Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)\, dy,\hskip .1in\text{(Discrete Mean-value property).}\tag{1}\label{1}$$ What can we say about the harmonicity of $u$? which kind of harmonicity do we have?

Recall that if \eqref{1} holds independently of $r_x$ (that is for every $r>0$ such that $B(x,r)\subset \Omega$), then $u$ is harmonic.

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    $\begingroup$ If $\Omega$ is bounded and $u\in C(\overline{\Omega})$, then the answer is yes, it is harmonic. This is a result of Volterra and Kellogg, check theorem $2.1.3$ in this paper $\endgroup$ Oct 25 at 13:35
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    $\begingroup$ For those who don't want to click through: @ViníciusNovelli linked to Llorente's 2015 paper in CPAA, which includes a broad survey of related results. In addition to the positive theorem of Volterra and Kellogg, it is also mentioned that boundedness of $u$ and boundedness of $\Omega$ are important, and dropping either can lead to counterexamples, and that the various cases (positive and negative) are treated by Hansen and Nadirashvili (as mentioned in the answer below). $\endgroup$ Oct 25 at 19:57

This question was addressed by Hansen and Nadirashvili in a series of papers, see, for example:

MR1315353 Hansen, W., Nadirashvili, N., On Veech's conjecture for harmonic functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 137–153.

and references there.


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