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 $S_{2\alpha} : L^p(\partial \Omega) \rightarrow H^{2\alpha-1}_p (\partial \Omega), 1 < p < 1$. As an application, we show the existence of the solution of the boundary value problem of the fractional Laplace equation.