On positive harmonic functions in cones and cylinders

Abstract

We first consider a question raised by Alexander Eremenko and show that if $\Omega$ is an arbitrary connected open cone in ${\mathbb{R}}^d$, then any two positive harmonic functions in $\Omega$ that vanish on $\partial \Omega$ must be proportional – an already known fact when $\Omega$ has a Lipschitz basis or more generally a John basis. It is also shown however that when $d\ge 4$, 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 $d\le 3$, 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 ${\mathcal{C}}_Y(\Sigma )=\mathbb{R}\times \Sigma$ (where $\Sigma$ is a relatively compact region of a manifold $M$), equipped with a suitable second order elliptic operator.