Calculus of Variations and Geometric Measure Theory

M. Bardi - A. Goffi

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

created by bardi on 23 Sep 2021
modified by goffi on 16 Apr 2023


Published Paper

Inserted: 23 sep 2021
Last Updated: 16 apr 2023

Journal: Advances in Differential Equations
Volume: 28
Number: 7-8
Pages: 637-684
Year: 2023
Doi: 10.57262/ade028-0708-637

ArXiv: 2109.11331 PDF


The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper we proved an abstract result and discussed operators on the Heisenberg group. Here we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and zero-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

Keywords: Carnot groups, ergodicity, degenerate elliptic equation, Liouville theorems, Hörmander condition, Grushin plane, Fully nonlinear equation, subelliptic equation, Khas'minskii condition, degenerate diffusion