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TASK: Prove that a function $u\in C(\Omega)$ is harmonic if in every point $x \in \Omega$ it suffices mean value property $$u(x) = \frac{1}{|\partial B(x, r_j)|} \int_{\partial B(x, r_j)} u \, \text{d}\mathcal{H}^{n-1}$$ for some sequence of balls $B(x, r_j)$ for descending sequence of radii s.t. $ r_j\rightarrow 0$.

NOTICE: This question suggests that it would work even with a finite number of radii, but the theory required to prove it is too complicated (since the task was given within an introductory PDE class) and distant from the initial task.

MY ATTEMPT: I was trying to prove by contradiction by assuming that $$\exists x \in \Omega \, \exists r > 0 : u(x) \neq \frac{1}{|\partial B(x, r)|} \int_{\partial B(x, r)} u \, \text{d}\mathcal{H}^{n-1}$$ and taking $r_j<r$ and constructing a function $v$ that satisfies

$$\begin{cases} \Delta v = 0 &\text{on } B(x,r_j) \\ v = u &\text{on } \partial B(x,r_j) \end{cases}$$ thus getting a function $v$ such that:

  • $\Delta v = 0$;
  • $u(x)=v(x)$ (thanks to mean-value property on the boundary);
  • $\forall r_j < r : \int_{\partial B(x, r_j)} u \, \text{d}\mathcal{H}^{n-1} = \int_{\partial B(x, r_j)} v \, \text{d}\mathcal{H}^{n-1}$.

And yet still I couldn't see how to extrapolate the mean-value property on $\partial B(x,r)$ of $v$ onto $u$.

Thanks in advance!

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Sketch of an approach: To keep things simple, suppose $u$ has the sequence mean value property in some $B(0,R), R>1.$ Try to show $u$ is harmonic in $B(0,1)$ by looking at $u-v,$ where $v$ is the Poisson integral of $u$ restricted to the unit sphere. The idea is to show $u=v.$ We already know $u=v$ on the unit sphere. If $u-v>0$ somewhere, then $u-v$ has a positive max $M$ somewhere in $B(0,1).$ Choose a point closest to the unit sphere where $u-v=M.$ You should run into a problem there ...

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