Comparison principles for degenerate sub-elliptic equations in non-divergence form

TL;DR

This paper establishes a comparison principle for viscosity solutions of degenerate sub-elliptic equations in non-divergence form, including the sub-elliptic infinity Laplacian and normalized p-Laplacian, under H"ormander's condition.

Contribution

It extends the comparison principle to a class of degenerate sub-elliptic equations in non-divergence form, incorporating invariance under nilpotent Lie groups.

Findings

Proved the comparison principle for viscosity solutions.

Applicable to equations involving sub-elliptic infinity Laplacian and normalized p-Laplacian.

Established results under H"ormander's rank condition.

Abstract

We prove the comparison principle for viscosity sub/super-solutions of degenerate subelliptic equations in non-divergence form that include the sub-elliptic infinity Laplacian and the normalized p-Laplacian. The equations are defined by a collection of vector fields satisfying H\"ormander's rank condition and are left invariant with respect to a nilpotent Lie Group.