Strong Comparison Principle for $p-$harmonic functions in Carnot-Caratheodory spaces

TL;DR

This paper extends the strong maximum and comparison principles to $p$-harmonic functions in Carnot-Carathéodory spaces, broadening the understanding of sub-elliptic PDEs under H"ormander's condition.

Contribution

It generalizes classical maximum principles to non-homogeneous sub-elliptic $p$-Laplacian equations in Carnot-Carathéodory spaces, using an extension of Bony's propagation of support argument.

Findings

Established a strong maximum principle for $p$-harmonic functions in Carnot-Carathéodory spaces.

Proved a strong comparison principle for solutions of the non-homogeneous sub-elliptic $p$-Laplacian.

Extended Bony's propagation of support argument to $C^1$ solutions under H"ormander's condition.

Abstract

We extend Bony's propagation of support argument \cite{Bony} to $C^1$ solutions of the non-homogeneous sub-elliptic $p-$Laplacian associated to a system of smooth vector fields satisfying H\"ormander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf \cite{Tolksdorf}.