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

Question

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?

Answer 1

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.