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

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.