English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The paper treats second order fully nonlinear degenerate elliptic equationshaving a family of subunit vector fields satisfying a full-rank bracketcondition. It studies Liouville properties for viscosity sub- andsupersolutions in the whole space, namely, that under a suitable bound atinfinity from above and, respectively, from below, they must be constants. In aprevious paper we proved an abstract result and discussed operators on theHeisenberg group. Here we consider various families of vector fields: thegenerators of a Carnot group, with more precise results for those of step 2, inparticular H-type groups and free Carnot groups, the Grushin and theHeisenberg-Greiner vector fields. All these cases are relevant insub-Riemannian geometry and have in common the existence of a homogeneous normthat we use for building Lyapunov-like functions for each operator. We giveexplicit sufficient conditions on the size and sign of the first and zero-thorder terms in the equations and discuss their optimality. We also outline someapplications of such results to the problem of ergodicity of multidimensionaldegenerate diffusion processes in the whole space.

Liouville results for fully nonlinear equations modeled on H\"ormander vector fields: II. Carnot groups and Grushin geometries