Comparison theorems for weak solutions of nonlinear maximally sub-elliptic PDEs

TL;DR

This paper proves a comparison principle for weak solutions of a broad class of degenerate subelliptic PDEs on manifolds, extending previous results from Carnot groups to more general settings.

Contribution

It establishes a comparison theorem for viscosity sub- and supersolutions of maximally subelliptic PDEs on manifolds, generalizing recent results to broader classes.

Findings

Comparison principle holds for a larger class of degenerate subelliptic PDEs.

Strengthens previous theorems from Carnot groups to general manifolds.

Demonstrates that maximal subellipticity enables classical comparison results.

Abstract

We establish a comparison principle for viscosity subsolutions and supersolutions of a broad class of second-order quasilinear, maximally subelliptic PDEs on general manifolds. In fact, we prove the comparison theorem for a larger class of degenerate subelliptic PDEs. Our result strengthens a recent theorem of Manfredi-Mukherjee, which was established in the setting of Carnot groups. Our main aim is to highlight that maximal subellipticity allows one to obtain a comparison principle for weak solutions in close analogy with the classical elliptic theory.