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Liouville results for fully nonlinear equations modeled on H\'ormander vector fields. I. The Heisenberg group
- Publication Year :
- 2020
-
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 H\"ormander 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.<br />Comment: 21 pages
- Subjects :
- General Mathematics
Degenerate energy levels
Mathematical analysis
Mathematics::Analysis of PDEs
Function (mathematics)
Nonlinear system
Operator (computer programming)
Mathematics - Analysis of PDEs
Heisenberg group
FOS: Mathematics
Vector field
Ball (mathematics)
Constant (mathematics)
Mathematics
Analysis of PDEs (math.AP)
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....6cac583e6dc438cb4112faf7d7d7ce05