A Nonlinear Mean Value Property for Monge-Amp\`ere

TL;DR

This paper establishes an asymptotic nonlinear mean value property for solutions of the classical Monge-Ampère equation, extending the concept of mean value characterizations from harmonic functions to a nonlinear PDE.

Contribution

It introduces a novel asymptotic nonlinear mean value formula specifically for the Monge-Ampère equation, bridging a gap in the literature of mean value properties for nonlinear PDEs.

Findings

Proves an asymptotic nonlinear mean value property for Monge-Ampère solutions

Connects mean value properties with nonlinear PDEs and game theory concepts

Expands understanding of mean value characterizations beyond harmonic functions

Abstract

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized $p-$Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. Currently, asymptotic nonlinear mean value formulas are rare in the literature and our goal is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Amp\`ere equation.