Asymptotic behavior of positive harmonic functions in certain unbounded domains

TL;DR

This paper investigates the asymptotic behavior at infinity of positive harmonic functions in a broad class of unbounded, non-smooth domains, including Lipschitz cylinders and cones, providing estimates that describe their growth or decay.

Contribution

It introduces new asymptotic estimates for positive harmonic functions in non-smooth unbounded domains with Lipschitz-like structures, extending previous understanding to more complex geometries.

Findings

Derived asymptotic estimates at infinity for harmonic functions

Applicable to domains resembling Lipschitz cylinders or cones

Includes various paraboloids and horns

Abstract

We derive asymptotic estimates at infinity for positive harmonic functions in a large class of non-smooth unbounded domains. These include domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g., various paraboloids and horns.