Characterization of positive superharmonic functions in a half-space

TL;DR

This paper provides a new representation formula for positive superharmonic functions in a half-space, leading to improved comparison principles and positivity results.

Contribution

It introduces a novel representation formula for superharmonic functions in a half-space, enhancing understanding of their structure and properties.

Findings

Derived a representation formula for superharmonic functions

Established comparison principles for these functions

Proved new positivity results

Abstract

We prove a representation formula for superharmonic functions on the half-space $\mathbb{R}^N_+ := \mathbb{R}^{N-1}\times]0,+\infty[$. As a consequence, we derive some comparison principles and various positivity results.