Is there a version of mean value property for $p$-harmonic funcions? We know by mean value property that harmonic functions satisfies the equalities
\begin{equation}
u(x) = \dfrac{1}{|B_r|}\int_{B_r}f dx = \dfrac{1}{|\partial B_r|}\int_{ \partial B_r}udS.
\end{equation}
I'm wondering if we have a similar Mean Value Property for $p$-harmonic functions. That is, instead we have $\Delta u =0$, for functions satisfyiing 
$$
 \Delta_p u = \mbox{div} (|\nabla u|^{p-2}|\nabla u| ) = 0.
 $$ Is there some equality like above? 
 A: A function $u$ is harmonic if and only if for every $x$ in the domain
$$
u(x)
=
\frac1{|B(x,\epsilon)|}\int_{B(x,\epsilon)}u(y)dy+o(\epsilon^2).
$$
This is also true without the error term $o(\epsilon^2)$, but it is important for comparison.
Similarly, $u$ is $p$-harmonic if and only if it satisfies the asymptotic mean value property
$$
u(x)
=
\frac{p-2}{p+n}\cdot\frac12
\left(
\max_{y\in B(x,\epsilon)}u(y)+\min_{y\in B(x,\epsilon)}u(y)
\right)
+
\frac{n+2}{p+n}\cdot\frac1{|B(x,\epsilon)|}\int_{B(x,\epsilon)}u(y)dy+o(\epsilon^2).
$$
That is, the function is asymptotically a linear combination of average over the whole ball and the average of the largest and smallest value.
Now the error term $o(\epsilon^2)$ is indeed required.
For which values of $p$ this holds and in what sense is a good question.
The asymptotic mean value property can also be used to study $p$-harmonic functions using stochastic game theory.
To get started, take a look at these papers:


*

*On the definition and properties of p-harmonious functions

*On the Mean Value Property for the p-Laplace equation in the plane

*Harnack's inequality for p-harmonic functions via stochastic games
If someone knows better papers to look at, let me know.
