On the Dirichlet problem for fractional Laplace equation on a general domain

TL;DR

This paper investigates the Dirichlet problem for fractional Laplace equations on general bounded domains, establishing existence, properties, and uniqueness of solutions using Green's functions and Poisson kernels.

Contribution

It introduces a method to construct Green's functions and Poisson kernels for fractional Laplace equations on arbitrary domains, proving their key properties and solution uniqueness.

Findings

Existence of Green's function via Perron's method

Construction of Poisson kernel from Green's function

Uniqueness of solutions to fractional Laplace Dirichlet problems

Abstract

In this paper, we study Dirichlet problems of fractional Laplace (Poisson) equations on a general bounded domain in $\mathbb{R}^n$. Green's functions and Poisson kernels are important tools needed in our study. We first establish the existence of Green's function by an application of Perron's method. After that, the Poisson kernel is constructed based on the Green's function. Several important properties of Green's functions and Poisson kernels are proved. Finally, we show that the solution of a fractional Laplace (Poisson) equation under a given condition must be unique and be given by our Green's function and Poisson kernel.