A Dirichlet problem of the fractional Laplace equation in the bounded Lipschitz domain

TL;DR

This paper investigates a Dirichlet problem for the fractional Laplace equation in bounded Lipschitz domains, establishing existence, boundary conditions, and integral representation of solutions in fractional Sobolev spaces.

Contribution

It provides a new existence and representation result for fractional Laplace Dirichlet problems in Lipschitz domains, extending previous work to less regular boundaries.

Findings

Existence of solutions in fractional Sobolev spaces

Solution representation via integral operator

Boundedness estimate for the solution in Sobolev norm

Abstract

In this paper, we study a Dirichlet problem of a fractional Laplace equation in a bounded Lipschitz domain in $ \R, n \geq 2$. Our main result is that for the given data $F \in \dot H^s(\Om^c), 0 < s<1$, we find the function which satisfies that $\De^s u =0$ in $\Om$, $u|_{\Om^c} =F$ and $|u|_{\dot{H}^s(\R)} \leq c |F|_{\dot H^s(\Om^c)}$. Furthermore, we represent the solution with an integral operator.