July/August 2023 Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries
Martino Bardi, Alessandro Goffi
Adv. Differential Equations 28(7/8): 637-684 (July/August 2023). DOI: 10.57262/ade028-0708-637

Abstract

This 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, 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 0-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.

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Martino Bardi. Alessandro Goffi. "Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries." Adv. Differential Equations 28 (7/8) 637 - 684, July/August 2023. https://doi.org/10.57262/ade028-0708-637

Information

Published: July/August 2023
First available in Project Euclid: 10 April 2023

Digital Object Identifier: 10.57262/ade028-0708-637

Subjects:
Primary: 35B53 , 35H20 , 35J60 , 35J70 , 35R03 , 49L25

Rights: Copyright © 2023 Khayyam Publishing, Inc.

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Vol.28 • No. 7/8 • July/August 2023
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