An asymptotic mean value characterization for 𝑝-harmonic functions

This is the authors' abstract. We don't add key points for this paper.

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 Δ _p u = div ( |∇ u|ᵖ⁻² ∇ u)=0 with 1 > p ≤ ∞ 1> p ≤ ∞ in a domain Ω Ω if and only if the expansion \[ u ( x ) = α 2 { max B ε ( x ) ¯ u + min B ε ( x ) ¯