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

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: I. The Heisenberg group

created by goffi on 12 Jun 2020
modified on 03 Jan 2021


Published Paper

Inserted: 12 jun 2020
Last Updated: 3 jan 2021

Journal: Mathematische Annalen
Year: 2020
Doi: 10.1007/s00208-020-02118-x

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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