A positive homogeneous fractional harmonic function in a Lipschitz cone

TL;DR

This paper constructs a unique positive, homogeneous solution to a fractional Laplace equation within a Lipschitz cone, expanding understanding of harmonic functions in irregular domains.

Contribution

It introduces a method to construct and prove the uniqueness of a positive homogeneous fractional harmonic function in Lipschitz cones.

Findings

Existence of a positive fractional harmonic function in Lipschitz cones

Uniqueness of the constructed function up to multiplication

Homogeneity degree between 0 and 2s

Abstract

We construct a positive function $u$ supported and solving $(-\Delta)^{s}u=0$ in a Lipschitz cone. Such a function is unique up to a constant multiplication. Moreover, we show that it is homogeneous of some degree $0<\alpha<2s$.