A Liouville-type property for degenerate-elliptic equations modeled on Hörmander vector fields
Stefano Biagi, Dario Daniele Monticelli, Fabio Punzo
TL;DR
The paper studies Liouville-type properties for degenerate-elliptic equations modeled on Hörmander vector fields with drift and potential, establishing sharp conditions under which every bounded smooth solution must vanish. By combining hypoellipticity, maximum principles, and a carefully crafted energy inequality built on an exhaustion function $\mathcal N$ and the surface measure $\mathcal S(r)$, the authors derive a divergence criterion $\int_{\rho_0}^{\infty}\frac{1}{\mathcal S(r)}\exp\{\Lambda (\int_{\rho_0}^r \sqrt{\hat q(s)}\,ds)^2\}\,dr = \infty$ that guarantees the Liouville property for $\mathcal L u=0$. The results are specialized to Carnot groups, where $\mathcal S(r)$ is explicitly computable, yielding concrete thresholds (e.g., in Heisenberg groups) and explicit forms of the Liouville property. Optimality is shown by constructing counterexamples when the integral condition fails, demonstrating non-uniqueness via barrier arguments and potential theory. Overall, the work links subelliptic geometry with Liouville-type rigidity, providing sharp, geometry-informed criteria for the uniqueness of bounded solutions in degenerate settings.
Abstract
We obtain Liouville type theorems for degenerate elliptic equation with a drift term and a potential. The diffusion is driven by Hörmander operators. We show that the conditions imposed on the coefficients of the operator are optimal. Indeed, when they fail we prove that infinitely many bounded solutions exist.