# Does the pointwise mean value property imply harmonicity?

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.

• 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 Oct 25 at 13:35
• 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). Oct 25 at 19:57