An asymptotic mean value characterization for 𝑝-harmonic functions

Abstract

We characterize p p -harmonic functions in terms of an asymptotic mean value property. A p p -harmonic function u u is a viscosity solution to Δ p u = div ( | ∇ u | p − 2 ∇ u ) = 0 \Delta _p u = \mbox {div} ( |\nabla u|^{p-2} \nabla u)=0 with 1 > p ≤ ∞ 1> p \leq \infty in a domain Ω \Omega if and only if the expansion \[ u ( x ) = α 2 { max B ε ( x ) ¯ u + min B ε ( x ) ¯ u } + β | B ε ( x ) | ∫ B ε ( x ) u d y + o ( ε 2 ) u(x) = \frac {\alpha }{2} \left \{ \max _{\overline {B_\varepsilon (x)}} u + \min _{\overline {B_\varepsilon (x)}} u \right \} + \frac {\beta }{|B_\varepsilon (x)|} \int _{B_\varepsilon (x)} u \,d y + o (\varepsilon ^2) \] holds as ε → 0 \varepsilon \to 0 for x ∈ Ω x\in \Omega in a weak sense, which we call the viscosity sense. Here the coefficients α , β \alpha , \beta are determined by α + β = 1 \alpha + \beta =1 and α / β = ( p − 2 ) / ( N + 2 ) \alpha /\beta = (p-2)/(N+2) .