Positive harmonic functions on comb-like domains

TL;DR

This paper studies positive harmonic functions on complex domains with boundaries formed by parallel hyperplanes, providing a characterization of domains supporting minimal harmonic functions with specific exponential growth based on hyperplane spacing.

Contribution

It offers a new characterization of domains with hyperplanes where minimal harmonic functions with exponential growth exist, depending on the spacing between hyperplanes.

Findings

Domains with certain hyperplane spacing support minimal harmonic functions with exponential growth

Characterization of positive harmonic functions on comb-like domains

Conditions for existence of minimal harmonic functions based on boundary geometry

Abstract

This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical semi-infinite lines.) It characterizes, in terms of the spacing between the hyperplanes, those domains for which there exist minimal harmonic functions with a certain exponential growth.