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 Hörmander 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.
Highlights
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
A consequence of mean-value formulas or, more generally, of the Harnack inequality. Such result holds for classical solutions to more general uniformly elliptic equations, provided the zero-th order coefficient has the appropriate sign for the maximum principle, and the equation is homogeneous, see, e.g., the monograph [26]
For linear degenerate elliptic equations, mean-value properties and Harnack-type inequalities were proved in many cases, typically for vector fields X that generate a stratified Lie group, and Liouville theorems for solutions to such equations were proved, e.g., in [12,17,29,30], see the references therein
Summary
Before showing our main results, we present the proof of a Liouville-type theorem for classical C2 subsolutions to linear uniformly elliptic equations in the Euclidean framework, which serves as a guideline for our proof in the nonlinear and subelliptic setting. Remark 2.3 The same result remains true if L is replaced by a degenerate elliptic operator LX u := − i, j Xi X j u + b(x) · DX u+c(x)u, provided the vector fields X satisfy the Hörmander condition and b : Rd → Rm, m ≤ d, is smooth, the proof being exactly the same thanks to Bony strong maximum principle for subelliptic equations. An example of such result is [33, Proposition 3.1]. We refer to [37] for a control theoretic interpretation of the Liouville property for Ornstein–Uhlenbeck operators
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
