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arXiv · 2006.06612

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

Martino Bardi·Alessandro Goffi
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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.

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Martino Bardi, Alessandro Goffi. 2020-06-11. Liouville results for fully nonlinear equations modeled on H\"ormander vector fields. I. The Heisenberg group. https://arxiv.org/abs/2006.06612

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