论文标题
Liouville在HörmanderVector Fields上模拟的完全非线性方程的结果。 I.海森伯格集团
Liouville results for fully nonlinear equations modeled on Hörmander vector fields. I. The Heisenberg group
论文作者
论文摘要
本文研究了在主要假设的主要假设下，研究了完全非线性退化的椭圆形PDE的粘度亚和超溶液，即操作员有一个满足Hörmander条件的广义亚基载体场。给出了一系列足够的条件，使得上述所有界面都是恒定的。它包括从一个大球中出现的超级溶液，在无穷大爆炸中爆炸。因此，对于大量的运营商，该问题降低为找到这种类似Lyapunov的功能。这是针对生成海森伯格组的向量字段进行的，在方程式中的第一和零订单项的符号和大小上提供了明确的条件。条件的最佳性通过几个示例显示。本文的续集将这些方法应用于其他Carnot组和Grushin几何形状。
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 Hörmander 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.
