在Hörmander向量场上建模的完全非线性方程的Liouville结果：II。卡诺群与Grushin几何
Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries
摘要：
本文讨论了具有满足满秩括号条件的子向量场族的二阶完全非线性退化椭圆型方程。它研究了在整个空间中粘性子解和上解的刘维尔性质，即在从上到下的无穷远处的适当范围内，它们必须是常数。在以前的文章中，我们证明了一个抽象的结果，并讨论了海森堡群上的算子。在这里我们考虑各种各样的向量场：卡诺群的生成元，特别是H-型群和自由卡诺群，Grushin和Heisenberg-Greiner向量场的生成元，以及步骤2中的生成元。所有这些情况下是相关的次黎曼几何和共同存在的齐次规范，我们使用的建设李雅普诺夫函数为每个运营商。我们给出了方程中一阶和零阶项的大小和符号的明确的充分条件，并讨论了它们的最优性。我们还概述了这样的结果在全空间的多维退化扩散过程的遍历性问题的一些应用。
英文摘要：
The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper we proved an abstract result and discussed operators on the Heisenberg group. Here we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and zero-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.
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