A ug 2 00 6 Positive Harmonic Functions on Denjoy Domains in the Complex Plane by Vladimir

Abstract

We use the following standard terminology. We denote by Denjoy domain an open subset Ω of the complex plane C whose complement E := C \ Ω, where C := C ∪ {∞}, is a subset of R := R ∪ {∞}, where R is the real axis (see [13]). Throughout the paper we rely on the following assumption. Each point of E (including the point at infinity) is regular for the Dirichlet problem in Ω. Denote by P∞ = P∞(Ω) the cone of positive harmonic functions on Ω which have vanishing boundary values at every point of E \ {∞}. Independently, Ancona [4] and Benedicks [7] showed that either all functions in P∞ are proportional or P∞ is generated by two linearly independent (minimal) harmonic functions; that is, either dim P∞ = 1 or dim P∞ = 2 respectively. In other words, it means that the Martin boundary of Ω has either one or two “infinite” points.

Cite this paper

@inproceedings{Andrievskii2006AU2, title={A ug 2 00 6 Positive Harmonic Functions on Denjoy Domains in the Complex Plane by Vladimir}, author={Vladimir Andrievskii}, year={2006} }