关于锥与锥中的正调和函数
On Positive Harmonic Functions in Cones and Cylinders
摘要：
我们首先考虑一个问题提出的亚历山大Eremenko和表明，如果$Ω$是一个任意连接的开锥${\mathbb R}^d$，然后任何两个积极的调和函数在$Ω$消失的$\partial Ω$必须是成比例的-一个已知的事实时，$Ω$有Lipschitz基础或更普遍的约翰基础。然而，它也表明，当$d \geq 4$，可以有一个以上的马丁点在无穷远的锥，虽然非切向收敛到规范马丁点在无穷远总是成立。相反，当$d \leq 3$时，无穷远处的马丁点对于每个圆锥是唯一的。这些与维数有关的性质与M. Cranston和T. R. McConnell关于平面域中条件布朗运动的寿命的结果以及R。Bañuelos和B.戴维斯我们还调查的性质产生的马丁点在无穷大以及马丁边界上的影响，导致存在的约翰削减的基础上的锥或其他正则性假设。主要结果及其证明推广到圆柱${\mathcal C}_Y(Σ)= {\mathbb R} \times Σ$ -其中$Σ$是流形$M$ -的一个相对紧的区域，配备了一个适当的二阶椭圆算子。
英文摘要：
We first consider a question raised by Alexander Eremenko and show that if $Ω$ is an arbitrary connected open cone in ${\mathbb R}^d$, then any two positive harmonic functions in $Ω$ that vanish on $\partial Ω$ must be proportional -an already known fact when $Ω$ has a Lipschitz basis or more generally a John basis. It is also shown however that when $d \geq 4$, there can be more than one Martin point at infinity for the cone though non-tangential convergence to the canonical Martin point at infinity always holds. In contrast, when $d \leq 3$, the Martin point at infinity is unique for every cone. These properties connected with the dimension are related to well-known results of M. Cranston and T. R. McConnell about the lifetime of conditioned Brownian motions in planar domains and also to subsequent results by R. Bañuelos and B. Davis. We also investigate the nature of the Martin points arising at infinity as well as the effects on the Martin boundary resulting from the existence of John cuts in the basis of the cone or from other regularity assumptions. The main results together with their proofs extend to cylinders ${\mathcal C}_Y(Σ)= {\mathbb R} \times Σ$ -where $Σ$ is a relatively compact region of a manifold $M$-, equipped with a suitable second order elliptic operator.
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