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 ∆pu = div(|∇u| p−2∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and only if the expansion u(x) = α 2 max Bε(x) u + min Bε(x) u + β |Bε(x)| Bε(x) u dy + o(ε2) holds as ε → 0 for x ∈ Ω in a weak sense, which we call the viscosity sense. Here the coefficients α, β are determined by α + β = 1 and α/β = (p − 2)/(N + 2).