# Is there a version of mean value property for $p$-harmonic funcions?

We know by mean value property that harmonic functions satisfies the equalities $$u(x) = \dfrac{1}{|B_r|}\int_{B_r}f dx = \dfrac{1}{|\partial B_r|}\int_{ \partial B_r}udS.$$ 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?

• Recall that the mean value property is an if and only if. If you work the other direction showing that the function is harmonic, you might be able to constrain the mean value generalization to get $p$-harmonicity. Jun 20, 2018 at 16:50
• My guess it that this is not known. For divergence form linear elliptic equations there are mean value type theorems, but they are already significantly more complicated than for harmonic functions. Google something like "Ivan Blank mean value theorem domains". But p-Laplacian is degenerate which makes this seem (to me) to be a hard problem
– T_M
Jun 20, 2018 at 17:02
• @T_M It is known. See the answer below. There is an asymptotic mean value property, but it cannot be written solely in terms of an average. Jun 20, 2018 at 19:38
• @CameronWilliams For $p$-harmonic functions you need to have a bit of average of minimum and maximum in addition to the mean. The asymptotic mean value property (see below) holds with only the mean if and only if $p=2$. Jun 20, 2018 at 19:39

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.