Asymptotic statistical characterizations of p-harmonic functions of two variables

TL;DR

This paper establishes asymptotic formulas involving local statistics that characterize p-harmonic functions of two variables, extending classical mean-value properties and highlighting their limitations in non-asymptotic contexts.

Contribution

It generalizes mean-value properties to p-harmonic functions via asymptotic formulas and proves their characterization in the viscosity sense for 1 < p < ∞.

Findings

Asymptotic formulas characterize p-harmonic functions.

Formulas do not hold non-asymptotically in general.

Extension of mean-value properties to p-harmonic functions.

Abstract

Generalizing the well-known mean-value property of harmonic functions, we prove that a p-harmonic function of two variables satisfies, in a viscosity sense, two asymptotic formulas involving its local statistics. Moreover, we show that these asymptotic formulas characterize p-harmonic functions when 1 < p < \infty. An example demonstrates that, in general, these formulas do not hold in a non-asymptotic sense.