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

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.