An asymptotic mean value characterization for $p$-harmonic functions

We characterize $p$-harmonic functions in terms of an asymptotic mean value property. A $p$-harmonic function $u$ is a viscosity solution to $\Delta_p u =$   div$( \vert\nabla u\vert^{p-2} \nabla u)=0$ with $1< p \leq \infty$ in a domain $\Omega$ if and only if the expansion

$$u(x) = \frac{\alpha}{2} \left\{ \max_{\overline{B_\varepsilon (x)... ...ert B_\varepsilon (x)\vert} \int_{B_\varepsilon (x)} u d y + o (\varepsilon^2)$$

holds as $\varepsilon \to 0$ for $x\in \Omega$ in a weak sense, which we call the viscosity sense. Here the coefficients $\alpha, \beta$ are determined by $\alpha + \beta =1$ and $\alpha /\beta = (p-2)/(N+2)$.