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

Abstract:

We characterize $p$-harmonic functions in terms of an asymptotic mean value property. A $p$-harmonic function $u$ is a viscosity solution to $\Delta _p u = \mbox {div} ( |\nabla u|^{p-2} \nabla u)=0$ with $1< p \leq \infty$ in a domain $\Omega$ if and only if the expansion \[ u(x) = \frac {\alpha }{2} \left \{ \max _{\overline {B_\varepsilon (x)}} u + \min _{\overline {B_\varepsilon (x)}} u \right \} + \frac {\beta }{|B_\varepsilon (x)|} \int _{B_\varepsilon (x)} u d y + o (\varepsilon ^2) \] 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 $\alpha /\beta = (p-2)/(N+2)$.