JournalsrmiVol. 28, No. 1pp. 201–230

On positive harmonic functions in cones and cylinders

  • Alano Ancona

    Université Paris-Sud 11, Orsay, France
On positive harmonic functions in cones and cylinders cover
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Abstract

We first consider a question raised by Alexander Eremenko and show that if Ω\Omega is an arbitrary connected open cone in Rd\mathbb{R}^d, then any two positive harmonic functions in Ω\Omega that vanish on Ω\partial \Omega must be proportional – an already known fact when Ω\Omega has a Lipschitz basis or more generally a John basis. It is also shown however that when d4d \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 d3d \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 CY(Σ)=R×Σ\mathcal{C}_Y(\Sigma )= { \mathbb R} \times \Sigma (where Σ\Sigma is a relatively compact region of a manifold MM), equipped with a suitable second order elliptic operator.

Cite this article

Alano Ancona, On positive harmonic functions in cones and cylinders. Rev. Mat. Iberoam. 28 (2012), no. 1, pp. 201–230

DOI 10.4171/RMI/674