A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$

TL;DR

This paper establishes a Liouville theorem for positive lpha-harmonic functions in the upper half-space, showing they must be proportional to a power of the distance to the boundary, thus characterizing all solutions.

Contribution

It proves a classification result for lpha-harmonic functions in b space, demonstrating all positive solutions are of a specific power form, extending classical harmonic function results.

Findings

All positive lpha-harmonic functions in b space are of the form Cx_n^{lpha/2}.

Solutions vanish outside the half-space.

The result generalizes classical harmonic Liouville theorems.

Abstract

In this paper, we consider $\alpha$-harmonic functions in the half space $\mathbb{R}^n_+$: \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x)=0,~u(x)>0, & x\in\mathbb{R}^n_+, \\ u(x)\equiv 0, & x\notin \mathbb{R}^{n}_{+}. \end{array}\right. \end{equation} We prove that all the solutions have to assume the form \begin{equation} u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+}, \end{array}\right. \label{2} \end{equation} for some positive constant $C$.