic functions?? – Yang Oct 03 '13 at 18:00
  • Yes. We have $f_n(x_0) = \frac{1}{\omega_n r^{n-1}}\int_{S_r} f_n(x), d\sigma$ for all $n$. Taking the limit, the left hand side converges to $f(x_0)$ because of the pointwise convergence, and the right hand side tends to $\frac{1}{\omega_n r^{n-1}}\int_{S_r} f(x), d\sigma$ by the uniform convergence. – Daniel Fischer Oct 03 '13 at 18:03
  • Then can I conclude that f is harmonic? I think I can. – Yang Oct 03 '13 at 18:11
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    If f satisfies M.V.P, then f is harmonic. – nicksohn Dec 30 '15 at 15:13

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If you don't want to go through the mean value property, fix an arbitrary ball, and write the Poisson formula for each element of the sequence. Then the uniform convergence will imply that the Poisson formula is satisfied for the limit function.

timur
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