An asymptotic mean value characterization for 𝑝-harmonic functions
Juan J. Manfredi, Mikko Parviainen, Julio Daniel Rossi · Proceedings of the American Mathematical Society · 2009
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} ( | abla u|^{p-2} abla 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 ) ¯