Liouville results for fully nonlinear equations modeled on H\"ormander vector fields. I. The Heisenberg group

TL;DR

This paper establishes Liouville properties for fully nonlinear degenerate elliptic PDEs on the Heisenberg group, providing explicit conditions for solutions to be constant based on Lyapunov functions and H"ormander vector fields.

Contribution

It introduces explicit criteria for Liouville properties on the Heisenberg group and demonstrates their optimality, extending the understanding of nonlinear PDEs in sub-Riemannian geometries.

Findings

All bounded above subsolutions are constant under certain conditions.

Explicit conditions on the sign and size of lower order terms are provided.

The optimality of these conditions is validated through examples.

Abstract

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the H\"ormander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.