Calculus of Variations and Geometric Measure Theory
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M. Bardi - A. Goffi

Liouville results for fully nonlinear equations modeled on H\"ormander vector fields. I. The Heisenberg group

created by goffi on 12 Jun 2020
modified on 10 Nov 2020


Accepted Paper

Inserted: 12 jun 2020
Last Updated: 10 nov 2020

Journal: Mathematische Annalen
Year: 2020

ArXiv: 2006.06612 PDF


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.

Keywords: Heisenberg group, degenerate elliptic equation, Liouville theorems, H\"ormander condition, Fully nonlinear equation, subelliptic equation


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