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

TL;DR

This paper investigates the boundary integral operator for the fractional Laplacian in smooth bounded domains, establishing its bijectivity for certain parameters and applying this to prove the existence of solutions to boundary value problems.

Contribution

It demonstrates the bijectivity of the boundary integral operator for fractional Laplacian in smooth domains and applies this to solve boundary value problems.

Findings

Bijectivity of the boundary integral operator for 1/2 < α < 1.

Existence of solutions to the fractional Laplace boundary value problem.

Extension of boundary integral methods to fractional Laplacians.

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.