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 Hormander 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.

中文翻译：

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基于 Hörmander 矢量场建模的完全非线性方程的 Liouville 结果：I. The Heisenberg group

本文研究了完全非线性简并椭圆偏微分方程的粘度子解和超解的 Liouville 性质，主要假设是算子具有一系列满足 Hormander 条件的广义子单位向量场。给出了一组一般的充分条件，使得上面有界的所有子解都是常数；它包括一个大球的超解的存在，它在无穷远处爆炸。因此，对于一大类算子，问题就简化为找到这样的类李雅普诺夫函数。这是针对生成海森堡群的矢量场完成的，给出方程中一阶和零阶项的符号和大小的明确条件。通过几个例子显示了条件的最佳性。