# New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

## TL;DR

This paper develops new maximum and comparison principles for fully nonlinear degenerate elliptic PDEs, especially in subelliptic contexts like Carnot groups, using the concept of subunit vector fields and H"ormander conditions.

## Contribution

It introduces the notion of subunit vector fields for nonlinear degenerate elliptic equations and proves maximum principles and comparison results under H"ormander conditions.

## Key findings

- Maximum of a viscosity subsolution propagates along subunit vector field trajectories.
- Strong maximum and minimum principles hold under H"ormander condition.
- A strong comparison principle is established for Hamilton-Jacobi-Bellman type equations.

## Abstract

We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the H\"ormander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton-Jacobi-Bellman form, such as those involving the Pucci's extremal operators over H\"ormander vector fields.