A mean value formula of sub-p-Laplace parabolic equations on the Heisenberg group

TL;DR

This paper establishes a mean value formula for viscosity solutions of sub-p-Laplace parabolic equations on the Heisenberg group, providing new insights into their characterization and limitations.

Contribution

It introduces two equivalent definitions of viscosity solutions and characterizes them via an asymptotic mean value formula on the Heisenberg group.

Findings

Viscosity solutions are characterized by an asymptotic mean value formula.

The mean value formula does not hold in a non-asymptotic sense.

Provides a new framework for understanding sub-p-Laplace equations on the Heisenberg group.

Abstract

We derive two equivalent definitions of the viscosity solutions to the homogeneous sub-p- Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions in terms of an asymptotic mean value formula. Moreover, we construct an example to show that these formulae do not hold in non-asymptotic sense.