Boundary integral operator for the fractional Laplacian in the bounded smooth domain

Abstract

We study the boundary integral operator induced from the fractional Laplace equation in a bounded smooth domain. For $1/2<\alpha ?<1$, we show the bijectivity of the boundary integral operator $S_{2\alpha }:L^p(\partial \Omega )\to H_p^{2\alpha -1}(\partial \Omega ),1<p<1$. As an application, we show the existence of the solution of the boundary value problem of the fractional Laplace equation.