On the asymptotic mean value property for planar 𝑝-harmonic functions

Abstract

We show that p-harmonic functions in the plane satisfy a nonlinear asymptotic mean value property for p > 1. This extends previous results of Manfredi and Lindqvist for certain range of p's. div(|∇u| p−2 ∇u) = 0

A priori Hölder and Lipschitz regularity for generalizedp-harmonious functions in metric measure spaces

“…More information has been recently obtained when n = 2: it turns out that planar p-harmonic functions always satisfy the asymptotic p-mean value property at any point. (In [13] the result was proven for a certain interval of p's and in [3] for the whole range 1 < p < ∞).…”

A mean value formula for the variational p-Laplacian

“…In the above, C, α, β and ε are constants depending on n, but their values will not be important in what follow, except the fact that |ε| < (2n + 1) −1 , see equation (2.4) on page 3861 in [3]. Note that by (8.3), we necessarily have…”

“…In [19], this was proved to hold in the pointwise sense, in the plane and for 1 < p <p ≈ 9.52. Shortly after, this was extended to all p ∈ (1, ∞), in [3]. Linked to a mean value formula, there is a corresponding dynamic programming principle (DPP), which is the solution U r of the problem U r = A r [U r ] subject to the corresponding boundary conditions.…”

“…The curious reader might wonder why we are not able to prove the pointwise validity for the mean value formula for the full range p ∈ (1, ∞), as in [3]. To make a long story short this has to do with the fact that the mean value formula considered in [3] has quadratic scaling. It is therefore enough with an error term of order strictly larger than 2.…”

Asymptotic mean value properties for fractional anisotropic operators

“…On the other hand, the case p = ∞ offers a counterxample for the validity of (1.2) in the classical sense, since the function |x| 4/3 − |y| 4/3 is ∞-harmonic in R 2 in the viscosity sense but (1.2) fails to hold pointwisely. If p ∈ (1, ∞) and n = 2 Arroyo and Llorente [4] (see also [18]) proved that the characterization holds in the classical sense. The limit case p = 1 was finally investigated in [16].…”