PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 7, July 2012, Pages 2453–2463 S 0002-9939(2011)11181-X Article electronically published on November 21, 2011 MEAN VALUE PROPERTY FOR p-HARMONIC FUNCTIONS TIZIANA GIORGI AND ROBERT SMITS (Communicated by Matthew J. Gursky) Abstract. We derive a mean value property for p-harmonic functions in two dimensions, 1 <p< ∞, which holds asymptotically in the viscosity sense. The formula coincides with the classical mean value property for harmonic functions, when p = 2, and is a consequence of a representation for the Game p-Laplacian obtained via p-averaging. 1. Introduction A recent article by Manfredi et al. [6] (see also [9]) characterizes p-harmonic functions via a weak asymptotic formula which holds in a suitably defined viscosity sense. Inspired by their results and by our recent work [4], where we present a nu- merical algorithm for the Game p-Laplace operator based on the idea of p-average, we derive a generalization in a viscosity sense to two-dimensional p-harmonic func- tions, 1 <p< ∞, of the classical mean value property for harmonic functions. The variational p-Laplace operator is defined, for 1 <p< ∞, as (1) Δ p u ≡ div ( |∇u| p−2 ∇u ) , while the Game p-Laplacian, recently introduced in [7] to model a stochastic game called Tug of war with noise, reads as (2) Δ G p u ≡ 1 p |∇u| 2−p div ( |∇u| p−2 ∇u ) . A function u ∈ C 0 (Ω), with Ω ⊂R 2 a smooth domain, is called p-harmonic in Ω if it is a viscosity solution of Δ p u = 0 (see Definition 2.1). The focus of this paper is in providing a representation of p-harmonic functions that for the case p = 2 reproduces the mean value property. Nevertheless, it will be clear that our main interest is the Game p-Laplacian and that our approach sheds light on the local properties of the solution of the Game p-Laplace operator. The representation formula derived was suggested to us by the numerical approximation we propose in an upcoming paper [4]. An insight on the local properties of the Game p-Laplacian suggests that the value of a solution at a given point is related to the p-average on small balls centered at that point. The numerical solution that we construct, in the case of dimension n = 2, using this idea satisfies a discrete analogue of our proposed generalized mean value formula. We derive the following main results. Our first theorem finds an expansion for C 2 functions in terms of the Game p-Laplacian: Received by the editors November 1, 2010 and, in revised form, February 26, 2011. 2010 Mathematics Subject Classification. Primary 35J92, 35D40, 35J60, 35J70. Funding for the first author was provided by National Science Foundation Grant #DMS- 0604843. c 2011 American Mathematical Society Reverts to public domain 28 years from publication 2453